So, hello everyone. The topic of this lecture is going to be one of my favorite theorems and mathematics, both for the statement and for the proof. I think I'm not going to go into full details on the proof but I do want to say enough to give you an idea of what all the ingredients are and why the various. There's some various funny things that arise and I want to make them kind of make sense at the very least. So, but first let me just tie up some loose ends from last time. So, so we had D was a fundamental discriminant negative. And we had that the set of strict equivalence classes of positive definite binary quadratic forms of discriminant D. And there's one different interpretation of this in terms of quadratic the quadratic field F, which is a Q adjoining square D. So this was invite this that was naturally in bijection with a set of ideal classes. So, non zero ideals. In the ring of integers of F. I equivalent to J, if and only if there exists some non zero scalar alpha and F cross such that alpha I is equal to J. And this is so called the ideal classroom. Or well. And well actually yeah. So we got a little better than this we got a statement without just passing to the equivalence relation we actually described all strict equivalences between binary quadratic forms in these terms, and in particular I also that gives you a statement on automorphism so you know the special orthogonal group of any such final quadratic form will be isomorphic to the will okay module isomorphisms from the corresponding ideal to itself. And then you can see you can show that this is the same as the okay module isomorphisms from okay to itself. And this is the units in okay. And I don't want to go into all the derivations of this but this is also just the same thing as a group called okay. Oh, F became K. Well, you know what, that could happen more than once. So let me say that F equals K equals Q adjoining square root of B. So, the only field we're going to be talking about is is this one so I hope it won't cause any confusion. Right, so it's the group of roots of unity in K. It's independently of the ideal or another in other words independently of the quadratic form you talk about. And this is actually a very small group so it's plus or minus one if D is different from minus three and minus four. You know, Mu three so that I mean six the group of six roots of unity is minus three, and it's new for the group of fourth roots of unity if D is minus four. So it's some very explicitly understandable thing. So describing the automorphisms, but okay this is, this is kind of a parenthetical. Let's go back to the main point here. So we have these strict isomorphism, a strict equivalence class of binary quadratic forms is the same as this thing over here, the so-called class group. And the important thing about the class group is that it's an abelian group. So the set of strict equivalence classes of positive definite binary quadratic forms of discriminant D has a natural abelian group structure just induced from on this side, the product of ideals. It's actually a little bit complicated to translate it back over to the length of binary quadratic forms, although it can be done. Now I wanna say that this abelian group structure sort of explains a lot of, or gives the answer to a lot of natural questions that arise in this theory of binary quadratic forms. So there were some questions we had earlier that we can kind of answer in terms of the group structure. So question one. So given F and G binary quadratic forms, positive definite binary quadratic forms of discriminant D, when is F not strictly equivalent, but just non-strictly equivalent. So there's a priori more natural notion of just being isomorphic under an arbitrary linear change of, or invertible linear change of variables, not necessarily one line in SL2Z. And the answer is if and only if either, so, well, so either they are strictly equivalent, right? That's one possibility. If they're strictly equivalent, then they're conjugate under GL2Z. Or in other words, if their classes in this finite agreement groups are equal, then this happens. And the other way, the other only way it can happen is if the classes are inverses in terms of the abelian group structure. And I didn't put this on the problem set, but it is a very nice exercise. So if you're interested in this and you wanna think about it, you can do that. So in other words, the set of, so the set of isomorphism classes of binary quadratic forms without the strictness or without the orientation condition is just a quotient of the ideal class group, modern equivalence relation, which identifies every element of the group with its inverse. Now, this is a funny equivalence relation. So it's not modding up by normal subgroups. You don't get a group structure anymore once you do this. And so if you were to not do SL2Z equivalence, but GL2Z equivalence, then you'd lose this really awesome thing in this abelian group structure. So you, so this gives one justification for why you wanna look at strict equivalence and not utilize the morphism. Because this fundamental structure, it is just kind of not visible if you impose too many, to identify too many objects. Okay, and the other question, so given FG, again positive definite binary quadratic forms of discriminant D, when are F and G in the same genus? So that means that they're isomorphic over the piatic integers for all primes P. So we're kind of asking about the, yeah. So being in the same genus, but not being isomorphic is kind of like giving a counter example to the integral form of the Hasselman-Kauski theorem. So it's a very natural question about this. And this is also gonna be answered very nicely in terms of the group structure. So if and only if, so if you take F, that F is equal to G times some X squared for some X in a class group of OF. So if and only if F and G differ by a square. So actually, so that means that sort of the genus class group is just equal to the class group of okay modulo of squares. So that is a finite opinion group. So classifying according to genus, strict equivalence according to genus is kind of a nice and reasonable thing to do. And this is again, that I wanna emphasize that the thing, this kind of classification is the one that generalizes better to quadratic forms of higher number of variables. So this was just to give, I don't wanna explain why either of these is true. This one's actually quite difficult. So I don't take this as an exercise unless you're a Gauss, I don't know. But yeah, I just wanted to say that this group structure gives a lot of information. But now let's move to the actual public of the lecture which is your place class number formula, which gives another, the proof of which also goes through this correspondence but doesn't quite make use of the group structure per se but makes use of the alternative interpretation in terms of ideals. So your choice class number formulas. So from now on, there exists a version of it for arbitrary D but I'm just gonna take the simplest special case. So for now on take P a prime congruent to three mod four and we'll let D be minus P. So we're taking the case when D is one mod four and it's square free and the simplest kind of square free number is a, well, simplest kind of square free negative number is just minus a prime. So that's what we're looking at here. Then the theorem is as follows. It gives a closed formula for H minus P. So recall, this is the number of elements in the ideal class group of, so that's by definition. So this number is equal to and it gives a closed form expression for it, which I think is very remarkable. So I'm sorry. It gives a closed form expression. Yeah, so let me, okay. Let me write mu minus P for the number of roots of unity Q of going square root of minus P, which is just equal to two if P is different from three and six if P is equal to three. So, but just to have a uniform freedom. So then the correct thing you want to find a formula for is not H minus P in isolation, but H minus P divided by mu minus P. And I'll make a comment about why that's the natural thing in just a second, but the formula for this, Dirichlet's formula for this says that it's equal to minus one over two P times the sum from K equals one to P minus one of the Legendre symbol K on P times K. So this is just something you can just, you know, bang out by hand, right? I mean, given your prime number P, you just calculate all the Legendre symbols, multiply them by weight them by the integer, the least positive residue and then divide that minus one or two P times that. It's a priori and integer, right? It's not obvious that it's greater than or equal to zero. I mean, greater than equal to one actually, but it's positive. And I'm not aware of any proof, any sort of independent proof that this quantity is positive other than showing that it equals, or no, not an integer, a rational number of course.  Yeah. So it's a very, very strange, very strange kind of expression. And so I want to make a remark about why the right thing, the right, the thing that should have a formula is the thing on the left-hand side. So just a small remark. So H minus P over mu minus P is what's called the groupoid cardinality of the ideal class groupoid. So you don't just have an ideal class group.  And for a general groupoid, the groupoid cardinality is, so for a general groupoid C, a groupoid cardinality, a finite groupoid, is the sum overall, so X in C representing isomorphism classes. So you take a sum over a set of representatives for the isomorphism classes and then you wait by the automorphism group of that representative. And it's this number that sort of behaves well, like has good general properties in the world of finite group oids. So I don't wanna explain too much about why that's the case, but this is kind of just some general theory tells you that this is the correct thing to be looking at. And in the special case of the ideal class group, all of the automorphism groups have the same cardinality, namely this mu minus P. And so this just reduces to the count of the number of isomorphism classes divided by that common cardinality of the automorphism group. But again, when you move to higher numbers of variables, you're not gonna have a group structure anymore, but you still do have a group void of binary quadratic forms up to strict equivalences in the same genus. And the thing you have a formula for is exactly this group void cardinality and that's called the Siegel mass formula. So yeah, so this is kind of the correct general perspective here, but that's just a remark, we're not really gonna do anything with that. Right, so now, where does this formula come from? It comes from combining three separate ingredients based on the following analytic definitions. So the first definition is of the Dedek and Zeta function. So Zeta function F of S, it's the function of a real variable, let's say a function of a real variable S defined by an infinite sum. So you take the sum over all non-zero ideals in the ring of integers and one divided by the norm of the ideal to the S. And you should compare with the Riemann Zeta function, which is the sum overall, natural number is greater than or equal to one of one over M to the S. Which you could write, I guess, as a Zeta Q of S. So if you take the analogous definition for replacing F by the rational numbers, you'll just get this here. And the other definition, the other analytic function that's gonna be important for us is the so-called Dirichlet L series with respect to the quadratic character, chi. So this will be the Legendre symbol. So chi is the Legendre symbol viewed as a function from Z mod PZ cross to plus or minus one, and you extend by zero to a function from Z mod PZ to minus one, minus one comma one comma zero. So you define it to be zero on the missing residue class here, I mean zero. So you say it's declared that it goes from, that it sends zero to zero. Then the definition of the Dirichlet L series, it's like a twisted form of the Riemann Zeta function. So it's the sum overall N greater than or equal to one of chi of N over N to the S. And all of these converge absolutely, or both of these I should say converge absolutely for S bigger than one. Okay, now Dirichlet theorem is a consequence of three separate ingredients which relate to these analytic functions here. So ingredient one is that the Zeta function of F, oops, Zeta function of F is the product of the Riemann Zeta function with this quadratic L series. Ingredient two is that, well you may know if you've studied the Riemann Zeta function, that it has a simple pole at S equals, so it converges for S greater than one, a real part of S greater than one if you wanna use complex variables. And it has a simple pole at S equals one with residue equal to one. So let me rewrite this kind of in real variable language. So the limit as S tends to one of S minus one times the Riemann Zeta function is equal to one. And now let me write down the analogous expression for the, this quadratic field F. So this is, so let me just recall F is Q adjoining the square root of minus P. So this is equal to two pi divided by the square root of P times exactly this quantity, class number divided by mu minus P. So this is kind of really where this, where this expression appears in the whole story. And then the ingredient three is that, if you take the limit as S goes to one plus of just the Dirichlet L series, L S chi, you get an expression that is, now I have to look at my notes to make sure I get it right. Yes. Minus pi divided by P to the three halves, summing from K equals one to P minus one pi of K times K. Okay. Now this ingredient three, this is your problem set today. So I split this, this is a big long calculation which I split into five pieces and put those as the five problems on the problem set. Yes, so there it goes. In the ingredient three, where does S go in the limit? One, always S goes to one. One plus, right? From the rest. Yeah, one from above because it's only defined when S is bigger than one. Yeah. I see. Thank you. Now, in fact, the series defining, I mean the series is actually convergent at S equals one. So I could write this as L of, I mean this is actually equal to L of one chi but you always have to be aware that it's only conditionally convergent, not absolutely convergent. So you have to be kind of careful with this notion here. And just for uniformity, I'm choosing to write it as the limit of values at which it's absolutely convergent. Okay. So why do these three ingredients imply an interest-based formula? Well... The ingredient one is for what values of S? Is it for S as that real part of S is greater than one? Yeah, yeah, yeah. For S greater than one, yeah. Yeah, so it's a quality of like series, a formal series really. And then it's valid wherever everything converges. Yeah. So well, it's because, so we know this limit here is equal to this number here but given this expression here, we can calculate that limit in a different way. It's the same as the limit as S goes to one of S minus one times Zeta S times L of S chi. And then that limit will be one times this value. So you learn that this has to be equal to this. And then you cancel the pies, magically the pies cancel, the square root of P knocks this down to just a P. And you get the, and the two goes in the bottom, right? When you solve for this and you get exactly the Dirichlet class number formula. So as I've kicked ingredient three to you guys, so I'll just talk about ingredients some one and two. So for ingredient one, we defined it as this sum over non-zero ideals of the norm of the ideal for the minus S. But there, well, let me remind you actually something about the Riemann's data function first. So this is the sum of N greater than or equal to one of N to the minus S, but there's this famous Euler product that you can rewrite it as a product overall primes of one over one minus P to the, and now I should make sure I don't get it wrong. Yeah, minus S. So this is a famous identity for the Riemann's data function that follows if you expand this out as a geometric series. So this is one plus P to the minus S plus P to the minus two S plus dot, dot, dot. And then imagine just in your head, expanding this infinite product out. And what you'll find is that as a consequence of unique prime factorization, you'll be getting precisely these terms here by just applying the distributive rule or foil or whatever people call it. Yeah, for multiplying sums together, you have to match all possible terms with each other and you're exactly building a prime factorization of some number and then raise to the minus S power. Like this will be P to the minus two to the S and so on and so forth. So that's the Euler product factorization for the Riemann's data function. And the reason the Dedekind's data function is defined as it is, is so that you have an analogous Euler product factorization because we know that every non-zero ideal is uniquely a product of prime ideals or maximal ideals, I should say non-zero prime ideals. So you get the analogous Euler prime factorization overall maximal ideals of the ring of integers. And then also you use that the norm is multiplicative, but so we're trying to prove a product formula. I mean, a product formula for Zeta FS as a product of two things. So it's good that we now have Zeta FS self-written as a product. So now we group maximal ideals P according to, to which prime P they contain. So each maximal ideal, this was more or less on your problem set from last time, each maximal ideal contains a unique prime P. And so there was, yes, and there was this, yeah, so and if P splits, then there are two primes, P one, P two, as we say lying above P, which means that the ideal P generated by P is P one times P two. And if P is inert, then P itself is a prime ideal. So P is its own prime ideal factorization, even in this larger ring and P ramifies, then P is equal to the square, sorry, in the ideal generated by P is equal to the square of some ideal there. And now let's look at what contributions to this infinite product you get from each of these three cases. So from when P splits, so then the if P splits, then the norm of P one and P two are both equal to P because the norm of their product is the norm of P, which is P squared. So you get the Euler factors. So one over one minus norm of P one to the minus S times one over one minus norm of P two to the minus S. This just becomes, you get the Euler factor for the Riemann Zeta function and then itself again. And if P is inert, then well, it's P is a prime ideal and it has norm P squared. So we get one over one minus norm of P to the minus S, which is one over one minus P to the minus two S, which is one over one minus P to the minus S times one over one plus P to the minus S. Okay, sorry for making that a little. So then here again, we find we recognize the Euler factor for the Riemann Zeta function and then this multiplied by some extra term. And if P ramifies, then there's only one prime in consideration, but it has to have norm equal to P by the same reasoning. And so we actually get just one over one minus P to the minus S, which I'll write as one over one minus P to the minus S times one. So here again, we recognize the Euler factor for the Riemann Zeta function and then this term here. But you also saw in your problem set that the question of whether P splits is inert or ramified is detected by a character mod D. And in the case where D is minus P, this character is exactly the Legendre character mod P. So P splits if and only if the value of the Legendre character. Oh no, I shouldn't have been using P. Oh no, sorry. P was supposed to be our fixed prime at the beginning and now I'm using it for a variable prime. Well, sorry guys. So the character of P, if and only if the character of P is equal to minus one, oh sorry, the character of P is equal to one. P is inert if and only if the character of P is equal to minus one and P ramifies if and only if the character of P is equal to zero. And when you plunk all this together, you get that the Zeta function of S is equal to the Riemann Zeta function times the product over all primes P of one over one minus chi of P times P to the minus S. So it just matches up. And this by the exact same unique prime factorization argument and the multiplicativity of this character chi, this is just the L series. So there's also an Euler factorization for the L series and that is the explanation for ingredient one. It's kind of just a repackaging of this study of how primes split in this quadratic extension. Right. Okay, any questions so far? So now let's move on to the second ingredient. So ingredient two, this is kind of the, I guess maybe this is the, I don't know, they're all the main point in some sense but this is the one where this expression H minus P over mu minus P actually appears. So we want to know that the limit, oops, limit as S goes to one plus of S minus one times Zeta F of S is equal to two pi over square root of P times H minus P over mu P, mu minus P. And I'm not going to give again full details in this analytic evaluation, but I want at least to make clear where all of these factors come from. So whereas the product formula, the first ingredient was proved by making a multiplicative analysis of the Zeta function based on the Euler product, this will be based on an additive kind of analysis of the Zeta function. So we'll actually just directly use the definition. So Zeta F S is the sum overall non-zero ideals of one over norm of the ideal to the S. And now naturally enough, we're gonna partition the ideals into their various ideal classes. So we'll let I one up to I H minus P be representatives for the ideal class group. So that every ideal is equivalent to a unique one of these I sub Ks. So then this is equal to the sum over sum from I equals, little I equals one to H minus P of the sum overall non-zero ideals in the same class as I sub little I of one over norm of that ideal for the S. Okay, so it will suffice to show that, you know, if you multiply this by, so the limit of limit is S goes to one plus of each of these terms of S minus one times this is equal to the same thing without the H minus P term, right? So two pi over square root of P times new minus P because of each of these things has that same behavior on the limit is S goes to one plus if each of these has the same residue, you know, then when you sum them up, it'll just multiply that number by H minus P and that will be where this H minus P factor comes from. Okay, now I'm going to make my life a little bit easier and only explain what's going on in the principle ideal class. It's not substantially different for a non-principle ideal class, but it's just a little more technical to explain. So I wanna cut to the chase. So I'll just analyze the principle ideal class. So we'll see that limit as S goes to one plus of S minus one times the sum overall non-zero principle ideals over pi to the S is equal to two pi over a square root of P times new minus P. Okay, okay, now we're gonna make this factor go away because look, what is the condition here? We're in the principle ideal class that means we're generated by some element alpha, but alpha is not uniquely determined. A generator of an ideal is not uniquely determined. It's only determined up to multiplication by units. So let's just note then that what if we remove that, right? So what if we just summed overall alpha in okay minus zero of one over norm of alpha to the S? Well, then we'd be counting each principle ideal. This is the norm of same as the norm of the principle ideal generated by alpha, but we're now counting each principle ideal more than once. Namely, we're counting it by waiting it by the number of generators for that principle ideal, which is the exact same thing as the number of roots of unity. So this would be equal to mu minus P times the guy we're interested in. So sum I not equal to zero, I is equal to principle ideal generated by alpha over sum of alpha, one over norm of the ideal to the S. So actually it's suffices to see that sum of all alpha and okay minus zero, one over norm of alpha to the S. And then I have to multiply by S minus one and take the limit as S goes to one, that this is just equal to two pi over square root of P. Okay, right. And now I wanna say it's gonna pay to abstract just slightly. It's not gonna be important anymore that this okay is the ring of integers in a imaginary quadratic field. That's not gonna be relevant at all. The only thing that's gonna be relevant is that it's a lattice inside the complex numbers. So let's make a more abstract claim for any full lattice lambda inside the complex numbers, i.e. L is some Z omega one plus Z omega two, where omega one and omega two are linearly independent. So it's just, I mean, we have some picture like this, right? I mean, it's just a lattice as you would imagine it in the complex numbers. So it could be a square lattice or not, but in our case, it will be the ring of integers of our imaginary quadratic field. Then if you take the limit as S goes to one plus of S minus one times the sum over all nonzero elements in the lattice of one over, I mean, I can still write the norm of that element, right? Again, norm of alpha just means alpha times alpha bar or absolute value of alpha squared, right? So it makes perfect sense for an arbitrary complex number as well. And what's this gonna be? It's gonna be two pi divided by the area of the fundamental parallelogram. So if you take a fundamental domain for action by translation by elements in the lattice, and you get some rectangle like this and you calculate its area and you take two pi divided by that. So what is this abstract thing and apply the theorem? So to see this finishes the job, we need to know that area of the fundamental parallelogram is equal to square root of pi for lattice being O, a square root of P, sorry, O being a Q would go in square root of minus P. But in general, there's a formula for this area. You can calculate such an area as a determinant. So if you imagine, so there's this first basis vector, which I guess was omega one and the second basis vector called omega two, then this is also, then you get this fundamental parallelogram by applying to the standard unit square, the linear map, which is omega one in the first column and omega two in the second column. So this is also just the determinant of omega one, omega two. And we have an explicit basis for our ring of integers. So we can just calculate the determinant and it's just a small calculation that I'll avoid doing, but that's how you can check that you indeed get square root of P. In general, for D, you get square root of, I mean, minus D, I guess, square root of absolute value of P. Okay, so now we're down to something kind of, it doesn't look as scary, I suppose. So this abstract claim. And in fact, this abstract claim can be proved very similarly to how you show that the limit is, S goes to one plus S minus one times the Riemann Zeta function of S is equal to one. So the idea is, well, so the key claim is that you can, approximate the sum by an integral. So, right, so more specifically, I mean, I don't wanna get into too many details here again, but the claim is that if you do this here, this is close enough to, or actually if you do this and then multiply by this area of the fundamental parallelogram, that's close enough to the integral, sort of a plane integral over all alpha of one over norm of alpha to the minus S and then kind of, I guess, DX, DY. So now this is an integral over all of the complex plane. And the reason this should be believable is you can imagine, you can sort of approximate this function on all of the complex numbers, I mean, maybe minus the origin, but the, I mean, the integral will be, it's an improper integral, but it'll converge. As I said, there are some analytic details that need to be looked at here, but you can approximate this function, one over norm of alpha to the minus S, defined on the entire complex plane minus the origin, by kind of a step function, which just says, okay, if you're in this parallelogram, then just be constant on value given by the value of your function at this lattice point. And if you're in this parallelogram, so you always select the lower left, always make a constant function with the value of the lower left integer lattice point. And then, so you make some kind of two-dimensional step function. And on the one, that will be fairly close to this, but on the one hand, when you take the integral of that step function, you're exactly gonna pick up a factor of the area of the parallelogram times the sum over the values on those lattice points. Okay? And then you can actually calculate that the limit, well, you can actually calculate this thing. I mean, oh, shoot, I put a minus, I did the mistake that people always make there. They do, you know, there's always this choice where they write one over N to the S or you write N to the minus S. And I swear, I've seen like a million people out, right, one over N to the minus S just kind of had their hand making a mistake. And I've done that too, right? Yeah, so you can actually calculate this by, let me just explain the technique. So you integrate over circles. So you're doing a two-dimensional integral over a plane, but you do it instead by integrating over the circles. And on the circles, the value of the normal function is constant. And then you'll pick up a pi factor from the fact that these are circles. And then it'll actually reduce to the exact same calculation you have for the Riemann Zeta function. Sort of when you then integrate over the remaining variable, which is the radius of the circle. So you'll have like one over R to the S and then you're gonna be picking up some pi factor from integrating along the circle. And then you'll essentially just reduce to the claim for the Riemann Zeta function. Yes, so there it is. A bit of a, maybe a very simple question, but when you say we integrate, you integrate in circles, you mean that we just do that integral in like polar coordinates, right? Yes, exactly. Double integral, okay, thank you. Yep, yeah, so I should just say, I mean, I don't know why I didn't say that. I mean, that's something that people study in calculus, right? So you change coordinates and they're great using polar coordinates. That's what I'm trying to say. Yeah, thank you. Yeah, and then you'll pick up a pi factor and then you'll get the same calculation as you have for the Riemann Zeta function, which again, is just you can evaluate the integral and see that the claim holds. So this was only for the principal ideal class, but in fact, the non-principle ideal class is also reduced to the very same abstract claim. So it's just for a slightly different lattice. Yeah, the lattice is given by the inverse fractional ideal of the ideal representing your ideal class, but it's still just a lattice and you can again understand what the volume of the parallelogram is and everything works out. So my special gift to you on the last day is that we're ending early. So I wanna say also thanks to all of you. This was a really great experience for me having such a tentative and sharp students and contributing to the environment and making my lecture so much better. So I really do appreciate you guys. And yeah, I'd like to say thanks, but also to the people who haven't said anything. Thank you also for paying attention or not paying attention showing up.  Yeah, so Therios. I mean, I guess we should clap. Yeah, there's a- I saw you guys clapping. It's okay. You don't have to unmute and clap. Let's have questions instead. Don't mean to interrupt that, but I do have one tiny question. So we have this formula, right? This I think it's called the Dirichlet formula for the involving the class number in terms of this. Well, this M minus B, that's something that we've started in the tutorial sheets. And I'm just wondering how, like, how do we use this formula, right? Do you have any, like, are there any applications of that to compute stuff? Like, how can I use this to, do we use this to compute a class number or? Yeah, you could. I mean, it's a girl. You did it by hand, right? For the first two fundamental discriminants. Yeah. You can compute class numbers. So I wonder which one's faster. You can also use this formula to compute by hand. I mean, the nice thing with the by hand computation using the tools from lecture 12 that you gave to us is that we can also have explicit reduced forms. So I don't just get a class number, but I also get some reduced forms. And then we have that theorem by which primes are represented. So I get this extra nice stuff. Yeah, you get more from that version yet. But I think that probably this one is just faster. I mean, if you just want to know the class number, but it's true that you get more from the other approach. I don't know. First of all, I'm not an expert in this. Like, so I'm kind of, it's a little embarrassing that I've chose to lecture on this topic because it's more just a love I had as an undergraduate. I thought I'd give it onward to you guys undergraduates. I mean, it was one of my favorite things as an undergrad. So I thought it makes sense, but I'm not an expert on it. So I don't, I mean, I'm not good at giving answers to questions like this. What is this used for? Yeah, I don't know. No, that's fine. But I do, I do know that the, yeah, I don't know anything that it's actually used for. It's just sort of a gem, a little gem sitting there. And of course it also, I mean, these kinds of formulas like for the residue of the, you know, formulas for residues of Zeta functions or L functions or special values of L functions. This is a whole huge world in arithmetic geometry. And there are some amazing fascinating conjectures about them, which are, I mean, for the most part, completely out of reach. But they're kind of a driving, like generalizing this kind of formula a driving force for a lot of arithmetic geometry and with a number theory, it's a huge area. So I think it more, it's more like a finished product in a sense than a thing that you kind of use to do other stuff, to my understanding.  Let's try to do that in other situations as well. I see. Thank you. Yeah. But again, I'm very much not an expert. So don't trust my answer. That's just based on my limited understanding. So essentially in the lecture, we have shown a closed formula for this, finding the class number when the discriminant is minus a prime. The prime is equal to three more. So I was just wondering, because this class number one formula, class number one problem for negative discriminant seem to be a difficult problem. You said at the story, right? I wouldn't even have find solving this. So I was wondering, probably there doesn't exist a trivial formula or a simple extension of this formula for other discriminants, right? Which are a lot of alternatives. There does. Oh, there does. Okay. There does. And actually you can see using the result from the previous problems that describing non-trivial two torsion in the class group, that if you're not minus a prime P three, one four, then you can't have class number one. So in fact, I mean, this formula, it would be all you need to know to attack that problem. But I don't know any way of using this formula to attack that problem. It's, yeah. Yes, Eleftheres? Could you please repeat that last thing that you said in your answer about the class number one? So if P is not prime, then H minus P cannot be one. Right. Because P is not minus a prime or minus four, yeah? Then, oh no, it could be four, okay. Or minus four. Minus eight also has one. Yeah, or minus four times a prime, I should say. But you can actually rule that out. There's a way to rule that out as well. But that's not gonna be obvious from what I say. But if D is not, okay, let me say if D is one mod four and minus D is not a prime, then it has more than one prime factor, right? And then you saw in your problem set that if you could just take that, then that prime will ramify.  And then that math frack P is gonna be non-trivial in the ideal class group. So it's gonna, yeah, generate, in fact, those prime advisors will generate two torsion. And the only relation between them is that when you take a product of all of them, you get one in the ideal class group. So as long as you have more than one prime factor, you get non-trivial two torsion. I see, thank you. Yeah. Yeah, what about four times a prime? No, no, wait, no, no, it was D, D square free. The condition was D square free. Wait, no, no, I'm confused. I know, yeah, oh, right, right, right. No, D mod four square free.  I know you look at the primes dividing D. Okay, so indeed, no, no, the argument does handle everything. So I was just getting confused because the argument shows that if D has more than one prime divisor, then you get non-trivial two torsion. So this only leaves minus P, P congruent to three mod four minus four or minus eight. Those are then the only possibilities, yeah. Yes, so there it is. So we define this genus class group, right? This class, well, the ideal class group of F, modulo the squares. And I mean, if I recall correctly from the last lecture, we had that the ideal class group was, well, technically, I think I should say isomorphic to the Picard group of the ring of integers. Is there any, like, do people do that as well when studying the Picard group? Like, is there an interest in Picard group modulo the squares? I don't think there's any interest in it above, like the same interest you would have for Picard group modulo the cubes or Picard powers. I mean, just on abstract grounds, one way to understand the feeling group is to understand, you know, how many twos there are in it, so to speak, or how many threes there are. So not that I'm aware of is the answer. So the, I mean, this is, again, something very special to this binary quadratic case that the genus can be expressed, genus classes can be expressed in terms of, oh, you don't have the ideal class group really for higher dimensional quadratic forms. You don't have a group. So I mean, again, some sort of special phenomenon. I see, thank you. So I think in the last problem set, we have this black box that the class group modulo the class group square is an abelian group of order two power r minus one there. Yeah, yeah. And today you were talking about this. So basically those represent the isomorphic classes of quadratic forms that are isomorphic over ZP. So how does our come up? It comes up because you can detect, you can detect genus classes by means of a certain homomorphism to, well, so if you have a binary quadratic form of discriminant D, you can ask which residue classes mod D it hits, copram to D, let's say. And it can only hit the ones for which the value on which the value of this character chi is trivial. So that's something we more or less saw. So, and then it turns out you actually do get a homomorphism from the ideal class group to the kernel of that homomorphism modulo, the subgroup which is the image of the principle norms and which is the kind of the residue classes that are hit by the principle of formal discriminancy. And so, and then you can just, you can just calculate the size of that quotient group and you find it's the same thing. It's again, this two group of size, two to the number of prime factors of D minus one. And then you give a separate argument to say that the homomorphism has kernel exactly the squares of the ideals. And then that's how you kind of, so this is, yeah. So it comes up by, you figure out whether you're, so yeah, you figure out whether you're in the same genus or not by looking at, basically by looking, not at ZP for every prime P, but just directly looking modulo D essentially. And then you can cook up some invariance there. Is more, is the simple version of the answer. So being isomorphic over ZP for every prime P is the same as being isomorphic mod N for every N for every natural number N, that's some kind of Chinese remainder theorem from pathness argument. And then it turns out the QN to look at is just D. And that's where the kind of D pops up on that side as well. I see. Thank you. You're welcome. And I think in this book, primes of the form expert was M. Weisberg, which I now have looked at. It gives another way of understanding this genus theory in terms of more class field theory. So like a lot of what we said here is kind of amounts to class field theory for quadratic extensions of Q, but then you can understand this genus theory in terms of quadratic extensions of your quadratic field and applying class field theory to that. So yeah, so if you want a more advanced perspective on this kind of subject, you can look in that book, which is in the list of references. Cox theorem, yeah. Right, theorem group. Thanks, thanks, Freddie. That's the more low brow, like Gauss genus theory version. But yeah, the class field theory version, I think some of us were looking at it yesterday actually, including from Montagion, I think that's like chapter five, if I remember correctly. So he does the, he does both the lower and the high brow versions, yeah. It's amazing that Gauss could do all this stuff kind of by hand, just binary quadratic forms and has like a 19 year old or whatever. I don't know. He wrote this book when he was 21 or something. It was just because he only said with Medicaid, I don't know. Don't feel too bad guys, we haven't reached those heights yet. Yeah.


This is Covering the Spread. Here are your hosts, Jim Sonnis and Dr. Ed Fang. What is going on everybody? Welcome on into Covering the Spread. That is our sports vetting podcast here on the FanDual Podcast Network. Getting you set for week three of college football and breaking down our favorite bets on the board. My name is Jim Sonnis. I am a senior writer and analyst for numberfire.com. Joined here as always by Ed Fang at thepowerank.com. You can find him on Twitter at thepoweranked.  I lost our bets with the Northwestern Stanford game. So I am wearing a Stanford t-shirt and people listen to the audio only version. It's very red. It's very triage. It's supposed to be cardinal in color, but. Yeah, cardinal. Because it's the Stanford cardinal. Is this like a maze thing where like, if you were to say Stanford red, people would like attack you in the streets? Well, Stanford fans aren't that aggressive. They're too busy studying and starting companies and stuff. But yeah, Stanford red, I mean it would be more of an embarrassment to you, right? Right, exactly. Like spilling a drink on a girl on your first day. Right. And I can't go walk around Ohio State saying red, crimson and scarlet or whatever it is. So the Stanford cardinal shirt firmly on me. I am draped in my shade. Looks good, man, looks good. Yeah, I figured you would approve of this. I just arrived yesterday because I am too cheap to pay for Amazon Prime. So it took a while for it to get here, but finally I can pay you back. But Ed, tough stuff over at Stanford with their tackle done for the year. You know, first round draft type. They do get Costello back, but like that's just really rough circumstance to start a year. Well, yeah, but I mean the real problem is on the defensive side of the ball where a true freshman, Caden Slovis, is throwing for over, what? I think it was 11 yards for pass attempt. Yeah. So that's where my real concern is, I don't mind that they lost the USC and USC is actually getting a little bit of love in the markets, which I think is kind of interesting because I still don't really believe in Clay Helton, but yeah, so the defensive side of the ball is where I worry about Stanford a little bit. Do you think that game was more about USC or about Stanford, the way things played out there? Because like you said, USC actually is kind of, I mean, they're getting a lot of love this week in not super easy environment as they go to BYU. Right. Four point favorites right now. Do you think that game was more about USC or was it more about Stanford? It's tough to know. I saw bits and pieces of the first half. I really don't know. I mean, I can imagine Stanford's defense being bad. I can imagine USC's offense being really good. So I think without more data, I don't wanna say anything definitive about that. For sure. And that is not one of the games we're going through for today. We had a guest lined up, but unfortunately got pulled into a meeting at the last second. So we're just gonna lean on Ed's numbers here and see what they say for week number three. We'll also have our NFO preview for week two. Going up tomorrow, we'll break down three big games in the slate and we'll also talk Pat's Dolphins because the line in that game, abundantly interesting. So we wanna break down that one. If you wanna get that podcast and every podcast, as it goes up, make sure you subscribe to Covering The Spread here on the Fando Podcast Network. Do search for Covering The Spread on Apple Podcasts, Spotify, the Google Play Store, tune in, iHeartRadio, wherever you can find it. And while you're there, if you like what Ed has to say later today or what our guest says tomorrow, make sure you subscribe, rate and review the podcast. We do appreciate that as always. Ed, anything else on your mind here is before we dive into Covering The Past and look back at our college football episode from last week. Marcus Mariota. Oh man. So Jim came on my show and talked about Marcus Mariota. I thought he had to have a pretty good season. Three touchdowns, zero picks, 7.7 yards per attempt. And a big road win at Cleveland. But the problem is I'm also a Baker Mayfield guy. So. Well, hey, you can't have everything, man. You gotta pick one. And you know what? You didn't talk about Baker Mayfield on my podcast. That's true. Marcus Mariota on the football analytics show. Marcus got a little lucky because Derek Henry House did 75 yard touchdown, but he had a really good past to AJ Brown and I think people were kind of overlooking AJ Brown entering the year like as an addition to the offense. And again, they didn't have their starting left tackle in that game and the Browns defense is legit good. So I didn't come away from that game thinking that like Mariota crushed it, but I thought that he was going to do really badly in that game. So he exceeded my, well like, I like the Browns defense a lot. Interesting. Yeah, I mean I still have my questions about the Browns defense. I mean. Well, the defensive front specifically is good and the Titans didn't have their left tackle or their right guard. And so like in daily fantasy, I had a lot of Browns defense shares, which did not go that well, obviously. But I was, my expectations were low for Marcus because of the context around it, given that they were on the road against a good defense. This weekend, I think will be the best test we'll get though because they're at home facing Indy. And you talked about even before the Andrew Luck retirement, your concerns around Indy because of their defense. And I want to see what Mariota can do in that game if he can duplicate the numbers that he had against Cleveland, maybe get a little bit less lucky than he did there. I'll feel pretty good. So cautiously optimistic. Is that an okay way I can phrase that? Yeah, absolutely. All right. I mean, you know, to be very optimistic about any team after one week or two weeks in college football is probably overreaction. It can lead you down some really dumb paths as I have learned plenty of times in the past. But I think that it was better than I expected for the Titans at least in week number one. Now with that said, let's dive into last week and take a look back at our college football podcast. Some interesting things to discuss from our time with Drew Martin then covering the past. So last week on the week two college football preview, we had Drew Martin on the show and you can find him on Twitter at Drew Martin bets. And I thought that Drew had some really sharp insights especially with the two bigger games we discussed. He wanted Texas A&M plus 17 and a half actually got that number when it was plus 18 and a half and they did cover that. It was late, but it still counts. So he got that one. Yeah, it still counts.  He said that for LSU versus Texas that he didn't wanna bet it unless the line got to seven points. And what happened? LSU won by exactly seven. The number at least based on when I was watching it never got to seven. And Ed, you talked a lot about Texas before the season. You talked about liking LSU and Joe Burroughs making you look pretty smart for being into LSU at the beginning of the year. Yeah, I mean, I don't know if I talk much about LSU. I don't love LSU. I still think the program from coach O. Well, Dave Miranda, the DC doesn't have anything to prove in my mind. And that actually ended up being the poorer unit on the field in that game which is a little bit of a surprise. You know, maybe Texas's offense is good and I can definitely believe that. But, you know, Joe Burroughs was very good. I was not expecting that many points in that game. I was expecting about that result, you know, touchdown for LSU on the road, getting it done. They are the better team. But yeah, a lot of points. I don't know if we talked about the over on that, but not something I would have expected. Well, I thought that it was also interesting because like we had talked on this podcast about how people liked Texas more than they should have. And we saw that kind of in the line movement too where it opened, I think it was LSU minus three and a half and it closed at like minus six and a half. And that's pretty considerable movement. But I thought that was a very interesting game from a betting perspective. No, absolutely. And my number was exactly at LSU six and a half. So I was pretty happy that it closed there.  But they got it done. Yeah. And Joe Burrow, his Heisman odds are five to one now. So that's what a couple of good games will get for you. We talked about FIU versus Western Kentucky. Western Kentucky won that game outright. Same thing for BYU versus Tennessee. BYU won that game in overtime. Tennessee, yikes. I had Miami minus four and a half against UNC. UNC won that one outright. And I think at looking back, I may have been underrating UNC just because I didn't expect them to do this with Mac Brown, a very young quarterback. And I thought my expectations overall for UNC entry in the year, I was expecting this to maybe be kind of just a year in transition, but they've had two really impressive wins. Is that enough for you to be kind of high on UNC right now? No, not really. My numbers actually kind of like them, but I think the jury's still out. When you look at some of the numbers in this game, Miami had better yards per play, had almost 100 yards more. So if they play that game again with the same underlying statistics, it could have turned out a little bit differently. So yeah, I mean, I think it's two weeks, right? So let's wait a little while and see what happens, especially on a team like North Carolina. Well, that makes me feel better about betting Miami. So at least we have that. Thank you for that. I appreciate it for sure. I run a little model which predicts how many points a team should have scored based on yards per play and total yards in a game. Yards per play is the big variable there, just because it's so closely correlates with points scored, even with the noise of defensive and special teams and stuff like that, that can affect how many points you score and then total yards to get a sense for pace. So in that little model, it actually, if you put those underlying statistics into the model, Miami actually wins by seven. Oh wow. So I feel a lot better. There's a lot of deceiving things in games. There's a lot of interesting things that can happen on the scoreboard. And that's why I just like to look at the underlying efficiency metrics. And so that would lead you to say don't overreact in a positive sense on UNC's side, but don't also overreact in a negative sense on Miami's side, just because they lost that game on the road. Yeah, for sure. For sure. I mean, I didn't get a chance to see that game, but you know, I mean, it looks like Jared Williams was pretty good in completing passes and I think that projects better for Miami going forward. Not sure yet what to think about it with Carolina. And that matches what we saw from him against Florida too. So I think that seeing that lineup is always encouraging. So that was week two. We're going to dive into week three in just one second, but if you want to get in on the action, check out the Fandal Sportsbook and place your first bet today. If you lose, Fandal will give you up to a refund of up to $500 in site credit. Visit sportsbook.fandal.com for more details, terms and conditions apply, must be 21 plus and physically present in New Jersey or Pennsylvania, gambling problem, call 1-800-GAMBLER. Let's dive in now to week number three of college football and break down some of the funnest games. Is fun a word? Regardless, the funnest games on the board in week three. Covering the present. All right, let's take a look at week three in college football. And Ed, before we really dive into these numbers, I want to pick your brain a bit because you've got these models over at thepowerwreck.com and things are obviously changing throughout the season. Your models are evolving. We've got two weeks of data on most teams by now. Some teams have played just one game, but like we've got some data. This early in the season, how much of your model is data from 2019 versus the conceptions your model had of the teams entering the year? Yeah, I mean, look, the vast majority is still the pre-season model. I really like, especially what I've done over the past two years with my pre-season analytics, especially in college football, which is a combination of data and then what the markets think in terms of win totals. But you know, like about 10% is 10, 15% right now is what we've seen this year. And the way I do it is pretty simple. I kind of look at how a team performed on the scoreboard compared to what the markets closed that for that point spread. And then I also look at yards for play, how teams perform compared to what the markets expected.  Unfortunately, a team like Michigan has struggled in both departments, really tough game against Army. So they're a little bit lower. Definitely a game we'll be talking about next week, but that just gives you an example. And even, this is the hard part about handicapping, especially in college football, because teams can move really fast in their number. And you don't have to look further than what happened to Louisville last year. The models kind of always had Louisville. They could never catch up to just how bad a job Bobby Petrina was doing with that team. So it's always looking at the model. And it's gonna kind of get the mean behavior in some sense, but some teams you should kind of lean more on the preseason and some teams you should lean more on what's going on this season, like a Louisville last year. And I'm sure we'll talk about that as we go through some of these games. Well, I think the important thing with Louisville last year too, is that how bad they were last year will also influence the way they're viewed in the model this year. And it's kind of hard to adjust for that because they pretty clearly quit on Bobby Petrina last year. But like you're not gonna be able to like fully account for that. So do you try to just, is there a way that you can manually adjust the inputs from 2019 versus 2018 for various teams? Like, I guess, how do you handle Louisville in 2019? It's kind of what I'm getting to here. Yeah, for sure. You know, later in the season, so after I get about three, four weeks of data, like I can start doing metrics just based on this year.  What do the markets think this season, which is kind of a completely different predictor, right? Because it's more wisdom of crowds-based. And then what do the pure numbers say? So in terms of margin of victory, yards per play, probably success rate this year, what does that say just based on data from this year? So the number that I'll be talking about on the show, this is a combination of those three things or will be later in the season. But you get those separate components to kind of make your own judgments about which one you feel is most important. And that's obviously gonna be different for different teams. Right. And you kind of alluded to this before, but when you're trying to dig into 2019 numbers, is it mostly yards per play? Like, if you were giving advice to someone, trying to dig into 2019 data, what would you say for them to focus on from a numbers perspective? I mean, yeah, that's a tough one too, because I wouldn't have told you a lot over the last couple of years. Like, yeah, yards per play is great. Yeah. The more I look into it, the more I realize, you know, there's a lot of randomness that goes into yards per play. Yeah.  That can have a huge effect on your yards per play numbers. So that's why I've been starting to look at success rate numbers in all my football predictions. I incorporated it into my NFL model last year. I think it really helps. We'll be doing that for college football this year. And some of the work Bill Connolly has done has shown that success rate is more sticky from early to late season. So if you have a high success rate early in the season, you tend to have a higher success rate later in the season. So that, you know, when you're looking at small samples, it's better to lean on that than a yards per play. Okay, that's nice. It's kind of nice not to have a guess. I should just get to pick your brain, which is something that I very much enjoy. So let's continue doing that here. And break down three of the bigger games on the board for this week. We will talk about Iowa versus Iowa State. Not sure if I'm allowed to say the nickname of that game. El Asico, I guess I'll just say it. El Asico, we're gonna talk about that in covering the future. So we're gonna save that game for later with college game day out there. So we'll talk about that one later on. But let's start things off here with Oklahoma UCLA. As of right now at the Fandal Sports Book, Oklahoma a 23 point favorite. The total in that game opened, I believe it was at 66, but it's now 73 and a half. And UCLA is a team that last week, you said you really kind of wanted no part of. And I feel like they validated that last week. So what do you see in this game to Oklahoma and UCLA? Yeah, I mean, I think this is a perfect example where I wanna kind of throw out my preseason number on UCLA. You know, when I run this model, it has Oklahoma by 13. That kind of makes no sense to me. I mean, I just feel like UCLA might be a program that's struggling, it doesn't look like they, despite having almost all their starters back on the defensive side of the ball, doesn't look like they've really done that much on that side of the ball. And that's a real problem when you're playing Oklahoma, a team that looks like it hasn't lost a step even though they lost the first pick in the NFL draft and four drafted offensive linemen. And you know, on offense, you know, Chip Kelly's supposed to be the offensive guy, but Dorian Thompson-Robbins, the quarterback, you know, couldn't hit a throw against Cincinnati, really up the completion rate against San Diego State, but wasn't very efficient in terms of yards per play. So it's, he threw for 4.8 yards per play, which is yards per pass attempt, which is not gonna cut it. I just, for the record, so that number includes sacks. I include sacks. I count sacks as pass plays. This is not the case in official college football numbers. So that's something you want to be wary of. Like if my yards per rush is different from what you're seeing, it's because I've taken sacks out of the, I don't define that as a rush as they do in typical college football statistics. So I just want to make that clear. And that's anything that number fires numbers do for the NFL as they include sacks and all passing efficiency numbers, because like a sack is not a turnover, but it's like next up on like the damage scale along with there with like penalties. Like sacks put you well behind. And that's a pretty damaging thing. Right now, again, the line is minus 23 for Oklahoma. It opened at minus 21 and a half. So there has been a lot of movement there, one and a half points, 94% of the bets and 97% of the money on Oklahoma at the Fandle Sportsbook. So like, I wouldn't be surprised if this number moved more. Is it a situation where you would be willing to bet Oklahoma at that number, given that it may move even more? Or are you just saying you want no part of UCLA? Yeah, I think I want no part of any game involving UCLA. It's kind of like staying away from Louisville last year. You just kind of have no idea what's going on. Maybe you can find a beat writer that's got some insight into the team.  And I want him to succeed. I just don't see it so far. Yeah. And the problem is they're all facing Jalen Hertz who has looked amazing. And part of that has been, you know, yards after the catch, his receivers creating for him. But a 17.3 adjusted yards per attempt for Jalen Hertz. There was first 41 attempts. That number does not include the sacks, but it's kind of balkers. What he's doing right now. Yeah. And I haven't looked at it, but when is he going to face a real defense? I mean, All right. Let me put the schedule here because I am also interested in- Get to Iowa State. I definitely, I definitely- Iowa State's a very good defense. So that would be one of them for sure. Maybe as a defense. So I don't, you know, I mean, Jalen Hertz might win the Heisman because of strength of schedule here in some, right? I mean, TCU is probably going to put up a pretty good defense. So you've got Texas Tech, Kansas. No, no. Texas, the Red River shootout is October 12th. Okay. And then West Virginia, Meh, Kansas State. Iowa State is not until November 9th. Wow. TCU is November 23rd. And by that point, sentiment on Jalen Hertz as it relates to Heisman voting will not be set, but we'll have a pretty good idea. And like if people are like, okay, cool, this guy's a Heisman, it's going to be hard to dissuade them from that. So like, I think, I would say this, like Jalen Hertz, Heisman odds have come down. I think he was seven to one when I was looking earlier today. But if you like him, like if you want to bet him, you're probably not going to get a better number than that at any point. He is, yeah, he's seven to one right now. Kind of the same thing we were talking about with whalecapper, you know, look at the schedule and decide. And he's, by the time he faces Iowa State, he'll probably be like three to one or something crazy. So I would say that if you want to bet Jalen Hertz, I'm not saying you should want to, but like if you do want to do it now, essentially. Yeah. Yeah. Absolutely. Any thoughts on the total in that game? It's a 73 and a half. And it's gone up quite a bit from where things opened. I know number fires numbers like the under here, but what do you think? That's a pretty lofty number there. It is a pretty lofty number given UCLA hasn't shown much competence on the offensive side of the ball. Obviously you have to balance that with the struggles that Oklahoma has had on the defensive side of the ball. So yeah, it seems high just because of that, but you know, maybe Oklahoma gets there by themselves. Right. And probably not. But you know, I mean, if it's like 77. Yeah. It was they, they had 80 points against Houston. That one was 49 to 31. And then against South Dakota, it was 70 to 14. But I think that one thing we could see with Oklahoma is maybe they go a little bit more run heavy given Jalen Hertz being there and stuff like that. I think that could show up more when they're facing not South Dakota opponents. So I would look at that line. I don't have a good enough feel on it to bet it, but I think if I were to go one way, I'd probably go the under, but I don't feel good enough on that to actually take that number. Let's move on here to Ohio State against Indiana. Ohio State, it's 16 and a half point favorite here. The total is 61. Interesting game because Ohio State kind of getting its first, I don't want to say legitimate test because like they haven't faced necessarily bad teams, but Justin Fields has looked amazing the first two games. So Ed, do you think the way we've seen from Justin Fields is legit and here to stay going forward? Yeah, I mean, I talked to Bud Elliott a couple of years ago before he had ever reached the field, the college football field, and he just loved the quarterback class of Fields and Trevor Lawrence and J.T. Daniels. And everything we've seen so far has supported that. And I think as many questions and marks that I had about Ohio State coming into the season, it's still Ohio State and they get the benefit of the doubt until proven otherwise. And so far they look pretty good and I don't see why that doesn't continue against one of the lesser teams in their division. And we're seeing a lot of, again, money on Ohio State here at Fandall Sportsbook right now, 99% of the bets and 99% of the money on Ohio State. So the spread has moved from Ohio State minus 15 to Ohio State minus 16. The total in that game, pretty stagnant. It opened a 60 and a half and is at 61 right now. But that's like, it is a road game. It's a conference game. Do you view Ohio State as being strong enough to cover a number that is not huge, but getting bigger? Yeah, I mean, I certainly think they are, right? I mean, with Fields and the explosiveness of the receivers that they do have. I mean, oftentimes it does come down to explosive plays and big plays and we know randomness plays a big role in that. So, you know, that's often what happens when you're trying to cover a big number and not necessarily trying to cover, but playing in these games. And yeah, so it's, you know, I think it's pretty fair. My number's a little bit lower than that, but it's not something that jumps off the board at me. It's like, oh, the number's like Indiana, let's go bet that. It seems like it should be a little bit higher. Yeah, Ohio State, the first two games, 42 to nothing against Cincinnati and then 45 to 21 against FAU. And I think the big key there is kind of the nothing against Cincinnati because last year when Ohio State was vulnerable, it was because of their defense.  It seems like they have changed their approach defensively, playing a lot less man coverage, which is seemingly beneficial for them. So like, I don't dislike Indiana, but I think Ohio State's defense is intriguing. What do your numbers say about Ohio State's defense? Are they buying into the early season performance? Or does it seem a little bit flimky? I mean, nothing yet. I mean, I can't make schedule adjustments until after week three. I will note that Cincinnati was not strong efficiency wise on the offensive side of the ball. I mean, they were a 90th, something like that, something like 90th in my adjusted yards for play, bring back the same quarterback in Desmond Ritter. So yeah, I mean, obviously a good game for them to show out Cincinnati, but I wouldn't make too much of it. It's not like they were getting like a team like Ohio that really had an efficient offense last year in that non-conference game. Yeah, I think this game, if I were to bet it, I'd actually been most inclined to go over the total of 61. It is a large number, but I think that Ohio State's defense probably not quite where, we haven't quite seen it fully yet in the sense from a negative sense. And I also just have some respect for Indiana's program. So I think that this game would go over. I don't have a good enough read on the spread here to go either way, but I'd be intrigued by the over. Number fires numbers also behind the over on this game. And I believe it was, actually don't have it right here, but regardless, I believe that they were on the over. So I think that that's where I go.  Yeah, I mean, I'm gonna stay away from this game just because I had my doubts about Ohio State that hasn't been true yet through two games. And so let's see a little bit more, let's see Ohio State get to a little bit better competition. Expand the sample size, make the data a little better. I understand that for sure, especially in a team where you don't know what to do with them, that's probably the smart way to do it. Let's go here to Alabama at South Carolina, Alabama 25 and a half point favorite, total here is 61 and a half, and South Carolina did lose its quarterback, Jake Bentley. That was back in their season opener. I believe in the final play of the game against UNC. So starting true freshman, Ryan Helinski, he looked awesome last week, but slight difference, Ed, between Alabama and Charleston Southern. I don't know if I'm being, I don't wanna be dismissing Charleston Southern, but slight difference. So how do you view South Carolina in general right now through the first two games? I mean, they're the tough luck team of the year. I mean, you come into this year and you have to play the top, kind of consensus top three teams in the nation, Alabama, Clemson and Georgia. Yeah. Because you're in the SEC East and Clemson's arrival. So that sucks to begin with. So you need to get that weak one win against North Carolina. Right. That doesn't happen. And you lose your quarterback for the year. And Jake Bentley was a quarterback that, I mean, he's been starting for two years and heading into the bowl game in the 2017 season, the numbers weren't impressive. Like they were an FBS average type team. Bentley came out and had a really good game against Michigan in the bowl game. Was very efficient, hit a couple deep balls, helped South Carolina win that game. And then they were very good last year. They were 14th in my adjusted yards for play, even better in passing in terms of the rank. And now you don't have that leader anymore. I mean, that's a killer. And they're just not catching any breaks. I still obviously have a good defense. And this is an interesting one too, because it's hard. My numbers like Alabama by about 15, that's obviously needs a big adjustment for the quarterback position for South Carolina. Do you get to 25? Probably not. But similar to what we see with the Ohio State game, these top teams get the benefit of the doubt. And partially you can think of this in terms of turnovers. When you're up, when you're a big favorite of the game, you tend to be up late in the games. That means a worse opponent has to throw to try to catch up. That can lead to turnovers. Sorry, that can lead to interceptions. And that's what makes the spread in these big games a little bit more. You can't have a model that gets close games right, but gets these large spread games right as well because of the turnovers. Because there's not going to be a huge interception differential in a close game. But there is going to be in a large spread game, which is something that I'm working to quantify over the course of this season. So Alabama, when they're playing people that are not up to the level, and South Carolina is probably not without their starting quarterback, you're going to see these inflated spreads. And just because the numbers, my numbers to take the underdog doesn't necessarily mean you want to bet that. And I think that one thing that backs up your point is you look back at, we were talking about the Browns Titans games last week. And that was in the NFL. But I think it's a good parallel because early in that game, it was a very close game the entire way. Then there's a safety, Cleveland has its left tackle ejected. And then their backup left tackle gets hurt. And things spiral. And so Baker Mayfield is trying to be aggressive to make up this scoring differential. And aggressiveness in that situation is not a bad thing, but it does lead to picks. And that kind of compounds things. And that's why we saw things get so out of hand and Tennessee win that game so handedly. And that's what you're going to see in these situations too. And I think that you look at South Carolina, the question you ask is, can they keep themselves out of a grotesquely negative game script long enough to avoid that compounding issue? And if you think they can, then you bet them plus 25 and a half. If you think that Alabama comes out and throat stomps right away, then you probably need to factor in the compounding. So it's a question of, do you think South Carolina will be relevant early in this game? If yes, take that number. And I think that that's definitely a compounding factor. You talk about the difficulties in projecting games with spreads this wide. Does that push you towards betting first half lines? Because it seems like they've been rising in popularity. Do you have a take on that necessarily? Or is it just for reducing the sample size too much for you? No, I mean, it's not necessarily sample sizing. I mean, I just haven't looked into first half betting much because I haven't looked at the data in why the first half lines are usually more than half of what the game lines. And I know there's a good reason for that. But I just haven't looked into that, so I tend to not do that. But yeah, I mean, if you like the favorite to cover, you probably more like the first half line. Especially in these powerhouse teams against FCS programs, you can potentially like the first half. You just run it up in the first half. 42 nothing, then try out whoever's available for the second half. So I think it's a very interesting game. I don't have a good enough feel to bet the spread in that game. Again, if I had to go, I might just go South Carolina. But I don't feel good enough. I'm not going to bet on a back of quarterback as a true freshman against Alabama. So personally, I'm staying away. Any definitive ones from you here, Ed? No, not really. I do want to bring up a point where you talked about the fact that you do need to be aggressive when you're down. And Baker Mayfield was a perfect example of that in the NFL Week 1. And Kevin Cole, who's a pro football focus, brought up a really good point. We laud Aaron Rodgers for not throwing interceptions. But perhaps maybe that's not the optimal strategy. Like when you're down late in games, you shouldn't be just trying to protect the ball. You should be taking some risks downfield. And on the flip side, that says, all right, well, I mean, you give Baker a little bit of a pass for throwing those picks down late in the game. You have to be a little bit aggressive in that situation. And I don't think that's something that's really appreciated in the numbers yet, and certainly not my numbers, and something I'm working towards getting a better feel for. Well, it's not even just that. It's also that sometimes people will think that they are being risk averse and take a sack rather than trying to force a ball on the coverage. But like I said, like a sack is pretty damaging, too. And Aaron Rodgers' sack numbers last year were high, which is interesting considering the talent they have at left tackle. They have one of the best left tackles in football, and David Bakhtiari, he shouldn't have a high sack total. So it's not just the fact that he didn't throw picks. It's also the fact that he took a lot of sacks potentially in lieu of throwing those picks. And maybe the aggressiveness should have been higher. Like Peyton Manning was never a low interception dude. Like he took risks. And that actually brings up another point. I had Dr. Eric Giger on my podcast this week. And he's done some really interesting things to show how much of pressure rates from the defense, like the quarterback owns as opposed to the offensive line. And that's been really interesting, too, because you watched the Monday night game. And it looked like Deshaun Watson was almost like inviting the sack in some sense. Well, he's been the poster boy of that discussion. He and Russell Wilson have been the two who have been criticized the most for inviting pressure.  And he's done some interesting work. So yeah, I mean, I think it all kind of comes into a more modern understanding. And you know, Eric's done all that work on the NFL. It's interesting to apply that to the college level. I'm thinking about Michigan in particular. And Shay Patterson didn't have the best game against Army and oftentimes was dropped back to pass and didn't get the ball out, had to pull it down and run. Well, he got the ball out just as a fumble rather than a pass attempt. Like, can we get Shay Patterson some stick him? Like, it's a real issue, I think. Yeah, it's a real issue too. And I feel like a couple of years ago, I would have been like, yeah, you know, just protect the ball, you know, fumbles are kind of random. But now the more we think about it, it's like, well, the quarterback has even more control over what's going on in the football field than we initially thought. It's not just the accuracy, it's, you know, this decision making is really affecting sack rates and everything, you know, related to that. Yeah. So it's making me look at football a little bit differently. And yeah, I mean, that's how we all get smarter and hopefully we can use that to make better bets. And I think that that can help, be helpful for live betting too. Like, if you see a game getting out of hand, but it's not a game where the team that's trailing will just roll over and die, like if they're gonna keep trying to come back, it's going to lead to extra aggressiveness. And I think that that can be a factor when you're live betting as well. So keep that in mind, try to keep a gauge on a team's desperation factor. And if you think they're going to get aggressive and force themselves into mistakes, account for that when you're making your live bets too. Well, and plus like, I mean, you guys do that for fantasy all the time, right? When you're looking for an offense, that's going to be down late in games. Right. A lot of opportunities to throw. Well, for me, it's actually defense. So like, talking about from like a fantasy perspective, I want my defense facing a lot of pass attempts because to score points in fantasy, you need sacks, interceptions, force fumbles, and what is the play on which those are most common? It's a pass. So basically you want to have the opposing team in negative game script so that they're throwing and that they're pulling a James Winston and toss and pick sizes. Exactly, and you could care less if they gave up a big play. It's great. In fact, that really helps you. Right, exactly. But I think it's important to keep that in mind for live betting too. So that will be a factor with Alabama, South Carolina. So I think that live in-game betting, definitely a fun angle to keep an eye on here. You know, check out how South Carolina does early on and that's going to dictate how things go for the rest of the game. Anything else you see on the board here you want to mention for week three in college football? And we're going to talk about again, Iowa Iowa State in covering the future. But anything else you want to tack on here before we move on to that? Yeah, no, not really. I mean, let's move on to Iowa Iowa State. All right, let's do it. In covering the future. Covering the future. All right, before we take a look at El Asico, we got to talk to you about odds fire. I was talking about some of those numbers during that last segment about how much money is on the total and stuff like that and how many people are betting on Ohio State, et cetera, et cetera. That all comes from odds fire and Ed and I always preach searching for the best value in betting on games. Well, look no further than the new odds comparison our engineers have developed over at numberfire.com. Odds fire is a premier odds comparison service across major bookmakers in the regulated US market. Compare odds, quickly identify the best value and even examine first party fan dual data all in one place. Never settle, always get the best odds. Check out the experience for free now on numberfire or at oddsfire.com. Gambling problem, call 1-800-GAMBLER. Let's dive in now to Iowa State against Iowa. I am from Minnesota originally. I have a friend, a couple of friends went to Iowa State. My friends, my mom is from Iowa. So like I have a lot of rooted, rooted or like vested interests in this game. You want to talk about this game because your numbers see an edge within it. So talk to me about Iowa Iowa State right now. It's Iowa minus two and a half total here is 44 and a half. What do you see with this one? Yeah, I mean, I've been a big fan of Iowa State this preseason. We've talked about that on the show. I see them as a dark horse contender to potentially win the big 12. And a lot of that is they bring back Brock Purdy, the quarterback who was fantastic as a true freshman last year. They also have a pretty good defense. They had the second best defense in the big 12 in terms of past defense and my adjusted yards per play. And they had a lot of guys returning with experience on that side of the ball. But then week one happened. They played Northern Iowa at home. The game went into overtime and they were actually down three and they fumbled the ball right near the goal line. And obviously if Northern Iowa recovers that the game is over and then it's a crisis in Ames. But Brock Purdy jumped on the football. They took it in for a touchdown on the next play. They ended up winning that game. Now, whenever you get a three point win over an FCS team, that's really not gonna impress anyone, right? And when I make adjustments to my model, like Iowa State's obviously getting dinged for that. But they look better in the underlying statistics. When you look at yards per play, they had 5.4 yards per play compared to 3.6 for Northern Iowa. I told you about this model earlier where I take yards per play and total yards in a game to project out a score. And by the underlying metrics, Iowa State would have won 31 to 11. But that's not what happened on the scoreboard, but it just gives you an idea for what happened in the underlying numbers. I'm also starting to look at success rate and here Iowa State also did a lot better. They had a 44% success rate compared to 33 from Northern Iowa. It was very asymmetric and I think you do have to be concerned about Iowa State in that they only had a 34% success rate on rushing. So a lot of their success came through the air and with Brock Purdy's arm. So they may not be able to run the ball again this year, an area that they struggled even with David Montgomery last year. So this preseason, I had Iowa and Iowa State basically at an identical rating, which means that with Iowa State at home, I would have made them about a three point favorite in this game. Iowa hasn't shifted much. They've done what they're supposed to do against Miami, Ohio and Rutgers. And Iowa State's definitely getting dinged for this game against Northern Iowa. But I don't think it's a five and a half point shift. I like Iowa State at home with the points plus two and a half. And I think it's gonna be a close game. Obviously a late score can definitely affect this game. But I mean, I think Iowa State can win outright. I don't, I still think they're a good team. The money line here is plus 115 on Iowa State. And I think that that's pretty interesting. 115, really? Yeah. Okay. That's interesting. Cause I would think it would be, it would be higher with the spread of two and a half. So that might. There's a big in there for sure. Okay. But I think it's interesting and hearing you talk about that, that UNI game, like I did get to catch part of that game and it didn't feel as close as it wound up being. And I think that your numbers kind of validate that. And it's also like, yes, it is an FCS team, but Northern Iowa is consistently among the best FCS teams in the nation. There's that little rivalry. UNI always plays Iowa or Iowa State really well whenever they play, like every single time UNI is ranked 10th in the FCS rankings right now. Like they're a good football team, even if they are an FCS team. It was week one for Iowa State playing without Hikim Butler, without David Montgomery. That's a pretty big changeover. So I, and they also have a week off. Like they had a week off after that. So they are rested. Iowa kind of rested because they played Rutgers. Like, did they actually play a game last week? But I think that, Well, they did. Yeah, allegedly. I'm not gonna confirm or deny that one. But Brock Purdy in that game, 41 attempts, or 278 yards, two touchdowns, no picks, played pretty well. And he's also like, I would say underrated athletically. I don't know if that's like, I don't know if that's like being outlandish, but like he did run for 308 yards last year. And that is while including sacks in that number. So I think that Brock Purdy is just a good quarterback who is at home and is getting points here. So I think that I would be okay with either the spread of the money line. I'd probably just go the money line honestly, but money line sounds pretty good. Yeah, plus 115, I think is a good number. So are we in agreement, Iowa State money line here?  I do both. All right, cool. So we've got Iowa State either the money line or the spread, whichever you're feeling more, but I think that's a pretty good one. And broadly, I think it's important just to note the context of that game where it was their first game without two big offensive pieces, facing a good team. Now they have a bye week at home. There are a lot of things surrounding that game that are different now. So I think it'll be a fun game. It's always like, it's always like kind of a slop fest. There's a reason it's called El Asico, but like they're fun. They're fun football games though. Yeah, and this definitely makes you think about the under two. Iowa has an excellent defense. Definitely going to be able to slow down Iowa State on offense. I mean, a lot of this game is going to be on whether Purdy can make plays with his arm. I don't have too much faith in Nate Stanley, the Iowa quarterback. So I don't see a lot of points in this game. It should be an ugly, slow game that comes down to the wire. I just like Iowa State's chances there. Another thing I want to mention, you did mention the bye week, which gives Iowa State rest. It also kind of screws with my model because it's one game is kind of being counted as two. When I make these kind of season adjustments, which is not really fair to Iowa State. Right. So, you know, let me, so yeah, so anyways, like it's probably overestimating the adjustment for Iowa State at this point. So a lot of things point towards the cyclones for this game. Alrighty, I'm sure that, you happy to hear that should be a fun game. College game day going to be there. So I was sad that the Syracuse game went the way it did last week because I live in Syracuse and I wanted to go to game day, but you could kind of tell that was coming. Like I was listening to the pregame shows and everyone was talking about college game day and being there the next week.  So hopefully things are fun out there in Ames. For my cover in the future at, I want to get your thoughts on this because when I first look, looking at the NFL this weekend, I love the under for the Chargers and Alliance in Detroit. And I want to get your thoughts on this because it's an interesting conundrum that I have here because the Lions are going to be a super run heavy team this year. And that's why I think that in general, they're probably not going to be in a lot of high scoring games. They played a pace up game last week against Arizona. So their slow nature didn't really show up, but they threw only 47% of the time on first down in the first half last week, which is a very low number. And they did that well, but the Cardinals defense is terrible. The Chargers defense does not have Derwyn James, but they're still much better. The Chargers will throw more often, but they're also missing their left tackle, which does ding their entire offense. Didn't do a ton last week against Indy despite the fact that's kind of middling defense. And Detroit's defense is not bad either, especially they can get some pressure on the passer, which could show up those issues at left tackle. So I generally love games and domes, and I tend to think highly of them, but I think this one will go under. But the problem, this is why I want to talk to you about it, Ed, is that I'm in the minority here because the total has already risen. It opened at 47. It's now 47 and a half at Fandall Sportsbook, and it is actually higher elsewhere, which is probably why 86% of the bets and 92% of the money at Fandall Sportsbook are on the over that is according to odds fire. So it very well could rise even more during the week, especially with this number being 48 elsewhere. I would expect it to get to 48 at Fandall Sportsbook at some point, but I personally feel very good about the under here. So I feel good about my number, but I'm in the minority here. So how do you handle those situations personally? How long are you willing to wait for that number to potentially get better before you pull the trigger and dive in? Yeah, no, I mean, I think that's a tough one. Right now, just kind of early season, Detroit had like the 21st, you know, a below average defense last year. Could they bounce back? Sure, with a little bit of health and some good bounces their way. So I mean, I definitely like your thinking about that. I think I would definitely wait a little bit while longer and see what happens when some of these limits start to go up and yeah, I would wait a little bit longer on that one. And I think that there's very good odds it gets to 48. So I wouldn't touch it until it gets that high because I think that the odds it gets there are quite high, but I just think this game is probably going to be a little more run heavy. And again, it kind of goes back to what we talked about before where the holding calls in run heavy games are going to be more apparent. And I think we saw a lot of those holding calls in week one and like, as a viewer, it was very frustrating to watch all these flags. Cleveland had like, I think it was 20 penalties. Like it's kind of insane. So I want to bet unders and gains that include run heavy teams. I expect Detroit to be a run heavy team this year. And they were when, you know, things were in their favor in week one. So I think I do like the under here, but just broadly, I just wanted to talk broadly about your thoughts on when you have like a number that you know has a possibility of getting better. So I agree. I think I should hold off there. Yeah, I mean, it's anyone's guess where the market's going to go, right? Like no one knows in particular. And like the best example was I talked to Chris Andrews, who's the director of the South Point sports book before the Super Bowl last year. And the day I talked to him, he had moved the line to I think New England by three. And then he quickly moved it back to two and a half. And you know, he came on my podcast and was like, yeah, this number is going to three. So I was like, great. This guy knows more about anything, more than anyone about where these games go. And I liked the Rams in that game. So why not wait until it gets to three? Right. Well, never got to three. Right. So that's how it goes sometimes. And it's the market. It's random. We don't know what other people are thinking yet. And we just kind of follow and see what happens. Yeah, I think it's very interesting that angle of it. We've talked with a lot of our guests about trying to get the best number. I think that things like Odd's Fire can help you get that number, just knowing where the money is at that time and saying, okay, maybe you should hold off because people are leading this way. That is all we have for today. But once again, we're going to be back tomorrow to preview week two of the NFL season, breaking down three of the biggest games. And including again, that Dolphins Patriots game, which should be fascinating discussion. Looking forward to that. Make sure you subscribe to covering the spread on Spotify, Apple podcasts, the Google Play store, tune in, stitch or wherever you get your podcast, you can find covering the spread. Ed, you mentioned a new podcast up at the football analytics show. Anything else going on? And where can they find that podcast? Yeah, the podcast is the football analytics show. Usually have a guest on every week. In a couple of weeks, I'll start doing some solo shows just because I like to get some of the good insights from my numbers as they start coming in and I can make schedule adjustments. Also, my email newsletter goes out every Thursday at noon. This is where I give you a sample of predictions that I talked about on the show, but are usually safe for paying members of the site. And I write up some analysis in there too. So if you like what I'm saying on the show, definitely check out the newsletter because that's where you can get more of it. And that's at thepowerrank.com and the podcast is a football analytics show. You can find Ed on Twitter at the Power Rank. I am at Jim Sonnis, J-I-M-S-A-N-N-E-S. You can follow me there in the Fandall Podcast Network at Fandall Podcast. Big thank you to Calvin Theobald for working the video side of things today. Cal, thank you for keeping us on the air as always. And thank you to those of you for tuning in for today. Back tomorrow to talk some NFL. We'll get you then and good luck with college football in week number three. This has been covering the spread right here on the Fandall Podcast Network.