Okay. Welcome back everybody to our second lecture, BC 314 Media and Technology. This lecture is not going to be too long, maybe, I don't know, maybe 10-15 minutes. We're going to spend a little bit of time just talking about social media. So in the previous chapter, or the previous two chapters, we talked about digital engagement, and we just said some guidelines on how we can optimize, how we can do our graphics and videos well, and how we can optimize them for search engines so that they become more visible in this vast ocean of content that people are putting out every day, there's so much of content being put out. So we just follow some guidelines in terms of title, description, tags and so on. The possibility of being more visible to people and being presented to people increases. Now, we're going to talk a little bit about social media and how, as from a church perspective, how we can leverage that. So we obviously, when we talk about social media, we need to answer a simple question. In terms of our target audience, as a church or ministry, it's okay, where are our audience? What are they engaging on, Facebook or Twitter or LinkedIn or any of those large number of platforms that are there? Where are they engaging and where do you want to engage people? So as a church, for us, we are primarily, we are looking, we are engaging with people primarily on these three platforms on Facebook, LinkedIn, Instagram. We're not doing too much on the other platforms, but depending on whom you want to reach, you will engage with people. So once you kind of decide on what social media platforms you want to be on, and we're not talking about the other platforms we already mentioned earlier, but once you decide on that, then what are some things to do? We need to do regular posts that are from the church or the ministry. So some of the things that we do, and I'm just sharing some ideas, not saying everybody has to do this, but just as a might isn't what we do, is in addition to the livestream which happens of our Sunday sermon on Sunday on YouTube, we post, we edit the sermon, that means we take out the worship parts and just the sermon part, we edit the sermon, and that sermon is put out separately on YouTube and other platforms. We also do sermon key points, which is a five-minute summary of the sermon. So that means, you know, those who say, well, I can't sit and watch the full 40-minute sermon, just tell me the key points for them. You know, this thing helps, they may be interested. And some people don't even want five minutes, they just want, give me one point on the sermon. So this is that sermon highlight, so we also do a 60-second highlight of the sermon. So this is what you can pack in 60 seconds, and that goes out on these channels. So we're giving people a lot of options. There is the full livestream that they can go and watch, or they can watch just the sermon, the full sermon, or they can watch five minutes of the sermon, or they can watch 60 seconds of that sermon, one minute of that sermon. So all these options are there, and these things go out, usually by Monday, they are released on these platforms. Then the other thing that we do is, usually by Friday of the week, we announced what is the sermon coming up for Sunday, the sermon title, although those kind of things work. The reason they do it is just to not only inform, but hopefully try to generate interest. Hey, this coming Sunday, I'm going to hear about this. So sometimes that sermon title might interest them, they will come with expectation. Sometimes they might think about inviting their friends. They'll say, hey, this sermon would be useful for them. So they'd invite them to come. So we do that every Friday. There's a graphic that goes out saying, coming on Sunday, this is the sermon title and so on. Now, of course, if it's a guest speaker, then we won't know what the guest speaker is preaching generally. So we just put the name of the person who's preaching. But if it's our own people, we know what we're going to be preaching. Then, typically, we also put out announcements of other events on our social media platforms. Sometimes we post testaments in an anonymous way. We don't mention names of people. So as and when this testimony comes, a little graphic with a brief summary of the testimony goes. Time-to-time happenings, that's services, volunteers, pictures about the life of the church or highlighting special events, et cetera, go up. I just think we get a feel of what's happening. We don't overload people with that. Obviously, we're not like putting every photograph online, but there's some little taste of what's happening at the event or at somebody's serving and so on. So these are kinds of things that we post on our social media platforms as a way to engage with people. Now, of course, we use LinkedIn in a different way. So whatever I said is what happened on Facebook and Instagram, but LinkedIn is targeting professionals. So for that, it's a different thing that we do. We post short articles related to people in the workplace so that that might be of interest to them. We have somebody who does that, one of our touch people who will do that. They'll take a summary of some text or something from our books and posts just to engage with people on LinkedIn and keep them interested and so on. Now, when we're using social media, some thoughts, here are just some good things to do. It's not a rule, but just some gentle thing. We try to make your messaging that is the look and feel of your posts, it should look and feel the same across all platforms because really you're creating a persona or a personality of who you are as an individual or who this entity is as a community. It's creating a persona. So it's important that there is a consistent look and feel, a consistent messaging, a consistent persona that comes across. People look at that and say, yeah, this is a church that does it like this. So maintain a consistent look and feel. Like we said earlier, the forms we use, the colors we use, the casts you use, how you're communicating comes across consistently. Now, one of the things that we've done intentionally is not to create a personality around the individual. So right from the very beginning, actually in the very beginning we were very strict. I told our media team, don't put pictures of me on the cover of any of the graphics. So we stayed away from creating a persona or a personality around the individual. We said it should be around the community, around the church. So in all the posts that we do, we stayed away from that. Only maybe a couple of years ago, the media team said, can we use pictures of the person who's doing the daily devotional? Just for the daily devotional, can we use them so that people know who's doing the devotional? And just for that, we have given permission. So only for our daily devotionals on the cover of the video, you will see the picture of the person doing the daily devotional. Otherwise, we said, no one else. So we don't want to create a persona around the individual. Let's do it around the church community. And that's our choice. A lot of churches and other ministries may do it differently. You'll find pictures of the pastor or the leader coming up everywhere, everywhere. That's okay, that's their choice. But our choice was stay away from individual and create around the community so that individuals will come and go, but the community continues. So that's long term. And so that's kind of what we are creating. Sharing content is like the word of mouth on social media. So I encourage people to share on their posts, on their timeline. We don't force people to do it. We just can request them, hey, if you like it, share it. Very important when you're doing your post is to use hashtags. That's part of the word or the keyword. But we're saying that this content is about this topic or in this category. And so that helps the search engines index the post and make it visible to other people. And when people search for a keyword or hashtag, that's related, it'll come up. So typically, you're using it as a word without spaces or numbers, no special characters. Like this or lower case. I think most people are familiar with it. So these hashtags are important because that's what helps that content, the graphic or the video that you're posted be found by the right people or interested in those kinds of things. And of course, it should be relevant to what you've posted. Typically, the suggestion is to use what three to five hashtags for every post. Don't put 100 hashtags. It normally doesn't use beyond for the first 15 hashtags. So the top three to five is what would be used optimally. So use that. And you can use hashtags that are related to your brand. So for example, we use all people's church bangles.  So you can kind of create some sort of a brand online by which people know what you're doing. Or you can use that. And you could include hashtags and titles and sentences. If you have space for it, you can do that as well. So if you want to know what are the right hashtags to use, you can use, there are some tools that help you find the right hashtags, like Google keyword planner or Instagram tag search. So they give you ideas on what tags to use, what tags would be most useful for your content. Here are the points here is that when you're doing your social media posts, of course, there will be people who will respond, who will put comments or so on. We ask questions, ask for more information, try and respond to these so that people know that you're active, you're interested in engaging with people. Again, we have certain people who are responsible for this, so they will handle it. I personally, I don't have time to even see what's happening, but generally there are people who are handling all of this. They will respond or they will see what's happening and look to that. Also generally, there's a best time to post. And again, you can study this up. I don't talk too much about this, but generally they say that there are certain days of the week at the time of the day when it would be most optimal to post. So if you post when your audience are not logged in, then what you post is going to go unnoticed and other things that are posted on the timeline come up and what you're posted goes way down on the timeline. So there is an optimal time that you can release content. And if you identify that for your audience, and now your audience online, what we pretend to post, so that when they are online, you post, you're more likely to see what you've posted. So that's a thing. And now some of our work, a lot of our online posts are automated. That means it's not like some physical person going and posting it. For example, like I mentioned, Sunday sermons, all that is all actually automated. There are scripts that have been written that actually keep pushing our content onto these social media platforms, all happening automatically. So things that happen every week, we have automated those posts onto our social media platforms so that it's not something manual. But other events that are happening differently week to week, somebody actually does those posts. We've automated a lot of those, so on. In addition to these posts, you can do paid promotions and advertising on social media platforms. So with a small amount of money, you can actually reach large numbers of people, it's very cost effective, and we do that. So on our Facebook, Instagram and LinkedIn, we do paid promotions of maybe our books, maybe Bible college admissions when we're taking in people, or certain events. We don't do promotions for everything, but on certain events, you would like to put in a little money and push it. I think this is very, very useful because you can reach hundreds of thousands of people for small amounts of money. So use that whenever you want. So even if you spend 5,000 Indian rupees or 10,000 or 20,000, you could reach lots of people and you can target them by demographics, by which part of the world they are in or the city they are in. You can target your ads so that you're making your event visible to your intended audience. The last two points is it's useful to measure results. Like you said, there are metrics we can look at our website. In a similar way, you can look at metrics for what's happening online. So YouTube analytics, Facebook analytics, insights on Facebook, these things are there, already available as part of the platform. Go and look at them, see what's happening. So we can see how many people are out. So every week, our team will send us a short update saying we have so many people watching, we have so many people viewing, etc. Real-time. So we are looking at it from so many different countries. So you can measure these results, you can see these results. And there are other monitoring tools also available. So there are insights available from within the platforms, but you can also look at it from outside some other monitoring tools that you could use. I think there was a question here. Let me see. Okay, so question. If you get hate comments from any of the advertisements, do we respond or ignore or hide it from them? Yeah, so generally, we hide. So for hate comments, we report it and we hide it. So we don't engage with hateful comments. There's no point. It just gets into non-humans. So we just report it, you flag it so that Facebook or whichever the platform is, they will handle it. And you hide it so you don't want others to see it. That's kind of how we respond to that. Yeah, okay. So social media is useful. But again, I don't know. I was just reading a recent article saying that Gen Z is actually moving back to reading physical books. So it's kind of like a trend that they're kind of tired of social media and there's a trend here to moving back to reading physical books and so on. That was just a recent research report that came out. I'm not saying every person is topping social media and moving this way. They're just finding that there is this shift back to reading books and coming back to those kinds of things, which is interesting. But to whatever extent we can reach and serve people online, we can make use of that. But I think people are finding social media just too cluttered to sometimes like John said pointed out very hateful and all kinds of things happening. Sometimes it becomes very destructive whereas when you're reading a book, you choose the book you want to read. You can read just that or the other advantages are there. So maybe there's a shift happening which we also need to be mindful about. Okay. So we're going to pause here for today. Next week, we're going to shift what we've been talking about. We're going to get into digital equipment. So I don't know if you'll like that, but we're going to talk about the equipment that we typically use for our work.  So I remember in those early days when we first started, we had, we got to buy, I had no idea about speakers and mics at zero. Hardly any knowledge. We had to go buy. I had to go buy the speakers and mix. I had no idea about these things. I just went to the shop and said, look, I want two speakers, I want three mics and how do I put it together? So I had to learn when you're starting a ministry. I had to learn how to buy. And then of course, as people started coming, we needed better equipment. And people would come and say, pastor, we need to buy, for example, I remember one of the people said, first we need to buy a snake cable. I had no idea what a snake cable was. What is that? So what is it? And then they explained, it is all these wires combined together and you have a long cable. And as like, okay, I wanted to do ministry and I didn't realize I had to make decisions about all these kinds of things. But it was a lot of learning. And then when you had video and camera and lots of these things. So I said, okay, maybe I don't need to know all the details.  You have to approve it. There are three options. Which one are you going to buy?  You don't have to be an expert. Enough to have a conversation with your team to know what to buy, what camera to buy, why you're buying that. Why is this camera better than that? These kind of things, doing presentation, what software, why the software, why not that. So what I want to do is from the next class, we'll go through software and equipment for cameras, videos, audio, live streaming. So we kind of go through those things. And I'm not saying we should be experts on it. You don't need to be an expert. Just have enough information so that you can have a conversation with your team and tell them, yeah, this is what we will use. And a lot of this is always changing. You're getting better cameras, better things. So people come to me and say, we need to buy better cameras, we need to add another camera, etc. So you have to have discussion, we have to talk to them, ask them the right questions and then make the decision. So from that perspective, I will be sharing these things. So next week, we get into digital equipment, a little bit on the software and the hardware, both from cameras, video cameras, to audio equipment, live streaming equipment, what they're using. And just keep that in mind so that it might be useful for your church and your ministry. So we're close for today. Thank you for your patient listening. I hope these things are useful and hope you'll be able to make use of it in your church and your ministry. Somebody could close in prayer and we'll discuss. Dear Heavenly Father, I come to you under the name of Jesus. I thank you for this day. Thank you for everything that we learned today about we thank you for the technology. We thank you for the development that the world is going through and for you fill us with your knowledge, with your wisdom and guidance. So therefore we could use these tools for the edifying of the church, for building up your kingdom in a right way Jesus. And we thank you for everything that we learned. We thank you for Pastor Ashish help us to be equipped even in this way Jesus so that we can work mightily your kingdom. We can be salt and light to the nations and nations. Thank you. We give you all the glory in Jesus name and prayer. Thank you, Rowan. Enjoy the rest of your day. I'll see you tomorrow. God bless.


Following their escape from a band of half-human yellow-skinned savages, Tarzan and his friends take to an underground river in a dugout canoe which carries them beyond a chain of mountains. Because of a waterfall, the party is forced to land and abandon their dugout. While seeking a means of reaching the river below the falls, they are ambushed and captured by a second party of yellow giants. Mounted on elephants, the entire company is transported to a distant walled city in the depths of the jungle. Upon their arrival, the eight man and his friends are left in a strongly guarded building while Mungo, leader of the yellow men, goes to report to Artea, queen of the giant savages. During his absence, the captives are discussing the possibility of escape when the door opens and the giant Mungo re-enters with instructions to conduct the captives before the queen. We are ready, Mungo. Lead the way. You come. What the devil? This ain't the door we came in. Careful of pitfalls, Tarzan. It looks already enough for us. Nearly a stone stairway. Probably need to be rolled. To the top of the wall. This house is built against the outer wall. At least we'll have a good view of this very interesting city, my friends. Couldn't we tackle Mungo, Tarzan, and make a try at getting away? It's dark enough to try scaling the outside of the wall once in the jungle. We would be retaken almost immediately, Mr. O'Rourke, and probably killed out of hand. Yes, probably. Handling Mungo would not be so difficult, O'Rourke, where he alone. Look over there beside that watchtower. A dozen or more guards are waiting for us. Yes, and besides, Terry, by the top of this wall must be all of thirty feet from the ground. Oh, well, it was just a thought anyway. If we leave things to Tarzan, Mr. O'Rourke, our chances for escape will be far better. It is unfortunate that we could not have dined before this audience with her savage Majesty. One dies so much better on a full stomach. Oh, Dr. Wong, how can you joke at a time like this? Dreadful one will probably have us all murdered, sent before entertainment. With the inner man well-satisfied my child, that should not be difficult to meet. However, it was a mere figure of speech. I should not worry too much about his queen. Mme. Mme. Moiselle, pod 2. I have been in much worse situation with Tarzan, and we have always won through. As you Americans say, let us not cross the bridge arm till we reach it, n'est-ce pas? Speaking of bridges, we seem to be crossing a sort of causeway, a stone bridge leading to that central building. We didn't see this from the edge of the valley, or even when we came into the city. It's on the opposite side from which we came in, behind that high-round tower. What sort of building do you suppose that is, the tower and the center? Couldn't have been built by these people. No, by any others, Jeanette. It is an excellent example of nature's own work. What it represents to these savages would be difficult to surmise. Well, we'll probably find out very soon. We seem to be going there, or at least to the building surrounding it. The Palace of the Queen, I suppose? I think so. I don't know what else it could be unless there are dungeons for us. Imagine. All this in the very heart of what we call darkest Africa. If it weren't for all those half-clothed, gigantic yellow-skinned people, one might very easily imagine oneself in one of the older European cities. There are more things in heaven and Africa than are dreamed of in white man's philosophy, my friend. That is very true, Monsieur Doctor. Very true. But this queen, Atea, we are about to intervene. As rural or such a place as this, she cannot be entirely as survived. Exactly my thought, Lieutenant. I believe we are to be very greatly surprised. I wonder what sort of woman we'll find. Probably be some huge yellow-skinned Amazon, like the rest of these women. I'm thinking, of course, that your question will be answered immediately. We're coming to the Palace. Or whatever this pile of rock represents. Oh, Pucquamungo! La-me-ge-tuc-tu, Atea! Joe Wong, look at those gates on chains. The place is almost impregnable. That way, sir, that is true. Against anything but modern siege guns. Not far now to Atea. You go in. Darno, does this place remind you of another place you've seen not so long ago? These wide corridors, these high ceilings. You mean the temple of Achea? Yeah. There isn't much difference between that and this. Where those corridors were gloomy and dark, these are well-lighted. And there are guards all over this place. And you see the pictures painted on the walls. Hunting scenes. Battle scenes. Just like those in Achea. Yet, there is a difference, mon ami. The Hésiharians were descended from the Egyptians. These people are... well-bendure-lone-no-what. As the rock says, Darno, we'll find that out soon enough. Those doors we're coming to at the end of the corridor. The guards have opened them. Arriving at the open doorway, the guards surrounding the captives halt. Led only by Mungo, Tarsan and his friends enter a large, brightly-lighted chamber. Crossing it, they approach a low platform. Seated in a sort of throne on the platform is the glamorous, veiled figure of a woman. The right and left of the dais stand several giants dressed in short, leopard-skin tunics. At the waist of each hangs a long, double-edged sword. In silence, Mungo and his captives approach the foot of the dais where they halt. The seated woman rises slowly, lifts the long veil covering her entire form and drops it at her feet. A little gasp of stunned surprise from each member of the white group as they stand and stare in amazement at the regally magnificent figure before them. The heavy veil, as it falls to the ground, discloses a form covered with a clinging gauze like robe of white, cut to reveal arms, shoulder, throat and breast of soft, ivory-tinted skin. A rope of diamonds encircles a slim weight. On small feet are sandals fastened with studs of gold. The woman's eyes are large and deep-set, dark and shadowy. A broad and noble brow, on which hair, blue-black as a raven's wing, grows low. The face is delicate with straight features. Surpassingly beautiful as Artea is, her face has stamped upon it a look of unutterable experience of deep grief, of evil passions. For a long, silent moment she stands, gazing thoughtfully, first at one, then the other of the white group. Presently, in a voice soft and alluringly husky, she addresses them. How come you strangers to the place of the yellow men? And what do you want here? How is it that you hold your lives so worthless as to place them in the hands of Artea, queen of Thor? You want to know how we came? By Jove, did you hear that one? She said, Thor, is it possible that we've been brought to... To the city we were seeking? Not only is it possible, Major, but very, very probable. You say I ought to know how you came? Why? Because we were brought here by Mungo. Ask him. I do not know. I ask the question of you, stranger. And I repeat, I do not know. Who are you? I am Tarzan of the apes. Tarzan of the apes. I do not understand.  I am king of the tribe of Kerchak. I have never heard of such a tribe.  And who are these others? Friends under my protection. Under your protection, yourself. You believe yourself able to protect them, as well as yourself from Athea? Why not? I shall answer that question later. Perhaps. The woman there, is she your mate Tarzan? No. Why do you lie to Athea? I do not lie. There is no reason for it. Speak, woman. Who are you? I am Jeanette Burton. You are Tarzan's mate. Speak the truth, lest you regret it. Tarzan answered that question, Athea. However, no, I am not. You are not without some beauty or face and form, and this Tarzan is a man. But no matter. You there with the narrow expression, Athea, who are you? I, gracious queen, am called Wang Tai. I am a doctor of science. Of what country are you? You do not resemble these others? I come from a country called China, Oatoya. I have heard of you. It is a great distance from Thor. What are you seeking here? My colleagues and I, pardon, Major Burton Ashley, Taran Thorok and Jeanette Burton, we were searching for the same city, Thor, when we became lost in the jungle. We were found by Tarzan and the Tarandano, but we were under the impression that Thor was a city of ancient ruins. Anciently there is Wang Tai, ruins? Thor has stood, as you see it now, people by a strong race of men for many generations. My house has governed here for a thousand years. I am the last of my line. When I die, if without an heir, the next ruler will be one of these yellow men. And how is it, old queen, that you speak English? The rulers of Thor have always spoken English. Many of my subjects speak it, as does Mungo, chief of my warriors, but enough for the present. Mungo! Wait a minute, Artea. What are you going to do with it? You will be quartered here in the palace until I have decided what I should do with you. Decide now. You are anxious to leave Thor. Who comes? One chief. Come in, Thongo. He comes from Keyes to Thokatea. Let him enter. Thongo, Thongo, Thongo. Harry Tarzan. One of the clawfoot devils. Yes, listen. Thuk-tu-wa-kambu-erok. Kambu-nuk-wa-tongo. Mumbu-tuk-wa-abonga-ana-luk. Look, Uncle Tim. Look, he's pointing at Tarzan and you and Terry. Just tell him, my dear. And this man, Thongo, chief of my outer guard, says that you, Tarzan of the 8th, and these others, kill many of his men. Is it true? Yes. According to the laws of Thor, a chief has the right to avenge the death of any of his men. In personal combat with him who has slain them. I cannot change the law, Tarzan. Thongo claims that right of you. When is it to be and where? Here, Tarzan, and now. I had hoped to keep you here with me.  He is a mighty warrior. I am not worried about him. Let him take care of himself. Thongo, don't. Don't. Step back with Jeanette. The rest of you, back to the wall. You have a weapon, Tarzan of the 8th? Yes. Then defend yourself. And may the gods of Thongo protect you. Aruk, Thongo! With a howl of rage, the yellow giant grasps his heavy cudgel and both hands swing at high above


Thank you, thanks for the, thank you for the invitations with my great pleasure to speak here, and today I will be introducing work from my two projects that are joined with Victoria Cantaro-Fafan, Alina Mantovan, Rachel Price, and Yunqing Town, even though there are two projects, but this really started the same problem from two different aspects. So let's start from the concept of ordinary and super singular elliptic curves. Let's go back to our first graduate course on elliptic curves. We were looking at an elliptic curve defined over a finite field. FQ, right? We can even like when we think about the characteristics, not two or three, we can even very concretely just write the representative curve with a short well-stressed equation given by y squared equals to x cubed plus ax plus b, where a, b in the coefficients are in the finite field, FQ. Then we can count the number of FQ points, and this number is given by one plus cubed minus a integer a, and this a is called the trace of Frobenius, is the trace of Frobenius action on the elliptic model of the curve. And then we learned a concept, a thing that we learned a concept that's called super singular, which is a phenomenon that only happens in this positive characteristic case. So the elliptic curve is called super singular if the characteristic p divides this trace of Frobenius, and it's ordinary if p does not divide a. So then in the case of because of the Hasselbaum, we know that this a is bounded between, it's bounded, the absolute value of a is bounded by twice square root of q. So in this case, we know that, well, if the elliptic curve is defined over a prime field at p with p greater than two, then this divisibility and the Hasselbaum together tells you that this elliptic curve is only super singular if and only this trace is equal to zero. So meaning that it exactly has one plus two many points. And then we also learned that if it could curve is super singular thing to bear as many special properties, for example, it has a much bigger endomorphism algebra. And then it had it behaves also like now we know that it's the number of actual points is given by this special value. It has all kinds of special properties, making making the very interest topic of study. On the other hand, then the question, the essential question of today is how many, how many of them are super singular versus how many of them are ordinary. So, since super single as I said have all these very, very special properties, it is, it is makes sense that to us to expect that maybe they're less of them, right. They're really like some some some sort of special ports. So, well, one way to, one way to think about how many of the curves are super singular versus ordinary, we need to put it to curves in a family. So the first way to the first perspective will take is, if I consider all the curves over at p bar, and vary my parameter AB really just in fp bar, and then, but then we get, then we can ask a number of asomorphism classes of super singular the curve versus ordinary to the curve. Right. And we know that over algebraically closed field, if the curves are parametrized by j invariant, right, I can literally just know how many as if I want parametrized as a multiple copies of the curves over algebraically closed field. I just need to pick a j invariant in fp bar, right, so there are usually many of them. So, then, in this case, we see that there are only finitely many among all this you can see many j invariant p bar, only fantasy of them, give you a super singular the curve. And we actually know that on the number of isomorphism classes over p bar is roughly, it's roughly p over 12. It's basically linear in p. And this is this is one way that we feel that there are less super singular super similar to rare phenomenon because, because over the whole modular curve or p bar. There is a open dense locus of repeat curves which are ordinary, and there are only a finite many sets of them are super singular. This is one way to think to ask the question how many to cover super singular how many are ordinary right by literally parameterizing all the curves or p bar. There is another very natural perspective, namely, I can also take anything curve over a global field, and then vary the set of primes is global field, consider the reduction of the curve at these primes. And ask that, at how many primes my reduction are ordinary, and how many primes reduction are super singular. So this is really a very characteristic kind of phenomenon if you're taking the global field to be a number, which is a perspective of taking today. So, let's fix it if you could be divided over a number filled L, then I can consider the set of primes. I'll call them q in L in the sense like you can think about fq as the rescue field for this prime that lies above P, a rational prime P which is a characteristic of your rescue field. And let's consider the places where he has good reduction, which is just all but fancy many places of L. Then, um, then when I when I take the reduction I get to the curve over the residue field, which is a finite field, then we can use the definition ordinary and super singular. I will say that he has already action or at Q, if, if the reduction is literally ordinary to curve or the residue field, and I'll also refer to q at the ordinary prime for same simple super singular. Then when I vary this set of primes. Again, it's a very natural question to ask that, what's the density of the ordinary versus super singular primes, if I take one of the curve or global field and it's a curve, right, and I can see all the primes of this. This number field and constant reduction, how many of them are giving me a super single reduction versus all reduction. Well, let me first tell you one case where we have a complete answer by a complete answer. I mean, not only we know the density will, if you give me a prime and literally can tell you whether the reduction is ordinary or super singular. That's the case when the curve global curve has complex multiplication. So by the work of Shimura Tanyama. If I have a difficult e defined over a number of L, and it has CM by an order in the quadrat imaginary quadratic field q or join squared minus D. Okay, so I have already drawn the field here. So my difficult e is defined over L, right. So if I have a prime q, that's the priming L that lies above the rational prime P. And I want to ask, what kind of reduction, let's assume that that the Q is a good prime of good action for E. Then I want to ask whether my reduction is ordinary or super singular. All I need to do is to look at the behavior P in this quote imaginary quadratic extension, q to q or join squared minus D. So what happens is that, is that P, so P splits in this, in this quadratic extension, if I'm the only if e, e reduction, I'll see more P reduce at P is ordinary. So this tells me that the, so this tells me that whether whether the curve has a, has a ordinary or super singular action at this prime q in L is just given by a Legendre symbol, which is just a minus D over P. And if you think about the other, let's take the other prime or something by quadratic reciprocity. It really gives me a congruent condition, right, in terms of P. So all I need to care about essentially roughly, it just the congruent condition P mod D. Okay, so this tells us that in this case and I said it's a complete answer. Not only I know the density of ordinary primes. I also know, I also know exactly which primes I have ordinary versus super singular reduction, right here. Since the congruent condition. So we know that, we know that the, for, yeah, so so we have explicitly write out the density, okay, by chapter of density theorem. Okay, great. So then how about the non-CM case, which is a more generic behavior for the curve defined over L, right. Okay, so then it is a, it is a theorem there, saying that for any particular E defined over a number filled L, it said ordinary primes always have density one or one half. So one half case. So you really should think about this as ordinary prime has density one, because the one half case only comes from the CM picture that we had. So imagine you're basically the point is if your base field was q, if the curve was defined over q, right, then you should think that half of the primes splitting this extension, and the half of the prime is inert in this extension. So I get a half. And also you can see that if I base change, if my base field contains squared minus D, then when I count primes by norm, I only see the speed primes. So, so, so really, you should think about it as if you have a unique curve E defined over a global number filled L, roughly the set of ordinary primes has density one. Okay, so, so this is one side of the question is, so now we know when we ask how many primes give me order reduction versus single reduction, right. Now we know that density one set of primes, especially for non-CM curve, density one set of primes give me order reduction. So it's so natural to ask how about the super singular, how many primes give me super singular reduction, what's the complimentary set, right. Um, well. So in the, if you think about the modular picture in the first slide, when we think about all the curves depend over fp bar, we know that also almost all the primes, give our ordinary, almost all, sorry, almost all the curves depend of p bar. And then the compliment set is, is finite, it's a finite set, right, but in here, the situation is different. It was, it was in our case thesis, that he proved that for, for lots of number filled L, especially, in one, one special case is when L at the miss at least one row in bedding, then for anything curve defined over L, there, there are usually many primes, where the reduction is super singular. So in this reduction perspective, right, you see that you have a density one set where you have all reduction, but then the super singular set turns out to be infinite compared to the over p bar from the modular picture, where you have an open dense set of gene variants being ordinary versus finally many being super singular. Okay, so, um, well, the takeaway this slide is that if you take a non-seminity curve, which again the generic case or the occurs. Over, let's just take it for simplicity, let's just take it over a basic of q, right, for any non-seminity curve e over q. And then it has all reduction as a density one set of primes, and then had super singular action also have a usually many friends. So this is the takeaway of this slide. And then the goal of my talk is to journalize the work of Sarah and Elkis to a billion varieties of high dimensional a billion varieties. So as an example, I will be talking about the decoding of a curve with the following defining equation as y to the fifth equals x times x minus one times x minus t. So take the smooth projective curve with this often model. And as you can see that this, this curve has generous for. So we will be starting a billion fourfolds. Okay. Okay. So first of all, let me tell you the generalization of Sarah sword in this case. Before, before I tell you just the work we did, let me first tell you the generalization of the notion of ordinary and super singular in the higher dimension of a variety case to set up the problem. Right. Okay. So, well, the generalization of order versus super singular, this look has become much more complicated when the dimension of your ability variety gets gets higher. So let me produce a notion of Newton polygon. So you can pull it out is a notion that you might have already seen in, in a class of algebraic number theory, where you have a polynomial with integer coefficients, and then you can choose a prime P, and, and, and look at the points with. And the draw the points of the, the, the, the, the indices being I which is the, which is the label of the index of the coefficient, and then also the periodic evaluation of the corresponding coefficient, and then draw the lower convex call draw a polygon out of it. So you learned that okay this tells you how this polynomial factors over QP right over the completion, and, and exactly the slope, the slope of this Newton polygon tells you the periodic evaluation of the roots of this polynomial. And very often you can use that to prove that your polynomial is a usable over Q, right. And, and that's it's really generalization of I sense that's criteria, right. So in here, when we see Newton polygon for a billion variety, we mean that if I have a billion variety a defend again or a finance field, then I can again look at the Curtis polynomial for Venus. Acting on the added model of a, and it draws the Newton polygon of this, of this integral polynomial. So, what we do is we will, we will, we will normalize this polynomial to be to be Monic. And also, I will index it in the opposite way meaning that the Curtis polynomial will really be a zero x to the 2g plus a one x to the 2g minus one plus data, and it is symmetric right so the last terms are a one q to the g minus one x plus q to the right which, sorry. Yeah, so it is it is so so so so so if you look at the coefficients right this is my a of 2g, this is my a of 2g minus one, this is symmetric. Okay, so we can draw the Newton polygon of this polynomial. And then because we know the leading coefficient is always one, and then the constant term is always q to the q to the g. What will happen is your Newton polygon will always all the polygons for the operating variety with this fixed dimension will have the same starting point and ending point right. And you can compare whether two Newton polygons lie above each other or not. And, and, and this whether one lies above each other not gives the set of Newton polygons with this fixed g a process structure. What is the work of art, saying that this post that structure turns out to give a stratification of the modular space of principle a polarized operating variety over at people are just like the modular curve case, we've seen that there are two locus is right for you to curve for D equals one one locus is the audio locus which is open sense, and then the super singular locus has a co dimensional one right it's stratification. And let's look at another example, which is, if I have genius to, and I consider a to consider the concept of building surfaces for people are then in this case well first of all the dimension of the modular space is is three, right, and I'm not drawing a three dimensional case so I'm drawing the two dimensional case but I want to remind you that the dimension of a two is three. Okay. So this is what we mean by a stratification. It means that there is an open dense three dimensional locus that is ordinary. Okay, and inside this locus, then there is a one, there is a co dimension one sub locus, which is what we call almost ordinary it has p rank one. And then inside this almost ordinary locus, there is a co dimension to sub locus, which are the super singular up in the surfaces. So this is what we mean by a stratification of a D. Okay, and in this case, again, as you can see that as a generalization with the curve case. So ordinary locus is always the generic locus for for general for general up in a variety inside a G. Okay. And the and it is ordinary if exactly when half of the Frobenius eigenvalues have p added value or q added evaluation zero, and the other half has q added evaluation one so it's the the Newton polygon for the big curve alternative curve is have once look zero and once look one. And then the ability of variety being super singular super singular is always the highest co dimension locus in a G, and it's super singular if all the Frobenius eigenvalues have q added evaluation or half. So if a curve case is it you can also say that say that if that they have both of the problem is eigenvalues will have q added evaluation half. So we're using q added evaluation to normalize it so that the Newton polygon will be a geometrically invariant. So if I enlarge my base field, right, my, my Frobenius will reach the power and by normalizing the slope or the, the, the, the eigenvalue won't have the evaluation the eigenvalue will stay the same. Okay, so that as you can so now this notion of ordinary super singular, but generalizing, which now allows us to ask a similar question, given a higher dimension of any variety defined over a number field, what's the density of ordinary versus super singular primes when I look at the reduction just like the curve case. Right. And the one thing I wanted to note is that in this case, as you can see trace of Frobenius is not enough to determine whether the opening variety is ordinary or not. Unlike the big curve case, right, you actually need to know, like, if you want to know the Newton polygon right really need to know the, the, the, all the valuations of all the eigenvalues really. Okay. So, what, what, what do we expect, what also do we expect from this question. Well, it is a conjecture that was raised by Sarah, saying that for any opinion variety a of dimension G over a number of field L the density of ordinary primes should always be positive. Okay, this is completely general is any opinion variety over any number field. And so that's why we say that it's an ordinary prime. And so that's why it's not an ordinary primes. It's just positive instead of saying it's going to be one. Right. So, if you think about the curve case right as I said for the sea out in the curve, you may have the 31 half but some of the people in politics should be, should be what we should expect always. Well, what do we know about this kind of sure. So first of all cats showed that this kind of sure holds for for our being surfaces. And then later on we are starting with able to explicitly analyze the density of the ordinary primes. So the set of ordinary primes has been to one or one half or or one quarter, and the one half and one quarter case also comes from a building surface with extra endomorphism just like the similar to see a particular case. So there are other more other results in this topic, and I just pick some examples. So one example is that pink showed that this connection holds for certain ability and varieties a with trivial endomorphism, but a particularly small monfortate group by small I really showed the generic monfortate group. And then also very recently, fatigue proved that for also prove this connection for a for a families of a billion varieties, whose endomorphism now the endomorphism is not true anymore, then the morphism will contain some imaginary product field, and then there are some other conditions for for a to satisfy. In this case he was also able to show that the set of ordinary primes has positive density. Okay, so now I'm ready to tell you our work. So join to his contour of a fan mantle when present on this is our current work ongoing work, which hopefully will finish soon. We were able to show that if, well this case is definitely finished is as we are trying to see how much how generalized our result can go. Okay, so for currency. With the current with the following defining equation what's the fifth equals x times x minus one times x minus t defined over in number field L. Let's consider generic is which is when the Jacobian of state does not have CM by CM I really mean so this curve is a generous for. As I said so it's up in Jacobian is up in a full fold by CM here I really mean it does not have CM by a CM field of degree eight a big CM. Okay, if I if I consider that case, then the set of ordinary primes for set of ordinary prime type of positive density. And if I have base change to L join the fifth to unity, or if it is already in your base field L, then this density will be one. Okay, so this is so this is the case for for this particular family we're starting. So I want to make a comment that note that in this. So, for a curve this in this family right over, over, over L joined into five or q joined into five. What happens is this curve has an automorphism, very concretely by mapping x y to x, comma, Zeta five why right, because one way you can view this curve is over over the field L join Zeta five. This is a degree five cyclic cover of P one, and there is this cyclic cover this this degree five cyclic group acting on the curve right, we will model that out to get P one. Okay, so then this automorphism induces a endomorphism of a decoding. And in particular, when the, when the decoding does not have CM really, or for generic member of this family of the opinion varieties. The endomorphism exactly is cured on top of it. And our work, and we expect our result to be for opinion varieties with with this essentially roughly this type of endomorphisms. So for example, one case we currently know how to do is, is that if the endomorphism is just like this case equals to a CM field of degree D, which in this case because this is the degree for CM field. So this is, so I just want to say that our, our general result will be for opinion varieties with certain types of endomorphism. Okay. And another comment I want to make is that as I said, this theorem is about a generic member of my family, mean that Jacobin does not have complex modification. When Jacobin does have complex modification, we can also very explicitly compute the set of ordinary primes using the Schmura-Tanyama formula just like the decode case and give a complete answer like the decode case. Okay, so really the non-CM case is the more difficult and more interesting cases and also generic case. Okay, so then let me tell you a little bit about the Newton polygon of this family and also the, yeah, and why, why this is a family that we chose to study. Okay, so if you consider the family of Jacobins for curves or Q bar, defined by this defining equation, right, then at a prime that is not equal to five, then the reduction is a, so then the reduction is a billion fourfold corresponding to a point in, in the, in, in A4, Fp bar. So note that, so I have a, I have a generous four curve. Of course, Jacobin is a principle formula as a billion fourfold, right. And if I look at all the possible Newton polygons with G equals to four, you can, you can draw them yourself. And there are, there are eight possible Newton polygons, so there are eight Newton stratos. So this is, this is why in general, starting the Newton strata of, of a billion fourfold is difficult because there are so many of them. But, but this is not any opinion variety, right, as I said, the, this opinion variety at the mid endomorphism of Q or joint data five. So because of the actual endomorphism, actually when you look at the Torelli image of, of, of these family curves. So it really like this image, it lies in a PL type Schmura sub variety of A4. And this sub variety S actually is one dimensional. This is really just a Schmura curve looks very similar to a, to the modular curve like the curve case. Okay, so the model of this family looks very similar to the curve. And it is a, yeah. So this, this follows the work of the limostro and the moanen. And then, following the work of Robert pot caught with research, or what, where the home and the, we know that the Newton strata for for you need to reach more variety. So in this case that as I said our S looks like very similar to Jane J line right, and it turns out that for any prime P that is not five. If I consider as a VIP bar. Again, just like the modular curve case. It has two Newton strata. One of them is open events, which we will call new ordinary so you can think it as a similarly to the ordinary locus in the J line of the curve or a P bar. And then there is a closed locus with fancy many points, which we will call it the basic locus. Again, it looks very similar to the super singular locus for the J line recurve, but note that even though this picture, this picture I'm drawing does not depend on P. For any P you have two locus, one of them is one of them is open sense new ordinary while with them is fancy many points basic for any P. But when you change P. So corresponding Newton polygon may change even though that all two locuses, but what's Newton polygon you should look as corresponds, or more concretely, what's the periodic valuation of the problem is again values are for these two locuses will change. Exactly depends on the congruent condition modified. Okay, so when P is one of the five, the new ordinary polygon is ordinary, and the basic polygon has p rank two. And when P is two, three or four more five, then what happens is the new ordinary polygon is not ordinary anymore. And the basic polygon has has p rank zero it is super singular in the actually I mean this is in this case. Okay, so I just want to emphasize that, even though for any P the pictures are the same, but what's Newton polygon, these points corresponds will be different. Okay, so now, after analyzing the modular. So as I said, like, so if you are starting a general operating four fold, there are eight possible you can spread us right, but then now in this case, I know that actually my operating four fold, when I reduce and each fixed P. There are only two possibilities. Okay, so this makes the problem slightly easier to study. Okay, so now let me tell you the general strategy. Not only for for our project, it is a general strategy on how to, how to, how to do these problems, like how to study the sensitive ordinary primes and generalize stairs, stairs work. Okay. Well, so first of all, let's talk about a observation is a very simple observation. Since we are only considering the density of primes, right, where I count primes by norm, or I can say other sweet price, why when I say other primes, who's resting field of prime. So this means that if I think about reduction and I want to start the density, or all I need to think about is on the sweet primes meaning that locally, or at the reduction, or I need to think about are the operating varieties defined over prime field. Okay, which is a simple observation by health. Okay, so now, let's think about it if you curve, if I know let's think about at what times P, my little curve of the maze or reduction. So as we already discussed that if my curve is defined over a prime field. It is ordinary. If I'm the only if, if I'm the only if my tracer for Venus is not a zero. Okay, so because coming from the coming from the hustle bomb. Okay, so this means that my prime my face build is prime. Okay. So this means that this tracer for Venus a. Okay, is a invariant of Frobenius, such that, first of all, it asserts whether if the curve is ordinary or not. Okay, I have one invariant that tells me whether my if you curve is ordinary or not. So second of all, when it's not ordinary. It takes a finitely many integral values, and this integral values are independent of P. So exactly a equals in this case is exactly a equals to zero. Okay, so when that if I have a invariant that tells me whether he is or not. So in particular, when it's not ordinary, it takes a value zero, whichever P it is, it takes a value zero, okay, the integral value that is independent P. And the same thing happens on the surface. Okay, so if you look at the characters for the more Frobenius property surface, as I said, it's symmetric over P is symmetric. It is ordinary. If and only if my middle coefficient, if you think about it in polygon is even only if my middle coefficient is a periodic unit. Okay, it's ordinary if P does not divide a two. So, then, again, if you use Hassabon, you see that a two is bounded by six, six times P, which means that if P is not ordinary, I can start the invariant a two divided by P. Okay. So if it is ordinary, then this invariant a to my over P has to be in this set of finite integral values that is independent of P. Okay, so for opening surface, there's also this very natural natural invariant. Which is a two over P. That also exerts when your ability surface is ordinary, and also takes finitely many integral values for nonordinary services that's independent of P. Okay, so this is the first step for doing this kind of problem is you want you want such invariant. Okay, because this invariant really gives you a function on the attic monotony group, such that the density of ordinary primes is bounded below by the ratio of the connect components of the attic model group, which is function is non constant because because this function turns out that it was on his proof is function actually is only constant when your value is interval. Okay, so by chapter of the city so this will already this will give you a, um, this will give you that they are, they are positive density set of primes, which have already reduction, as long as you could prove that there is at least one kind of component where your function takes a non constant value. Okay, so the first, the takeaway is the first step is to find one such invariant for the ability of writing you want to study. Okay, then the second step is that I said is now you need to, you need to study on the attic monotony group. So what value stuff might be very take. Well, so, which means you need to start the connect components also started to collect components of attic monotony. So for the curve case, this follows from stairs open image theorem. And then in the, in the obedience surface case, Southern, that he was able to explicitly compute the ratio relies on the work of, of the take a lot of Roger Sutherland on the definition of a subtle take groups for our building surfaces. Okay, so this information allows them to explicitly compute the density of ordinary primes for the curves and our building surfaces. Okay, so for our case, because of this cure join data five action, so this is this data five action. So, we were also able to define functions on Frobenius, based on, based on is the image in our war. So, based on whether which Congress class of mod five, the prime has, and then the value of these functions also exactly have the property that we want. It does this when my, when my ability of writing has a new order action. And also, when it's not to me or it takes a bounded integral value of independent OP. And also then the second step, we use the fact that we use the fact that we know, over which field that I think monotony is connected, and the vassals work on them offer a texture for our case. So really, in our case, roughly the analytic monotony group is not that different from the unit for the for the for the for the for the Shimura curve we have. So this is why we're going to conclude really that so our actual results will be that the status of a new order in France for any, for any Jacobi in this family has been to one. And then you recall that we said, ordinary prime has positive density. Well, because, as I already told you that when P is one more five, the new ordinary locus is ordinary. Right. So this means that if my new order has 31, then at the positive density, I am ordinary. And on, moreover, if I can't prime by norm, that if my base field contains data five, then all I can see other prime was characteristic or one more five. And this is why, when the, when data five is in your base field, the density of ordinary prime becomes one. Okay, so this is how you show that there is a positive density of friends that bears ordinary or new order reduction for a bigger variety that you're studying. Okay, so then the next, let's recall our case work started the super singular that the complementary set. Right. Okay, so I'll recall that our case shows that if you have any difficult if you find the work you, or, or more generally a large set of number of fields. There is this infinitely many primes at which the reduction is super singular. And then there are some analogous results for a billion surfaces with Quaternion multiplication. And so these are all of the surfaces case. And you, and now I want to talk about our family curves. These are again up in a full fold. Okay, so, but instead of working on any of them. Now we need more conditions. So let's take C to be a smooth product curve with a defining equation y to the five equals x times x minus one times x minus t. Okay, then I'm going to define a rational function called a J. That is a rational function in key. And I asked this J value to be in Q, and it is between zero over and the 27 or four, I will explain, I will explain what this, these conditions mean in the next slide. And this, so you can think of this as a condition on the Archimedean place for, for my family for my curve. And then, as the prime five I also have a special condition, I asked the reduction of CF file to be singular to be a better doctor. So again notes that this is a family of generous four curve. And my singular fiber actually looks like the intersection of two generous two curves at intersect as a node. So this is what my singular fiber looks like. Okay, I want to reduction of CF five to look like the singular five. So if these two conditions are satisfied, then we were able to show that there exists infinitely many primes on where Jacobian C or the miss basic reduction. So recall that we have previously we proved that at the density one set of primes Jacobian C or the miss order reduction. Right. So now we're we're generating our case in the sense that we are proving that is basically zero set of primes is infinite. So, let me let me next slide let me first explain to you what these conditions really mean. Okay. So, well, what I have this family of curves, some of you may already start to think that this is a defining equation looks very similar to the LeJongere equation for the three curves, where you have y square equals to x times x minus one times x minus lambda, where I can parameterize magic curve law by lambda. Right. But on the other hand, we also know that the information of lambda really gives you a two torsion point the magic curve. So lambda is not just parametrizing the isomorphism classic curve that the actual level structure. So same thing happened here for tea. Of course I can parametrize my curve by varying tea, but then by varying tea is not a one to one correspondence to the isomorphism classes of the curve in a family. So, just like the lambda to J map, which is a six to one map, I can just model this ambiguity coming from tea, just by defining what we call the generalized J function, which is this degree six rational function in terms of tea. And then just in just the same way as a correspondence between lambda and J, this is even the same function because you are modding out with the six, the six possible lambdas or possible teeth corresponds to a J. Okay, so then this J will uniquely determine the isomorphism class over Q bar and FP bar for my curve or for Jacob Jacobin for any P that is not equal to five. So, so J is a parameter for the course modular space S, which we already, we already showed up in the previous, in the previous slides of the family. And from this, this parameter we actually obtain a Q model for the for the Schmura curve, even though that's the reflex field for the Schmura curve is Q drawn data five. Okay, but it actually has a Q model, which is actually very important crucial for our work. So the theorem, when we ask this generalized J value to be in Q, it means that the field of modularity is Q. So you can, that's why you can think it as we are really generalizing LK theorem for the curve divider were Q. Okay, so, and then, well, what happens for five. Okay, apart away from any prime away from the prime five, the Schmura curve S has good reduction, and we've also already discussed that. In this case, if I consider S over P bar, then there are two Newton stratus, one of them is new order and one of them is basic. But as the prime five, S has a bad reduction, and there is only one Newton locals there and it is for singular, meaning that if you take any of your variety in my family and reduce more five, it'll be super singular. And this is essentially why we will need to impose the condition at five, I asked my seat to be a bad reduction at five. So, the condition of bad reduction can be replaced by some other condition, but in our proof, because of this fact of the five, we will need some condition five for our C, which we all explained in the proof, why we will need some condition. But here I can tell you that the reason we need a condition at five is exactly because the modular, the Schmura curve has a bad reduction at five. Okay, so one thing I really like about these projects is that I give you one a billion variety, like a handy one a billion variety defined over a number of years, and I want to start this reduction at different price. And it turns out that instead of starting this one a billion variety, I'm really I need to know which family is a billion variety behaves. And then I need to know properties of the modular space that this ability that parameterized the family was a billion right it is like this ability of writing even though it just one single I've been right it remembers which family belongs. Okay. So, and this is a prospecting to pick. Okay, so now I explain to you what the means for j to be in queue, and, and essentially where why we would need a condition five. Let me explain the weird, the weird interval zero and 27 or four where that comes from. Well, to do that, we need to think about the, let me tell you a little bit again more about the family where it's a very variety belongs. Let's think about the complex points of the Schmura curve. This S over C is a compact in a curve, as I said with reflex your cure is a five more over it. It is a, it is a triangle modular curve for the triangle group delta 2310. Okay, so it looks very similar to the, to the modular curve where we know that the modular curve X one is H as upper half plane mod S or to the right here, our curve over C is given by upper half plane modulo this triangle group. Then the fundamental domain of my S is two copies of a hyperbolic triangle which I draw here. So, so because our S is really just a P one right it's paralyzed by J this one function so it's a P one. And you can think that you take two triangles, a thing that looks the same and the glues them together along the side this gives you a sphere, which is a complex P one right. Okay, so what does the fundamental triangle looks like. Well, the fundamental triangle it has three vertices. Each, each corresponds to a curve or I've been a variety in my family with extra automorphism. Okay, so the vertex P correspond to a degenerate curve as I said it is two copies of a generous two curve. But note that my Schmura curve is actually compact. Okay, so because the generator coping for this genius for for this curve is still a billion fourfold. So, my, so, so this is not a cost. I want to emphasize this is not a cost. Okay, even though the curve it corresponds is singular but the ability in variety is good actually. It's good. Okay, so it corresponds to actually smooth up in a variety, not a semi up in a variety. Okay, so. So the point is because it's too coffee genius to curve extra extra automorphism coming from it I can take automorphism or one copy and fix the other copy. So that other smooth curves don't have this automorphism. And then there's a point q with a extra automorphism of order three with J value zero. And then there's another point are with extra automorphism of order two, with J value 27 or four. So you can think of this as very similar to the busy curves with CM by by cure to I and the cure to the three. Okay, so then, when I asked the J value for my curve I started to be between zero and the 27 or four means that I'm asking my curve the motive I point to be on this side. between this q and r. Okay, so the sides of this triangle is actually all the real points on my smaller curve. And, and these are geodesics in the upper half plane. Okay, so this condition is restricts our motive point to be on the side qr. And the reason here just a technical reason just in our current work. We know much better. We know more information about this geodesic this side that the other two. So we will, we will hope to work to get rid of this, this, this restriction, this is our current current restriction from our pool. Okay, so this is how I explained the serum. Let me tell you a little bit about the strategy. Again, let's think about the strategy of elk is cool. Because when you when we ignore all the details roughly, that's the same strategy, just like the case for the ordinary right is the same strategy always. So the proof for this for this serum is, is like the proof of there exists evenly many primes, right, how do you prove they're using many primes. Well, I assume there's a finite set of primes, and then I construct a new one. So it's similar here. So the point is another way to visualize it I always think it as like a fishing game, where there's an ocean of primes, and I'm building some fishing nets to catch some primes with a special property. Okay. Okay, so well what's the property do I want is exactly the property that my fix it because this is my fix me. I want to start it. Mod p is in the super singular locus of my motor a curve. Okay, so what are my fishing nets, what my fishing nets will be see Emily curves. The reason is the following. The reason is that I won't start it where he has super singular action, right, let me start with some of the curves where I know exactly where they have super interaction. And as I explained the CM was whether they have ordinary super direction comes from in the jungle symbol that we discussed before. Right. Okay, so things that I know I build some fishing nets which are standing curves. And what are the primes will catch me. Well, it exactly catches a prime where E, and it's super singular, sorry, this CME isomorphic modus p so this exactly when the intersect with the respect in the section right. So, so every CMD curve will catch me some primes by intersecting my E. Okay. So, okay, so I want, I want to catch infinitely many friends where I have super single reaction. This means that I want the prime that catch to be a super singular. So what do I need to do. Well, I need to guarantee that each fishing that will intersect a super singular locus. Okay, okay, every time I saw that this fishing actually doing something, right. But on the other hand, I want to catch infinitely many. Right, this means I need to make sure I have this infinite set of super singular prime, a CMD curve, I need to make sure they don't intersect at the same single point. So what I'm going to do is as I said, I assume the finance that I just need to make sure that the next one. I'm sorry, I just need to make sure that the next one. The next one will not intersect the same super singular prime as the previous one, right, then, then I'm done because I have infinitely many each will each one will catch me at least one super singular prime. And once I've been caught by this net is not the ones that have already been caught. Right. Okay, this is a general strategy, and this is also a general strategy for our work, which I don't have time to discuss. So in the last minute, let me explain one thing, which is the condition at five. Right. So the point here there is, how do I guarantee the new fishing that will not catch me a previous prime. The easiest way is, I want to say that for the order for all the primes that I already know of super singular action, I can construct a CMD curve, which has all the reduction at all the known primes, so that I can guarantee the new one I guess is actually new right. So this would work for any prime that is not at five. Because remember, for all the opening varieties in my family, they all have super singular action at five. So I can't, I can't use a Newton's strata to guarantee that the CM curve I construct, not intersect my original curve at five. So I need to use some other discrete invariant to tell me to guarantee that even though I have infinitely many fishing that they are not just all catching five. Right. So the current condition we impose is smooth versus singular. But we will also work on trying to, to use other discrete conditions instead of that one to be able to generalize to make our system more general. But this is behind why we need a condition at five. So I'm running out of time. So I don't have time to tell you the schedule approved. So let me just finish there and thanks for your attention.


I sold this kit for $354. The ears were cut off. Hey BoloBuddies, thanks for watching. Alright, in this video we are going to talk about big money plush Bolo's and also big money items that people bought on whatnot and sold on a platform. So people that sourced on whatnot. And I'm going to put these two videos together because I only have a handful of each on each post left from the other videos that I completed. So this is what the best plush Bolo post look like. And thank you to everybody that shared and I will be doing this again in the future. And here is the post for share your best whatnot Bolo. So we're going to get started here. We're going to start with the plush. And there's some really, really fun Bolo's here. I love plush. And if you guys would like to share, you can find these in the featured section of my BoloBuddies Facebook group. And there is a new format. If you want to learn more about that, you can check out this video or you can just go to the post and I explain it there. It's just a different way. It's a different format, same concept. So this one says I was at the Goodwill and I sell, I see the ugliest Harry plush and I moved it aside to look at something else. Then on my way home, it hit me. This was what Courtney has been talking about, realistic plush. Well, the next morning I had forgotten all about it. Two days later went into the Goodwill and there was that ugly Harry plush. Grabbed it up for $2.99, took it home, combed his hair, then listed on eBay, took about four months to sell. I had several offers during that time, but was holding out for something larger. Accepted $140, buyer paid right away. Woo-hoo! It is a Ditz Design Henhouse Realistic Silver Fox plush. Yes, awesome. Congratulations! I got this snow leopard at the Goodwill Benz half off day. When I got it home, I saw damage. The ears were cut off. The white was dingy and it was missing the tag. But Google Lens helped me find it. I listed in poor condition for $90 and just sold it for $70. The ears were cut off. I just spit. The ears were cut off. Let's do that again without spitting. All right. It is a snow leopard Home Save R Space SOS plush realistic. And yeah, right there. Sold it on Mercari for $70 plus shipping. You guys, Mercari is great for plush. I tell you all the time, if you're not on Mercari, definitely check it out. I have a referral link down below. You'll get $10 to shop when you join and another $20 to shop when you sell $100 worth of stuff. So that's $30 of free shopping money. And then they also give me some money to shop. So thank you when you use my referral link. I really appreciate that. Found at the Goodwill for $2. Listed same day and sold within 10 minutes for full asking price on Mercari. It is a vintage 1989 Thai TY 15 inch Patty Black Panther cat kitty. 15 inch no hang tag guys. $125. I said Thai.  It's not the little beanie baby. This is a 15 inch one jelly cat smudge the elephant bought it for five and sold it for $149. It doesn't say where they sold it, but it's an elephant and it's a jelly cat. And we all know jelly cat, jelly cat, jelly cat is a Bolo summer bread and butter, but some go for crazy money.  It finally happened to me bought this lovey several months ago from Goodwill paid $1.11 for it plus tax sold for asking price of 25. It's a little miracles Costco lovey brown puppy plush security blanket buddy rare hard to find sold on Poshmark. How cute is that? Poshmark is also great for plus and I have a referral link down below where you'll get $10 to shop if you use that bought at an estate sale for five sold within 24 hours for full asking price of $249 plus shipping. Swibco 1998 plush bear sherbert rainbow 17 inch. This is a big money Bolo. There is also one that is bold dark colors that can go for a lot of money as well. If you see the brand Swibco, definitely look it up. I have some smaller plush that are Swibco that are not of a that are not a Bolo. I got excited and I'm like, yes, I know this brand, but the smaller ones do not sell for as much. But I do recommend that you look up anything with this brand because you never know. I got this at the Ben's. So I have less than a dollar into it. It took a I took a best offer of 50 plus shipping. The sound function doesn't even work. That's awesome. This one comes from vintage spice on YouTube. She said jelly cat London wiggly phone plush toy stuffed animal and telephone no sound. And this is what it looks like. I don't know it. Some of those you can add a battery and some of them can't. This might be one that you can't found this realistic plush at the Goodwill paid $3.99. There were none listed. So I went high and put it up for $124.99 sold best offer in one month for $88. It is a vintage plush Avanti Collie pup by applause. Yes, realistic animals. If you see them and their plush, definitely look them up. And if you can get them for the right price, I would pick it up if you don't see any others listed listed high bought a big Christmas lot for 100. So I probably in this for about five bought a bag Christmas lot for 100. So I'm probably in this for five. Okay, so she pulled out particular items and guessing she has about $5 in this lot. The plush sold for $53.95 plus shipping. This one comes from Picker Money Flipper on eBay and YouTube. It's a 2009 Dandy collector's choice Rudolph reindeer plush set from CVS. It says Dasher and Dancer. And it looks like it's some of the other reindeer as well. This one comes from Piece of Scrap. And that is her YouTube and she sold this on Mercari. I paid 25 cents at the Goodwill sold for a full asking price of 21 shipped on Mercari. She said her kids didn't know who lamb chop was but the clerk did. It's a baby lamb chop plush hand puppet from 1990. There it is. Yes, I have sold some lamb chops. Definitely look up lamb chop also. This one is also from Piece of Scrap. She said I paid $2 at the Goodwill sold on an offer for 20 plus shipping on Poshmark. Again, Mercari and Poshmark a lot of these sales came from those platforms. This is what it looks like. I'm not sure who it is or what it is but it sold for 20. All right. Let's go over and look at the what not. I bought it on what not and sold it on a platform Bolo's. This first one says I bought this beautiful Traffari on what not from Lucky Day Shop and a lot of pick three brooches for $9 including shipping. So the brooch cost about $3 sold on eBay for a best offer of $400. It is a Crown Traffari, Alfred Felipe, World War II, Red Blue Enamel, Rhinestone Hummingbird brooch. Wow. So definitely go follow Lucky Day Shop on what not. And this one actually was shared by Catfeel and Thrifty who is also on what not and sells amazing jewelry. I mean some of the vintage jewelry she brings, I'm like, ah, these ladies are both awesome what not sellers. They also help me out moderating and they are great, great ladies. You're going to love them. So I'll post their store names down below, their what not names. I bought this Snapco head base on what not from Bolo Buddies. This one comes from Catfeel and Thrifty and she bought it for me. Now she did pay up a little bit for it. She paid $35.50 and she flipped it on eBay for $68 plus shipping. So she knew that she was going to make a profit and you know sometimes it's like retail arbitrage. You pay up a little bit because you know you can sell it quick. I bought this brooch on what not for $15 including shipping and sold it on eBay for $60 plus shipping. It's a vintage Vondell Wedgewood Jasperware 12 karat gold filled Angel Cameo brooch pendant and she got this one from Donatella on what not. Bought on what not for $10 and sold on eBay offer for $70. Going international as I seem to have a drop shipper working my jewelry listings for forwarding to orient for three months now works for me. So what she's saying is people are she has a drop shipper somebody is buying from her and I don't know if they're like then selling it higher and drop shipping it somewhere. I think that's kind of what she's saying but she's like I don't care because I'm making money if they make a little money that's okay and that's kind of how I feel. I set my price and if I'm willing to take their offer then I don't care who buys it. If they drop ship it and sell it for more than me that's fine. That's kind of what's happening on what not. She bought it for 10 and sold it for 70. So same idea you know we all source different places. Hattie Carnegie mushroom brooch and it's really pretty. Mushrooms seem to do well. Bought from Bolo buddies on what not for $7.48. I thought it was unique and no one was bidding on it. Figured it would at least be a $30 necklace.  Depending on the shell they can go for much more than what mine sold for. I sold mine for a best offer of 82 plus shipping after fees and cost of good. She made a profit of 63.82 and this one comes from seems so vintage. This is what the item looks like and she is seems so vintage on what not. So definitely check her out. It says bought this Polly Pocket Cinderella pumpkin coach on what not for $10. It was missing the Cinderella in prints so I didn't list it as at the highest end of comps and disclose the missing pieces. It sold in four days for a full asking price of $54.99 and the buyer paid shipping and there it is. $19.99 bluebird Polly Pocket. So even incomplete. Hi Courtney I have two items I bought on what not that turned out to be really great. This one comes from basket case Mama Fab Fines on eBay Posh, Mercari and what not. She said be back on what not soon killing the money pile first and she said in October but this post has been up a while so I just said soon so I'm not sure when she's coming back to what not. vintage depression glass pink fluttered no I'm sorry vintage depression glass pink fluted pedestal cake stand mid century and it sold for $40 but I don't know what she paid for it. I just know she got it on what not so she obviously made a profit. Second item was a lot of patches from glt treasures I think they were all military except for two were auto maker ones 10 for two sold three so far this is the best of all though a Chevy USA patch sold for 35 the other two sold for 18 all in less than 45 days a great buy for sure it is a vintage Chevrolet USA one taking charge iron on license plate patch embroidered sold for $30 and this one comes from basket case Mama Fab Fines paid $3 on what not in a storage unit category event had free shipping already sold both in a week she said I sell on what not as vintage chick pottery barn black cast iron curtain drape round double ring large sold it for 68 and the other one is a pottery it says oh it looks like she's selling them in lots of 10 the other one is 10 pottery barn black cast iron curtain drape round double rings sold for $74.90 so she paid three and it looks like she sold one lot for 68 and the other lot for $74.90 Bolo thrifting on what not bought these two takahashi broaches for $15 on what not that when they arrived I didn't really know what I had but they looked special I researched them and priced them at a lower end of comps range because I didn't have often they weren't often to say that word no papers on them there's a fascinating history behind these bird pens sold both within 48 hours for asking prices of 244 so 244 two separate prices there should have priced them higher I suppose but I'm also happy to give someone who will love them a good deal and this one comes from lisi dc who is coming soon to what not so make sure you give her a follow as well again I will put all of these what not names down below and it looks like it is a pheasant hand carved and painted and this is what it looks like and here is the other one it is a another bird brooch hand carved and painted okay this one also comes from seamstow vintage um that's her what not name she said I picked up several needle point canvases on what not from donatella paid $2.75 for each canvas this one was a kit so I grouped the pieces together to sell as a set I averaged this one at $2.75 as well since I counted as one listing I had several lowball offers but declined each because I knew the value I sold this kit for $354 and 53 cents buyer is all in on ebay my profit was $276.18 after fees and cost of goods positive feedback was received it is a so much fun needle point 3d betsy bird all piece canvas needlework and this is what it looks like this one comes from farm girl scavenger noelle and she is farm girl scavenger on what not and she says okay everyone so I purchased a lot of books from bolo buddies that's me on what not for I believe 15 plus shipping I knew what they sold for as I have sold them before and these are huge bolos she spells huge with a Y the one I am showing had um the one I'm showing had two of I had two of and one is still available however there were eight books total on the lot here is the one that sold and it sold for $59.49 all in $68.99 any old books from that new basic reader series and anything Dick and Jane are money think and do book part two the new basic readers Robinson Monroe um and I think I don't know if that's I don't know if that's the Dick and Jane or not I think there was a bunch of Dick and Jane she got eight of them it was a big bundle and she's like these are a bolo and I'm like okay well then buy them anyway here's what it looks like so check out farm girl scavenger noelle for lots of bolos she has a YouTube channel it is incredible if you want to learn about different different categories and she does a let's talk about it and she'll break things down for you guys and teach so check her out on YouTube I will link her down below all right you guys thank you so much for being here thank you for watching be sure to like comment subscribe and share and as always thanks for watching