It takes a big bait to catch a big fish. And that's why in today's video, we're fishing with giant live bait for monster fish. Two months ago, we met this stranger while pond fishing and he told us about this hidden pond that has some double digit bass in it. But we weren't able to land any of those double digit bass. So one month later, we went back to only catch some big mud fish. But that just wasn't enough. We wanted to catch an absolute lake monster. And that's why we're live bait fishing today at a new location in hopes that we can catch the monster we've been waiting for. But the first step to this process is to catch some giant live bait. Fletcher is going to be attempting to catch some live bait with a cast net while your boy Noah is going to stick to a more traditional method with just the rod and a reel. Now let's hop into it. All right boys, we're ready to get this started. We are doing something a little bit different than Fletcher today. We're going to be sticking to what we know using a rod and I'm going to try to catch as much bait as we can. Fletcher has actually had another spot and he's going to be trying to catch some different live bait. So we're going to go ahead and get started, get our poles rigged up and then I'm going to go through everything and show you guys exactly how we're about to catch these fish. All right guys, it is time to get started with the cast net. We got to catch some live bait here. And I think there's no better way to do that efficiently than a cast net. If we can find some fish up on the bank, try not to get it snagged in every single treble hook in the back of this first. That would be a good way to start. All right boys, we're going to set up camp right here. And I'm going to show you guys exactly what items we got. So for rods today, we got a kick in their bass TVX loose combo. If you guys want to check them out, kickintheirbass.com to be linked down below. And then I'll also have a mock smash with me as well. If you guys want to check out any of the mock combos you can check them out on mock. Before we get deeper into this video, only 27% of you all watching these videos are actually subscribed. So if you could take the time to subscribe button really helps the channel. I'd appreciate it. We got this green libeye bucket right here, which we're going to be putting all of our bluegill in. We got some green night crawlers. That is going to be our bait of choice for today. And then we also have the tackle box. So pretty much inside, we've got anything and everything you can think of. We got all different size hooks. We have some bluegill hooks there. We have some bigger hooks for some catfish. Some really big hooks there. I don't even know what we'd be using those for. We have some regular sinker weights, some split shots, and obviously a bunch of different types of bobbers. So we have everything that we would need. But first things first, let's go ahead and rig up these rods and get them ready for the bluegill. There we go. We got what we came for. So this is actually brackish water. So hopefully once we actually put them in the fresh water, they don't just get like shocked from like the, what is it, like slant? How do you say, salinity? Something like that for the salt content. I was always bad at English. There was a few things I was good at at school at and English was not one of them. But that's okay. We make fishing videos for a living now. So I don't think I needed it. So we already have a bobber on here. Overly sized bobber. We've got a tiny little panfish hook. I'm going to kind of downsize this bobber. So I'm actually going to take it off and put one of these smaller ones on. Gonna open up our tackle box here. We're just going to get a bobber like that. Pretty clean and basic, nothing big. That's really all we need for bluegill. This really varies too. It really depends on the day. I'm going to tell you, because there's some days I only put the bobber about eight inches up that line. There's some days I'll have it, you know, about a foot and a half. But we're just going to put it about a foot up the line. I think that's going to be our hot zone. Go ahead and strap this bobber on here. Boom, simple as that. And there we go. We got a simple bluegill rig. One other thing that we can do to make this even better. Just going to grab us one of these small split shots. We're just going to put this about halfway up the line between the hook and the bobber. Pretty simple. And that is all we need to catch some bluegills. So now our old green worms never have used these before. So let's take a look and just see what these worms look like. Oh boy, look at that. That is spicy right there. A little bit of green action. That's definitely dye. That's not something that's natural, but they do look pretty cool. Maybe the fish think they're pretty cool too. So let's go ahead and get one of these out. Let's kind of thread them up on the hook. Boom, just like that.  But that's usually what I do. I'll put it through a couple of times so it doesn't come off. And that's just what I've had the best luck on. I'm going to try to go out on the dock first and see if I can't cast over this railing. There's usually some bait and things swimming around here. All right. Let's get this thing in here. We got something. We got something. What do we got? I don't think we're going to be using largemouth bass today. We got like a pound and a half largemouth. That's one way to start. I'm surprised I didn't see him sitting there. It's pretty shallow. Just a cute little guy. We'll get him right back in there and try to actually find us something that we can use. Thanks, buddy. Just want to drop this bobber right off this ramp. Pretty basic here. Uh-oh. Looks like I got a bass coming up to it. Oh. Yeah, that's a bass. He is like ultra smart. He's just swimming all around the worm. I can see him in my glasses. We don't even need a dang bass. Maybe we throw this over here in this corner some more. Let's see. Right up there. I've had a lot of luck in the past right up on that little boat ramp. This might be proved to be a little bit more difficult than I thought. We'll try this other side. I have caught fish in here so we won't see if we get any in the cash net. I feel like the closer we get to the bank, the better chance we're gonna have. Oh, we got one. Nice. Oh, that's perfect size too. That's exactly what we needed. Boom, what a being. Is that a bluegill? Oh yeah, that's a little blue. I'm glad we kind of got like two different strategies going here because I feel like I might have a harder time catching some of like the really big live bait. No will probably have a better chance with the worms and like actually trying to catch them on a rod and reel than I will.  We're gonna put out some big bait as well in hopes of catching a giant. Far as just quantity is concerned. These guys are the perfect thing for the job. Every once in a while, it's pretty good to sit back, relax, you know, drink some coffee, you know, and just kind of chill out. Don't get me wrong, it's fun here and there. I'm more of a guy that likes to move, move, move in cast-a-ton though, for sure. Here it goes, nothing. Probably get a little bit more distance on that. Ugh, nothing man. All right, let's keep on moving and try to work down this bank line where we can slip in here where there isn't too much brush and stuff. Somehow I always get stuck with the cast netting duty when we film live bait videos for Noah. The man never wants to get his hands dirty, I swear. I've never seen smoother hands by a male species, moisturized and everything. I'm gonna tell you right now, you would not think that this is hard to catch a bluegill on a bobber but I have not seen anything and usually there's a lot of fish here. I have a lot of confidence in Fletcher right now, catching a lot of bait on a cast net but at the same time, I need at least a couple fish. At least three bluegill, good sized ones that we can use. Oh, finally. We got another one. Yeah, we got two, nice. They're stuck in the net a little bit. There it goes, one and two.  These are definitely usable by all means. I think this is also a good size for live bait. Come on, fishy. Oh, yes sir, fishy. We are on board and it is a bass. We would catch a bass when we're trying to catch bluegill. Okay, Bubba, calm down, calm down, showdy. That is by far boys, my PB bass. I think that's the biggest one I've ever caught. You guys see what I'm doing here.  Absolute toad. That is not what we wanna catch. I've asked this in a few other episodes and I don't know how you guys feel about it. I'm just a bass lover so it'd be very hard for me to do this but at the same time, I'm curious. Should we go live bait fishing with bass for a bass? Comment down below, let me know your feedback. I love bass, I would feel very bad but at the same time, I'm very curious. Well, there it goes.  I think you would, fish are crazy. That was a great textbook. Oh, we got something. Well, we got something, it looks different. Is that a shad? Is that an American shad? I believe it is, look at that. I don't know if I've ever caught a shad with a cast net before. Let's go ahead and stick that thing in here. We're gonna be careful with this guy. This is, I feel like, premium bait. I feel like the vast majority of bass fishing lures are designed to look just like this guy and just about that size. So, if they'll eat artificial stuff that looks like that, surely they'll eat a live one. Ooh, look at you. What is going on? Okay, boy, that is the longest worm. Let's get him on the hook. I actually got a little more. So, this is what I was saying the first time. A lot of people have told me this is how they rig their worm up. They'll like sit here, okay? They put their worm up on the hook and they'll almost thread it on like an actual artificial bait, like that. Which, that's an option. And then I can kind of do this. I can kind of double it over. Boom, give him the hook a little bit more. Cause at the end of the day, if we're catching bluegill, I don't want too big of a clump, but that right there should get a bite.  Just fill this thing up. Oh, let's go. Not the mother load, but we got another shad and a bluegill and another shad. Let's go. All right, guys. We got that five star bait today, boys. You know what happened last time when I got a bite? I started singing to him. Fishy, fishy. I gotta think of the lyrics first, boys. I'm not no singer. Hopefully Noah's delivering over there with some quality size, cause I got the candy. Look at those little floppy boys. Here you go, guys. Joy a new home. Fishy, fishy. Don't be wishy, washy. We did pull up the shell earlier. Do we have any critters in here? Oh, we got a little, some kind of crab. I don't know, is he alive? Oh yeah, he's kicking. We'll put, we'll put him in there.  Well, you know, you can keep your shell too.  Just slurp on the worm egg today. My singing today is definitely not on point and that might cost us from getting bites.  Oh yeah, that's gonna be it boys. That is all we need. Three more little shads and we probably have the best bait we've ever had for live bait fishing. I don't know how the catfish are gonna respond to these guys, but surely the largemouth are just gonna have a field day with them.  I think I'm gonna take a little route across the street. There's actually another pond that connects to this one. And we're gonna see if we can get better bites over there because over here it's just dead, man. I don't know if it's my worms or the fish aren't bite, but I'm kind of worried because Fletcher's probably gonna carry the job on the live bait. But let's go ahead and head over there and see if it's any different than this spot. Hold on, hold on, what is that? What is that? We got him. I think we got him, dude. Unless the net got messed up. No! They're laying there right on top of that fish. She's getting worse by the second here. Oh, we got something big. Oh, let's go! I think we got one of those horse mullet I was talking about. This is honestly a small one compared to some of the other ones in here.  And they're really, really hyper. All right, calm down. Relax, honey. All right, here we go. That is a mullet.  This is something that you'd like to throw for that. I don't know what else would eat something this size in this pond besides a giant tarpon. Maybe a huge catfish. To be completely honest, we have absolutely no use for this fish right now with what we have. So we're just gonna let him go on his way. But we appreciate getting a chance to check you out up close and personal. There you go.  All right guys, on that note, I think it's time to go check the rods, see what Noah's up to, and see if we have any big fish over at the dock. Boys, I do not know what is going on. These fish are not one to bite. We were just trying our best. And I just haven't seen a fish. Like I haven't seen anything at all. Like I got polarized glasses on right now. If there were fish up here, I should be able to see them technically, but there's really nothing. This is pretty crazy. I don't see Noah on the dock. Oh, there he is. He's out there to the left. Trying to catch a fish on that little pipe. What's up? Greetings. How's the bite? Totally smoked him. We got zero bluegills. We got zero, but hold on, hold on, hold on, to make it even better. I caught a bass. Okay. He got something. That's not gonna help us, but. I mean, yeah, I didn't keep him. Hey, at least I'm doing the heavy lifted over here. Come check out what I got. What do you got, dude? Dude, we got a palethora of stuff. Oh my gosh. Is that a shad? Yeah. He looks like he's doing, he's going to have a hard time. All of them have had a hard time, haven't they? Yeah, that sucks, dude. We caught some shad. It looks like they might now be cut bait. This is what I'm thinking. So we're gonna go to a whole different location, Fletcher, to catch whatever big fish with the big live bait. All right. First of all, what we do is we go over there, we start some cut bait, we set out some rigs, maybe even some live bait, and then we can still attempt to catch some of the bluegill on the rod. All right. You think about that? You think that's a good game plan? Yeah, that sounds good. I mean, you've got enough bait for us to start. So let's go ahead and get out to our next spot and see if we can go catch some really big fish. And also, some even bigger live bait. We have a raft. Yes, sir. Okay, let's get this stuff out of the golf cart and down to the dock. All right, you've got the rods. Do you want your net? Yeah, we'll take the net, too. I caught me a wild fish. It's my pee pee. Yeah, nah. No, we're good with it every now and then. Not the butterfly net? Come on, now. No? All right, this is a heavy mess. You got her? I should. Oh, yeah. Oh, yeah. Rollin' down, boy. Where am I at? Come on in, come on in, come on in. Well, you ready to catch that big juicy mama? Big juicy. You got it there, man. I'm trying not to spill the water, you know? All right, boys, so first step to this process, we want a big bluegill. That's the purpose of this video is giant live bait. So we need, we have some other options, but we need a big bluegill first. I feel way better about the big fish. But to catch a big fish, we need a giant bait, right? Yes, sir. That was like the ideal depth for bluegill. So I just run it about a foot, foot and a half above the hook. I'm gonna flick up here against the wall, just up. That was on a group. Oh, okay, that's gonna work. I think that's gonna get the job done. Put that down. Guys, I got one ticking me. I got one ticking me. Yes, look, he's running with it. There we go, baby. There we go, baby. Yes, it is. That is a big bluegill, too. Look at that, boys. That is exactly what we need. We came to this marina side, and I have a feeling that there's a lot better fish in this location. So we got one big bluegill. Yes, sir. To work with here. That's a freaking massive one, too, man. Can you get him right on the end of the dock? Right on the end. He was probably right under that little float, and that is our first big bluegill. Yeah, that happened quick. A little beat up, man. That was his tail and everything. Pretty fish, though. Thank you, sir. Hopefully he enjoys this new home for the meantime.  All right, boys. So our game plan is pretty much to get that big bluegill that we just caught.  We're gonna cast him out there, kind of let him roam around and get comfortable and wait for a big bass to bite him.  We have some cut bait. We also have some really stinky catfish bait over by the golf cart that we can go pick up here in a second. And I think we're gonna end up catching a lot of big fish. But first things first, let's go ahead and rig that big bluegill up and let him swim around. I'm thinking one of these big circle hooks should be the perfect tool for the job. All right, boys. I had to come back here. I wanna get a few things first. We're gonna get these pliers. We also have a really big pair of pliers.  And we also have some of this stinky catfish bait. We have another one here that's also supposedly vanilla flavored.  I do not wanna eat vanilla flavored catfish bait. Yeah, that's some other supplies that we got that we're gonna bring out to play around with. There we go. That should be good. I'll let Noah decide what he wants, but we're probably about a foot and a half right there.  And there's a few ways you can do this. I'm not sure which way I wanna go about it yet, but let's just get him out of the little bucket right there. All right guys, so we're trying to figure out exactly where to hook this. So this is a perfect question for y'all. We're gonna try our best today, but where would you guys hook this bluegill if you were to hook it? Where I'd like to hook it is right there behind the fin. There's a line right here that you don't wanna hit and kind of see it on the bluegill. So Fletcher probably right, somewhere right there behind the fin.  Yeah, I think that should be a good spot. Good tender spot, tough spot, not even tender. Oh, he's in there now. Let's go ahead and give her a cast. No, I'm gonna let you do the honors here. Wherever you wanna put him. I gotta be careful. Where do you think? Are you thinking of just way out here? I don't know, man. I feel like you gotta put him next to the dock somewhere. Maybe this way, dude, like kind of towards that boat. He'll still have room to run to the right if he wants to. Maybe put him like right between these two docks, kind of in the middle. Okay, this rod is so flimsy, bro. Give it a sling. There we go. Money. That's money.  Oh, I think I just had something on here, guys. My bobber got pulled like way underneath the dock and a lot of my worm just got ate.   But yeah, we'll give him some line to move around. Maybe we'll get a really big bass on that. Something take your bait? Yeah, something. I was like, my bobber was just gone. It was like all the way up underneath the dock. Was it really? Yeah. We're gonna put it back.  Put just like a little feeler out there. All right, Fletcher, I'll be right back. I gotta grab our filet knife. We need this filet knife for some of the cut bait that we're about to do since we got the bluegill out there. All right, Fletcher. What you thinking? Got the knifie. You got the knife? Okay, now we just gotta pick which bluegill or bait in particular. I know you got a bunch of different things in there. We're gonna do him through the nostril. Fish actually tried to eat other fish head first so they go down their throats without the fins popping back up and getting caught. So I think that should do the trick. Go ahead and get the bobber on this. There we go. Right there in that shade line. I think it looks good, bro. I think Swam off really good. She's still kicking great. All right. All right, so we got our worm. We got our big bluegill. We got our shad. We got a variety. All right, I guess I'm the grim reaper today. Sorry, boys. Looks like it's your bad day. Well, Jimmy, it was nice knowing you, pal. I'm gonna rig it up here. This is, like I said, not the best piece, in my opinion, but it's something to start off with. I'm gonna actually toss this one right here in this corner. I think it's a great spot for a big cat to be sitting, so toss that bluegill head, cut bait, or in that corner that, I think that's gonna be a good spot. So what I'm gonna do here is I'm gonna reel up ever so slightly so my line doesn't have a ton of slack. I don't think I've ever fished with cut bait shad. I'm pretty sure this should go pretty good here. Nice, nice and juicy. Oh yeah. That's the good stuff. Oh yeah, Fletcher. That's it. Oh yeah. That's what they like. Yo, the catfish gon' eat that, boy. What do you think we should put this now? Maybe other dock or all right here? You could go to that dock and then, if we don't have any luck here, we can always run to that side. Okay, I might go put them straight across in that hole. So let's pop right here. We're gonna throw this right in the corner.  Like that is gon' produce us a big ol' kitty. So it's been a few minutes guys and we haven't had any luck yet. Our fish are still swimming around well. Our shad has been moving all over the place. Our bluegill has kinda moved back into this corner and I'm thinking it's time that we just go ahead and pick up this worm. Seems like the fish aren't ultra interested in it and we don't really need a bluegill until we lose the big one. What I wanna do is actually use some of this, like I guess it's not artificial, but manufactured catfish bait and toss that somewhere around here and see what kind of results that can deliver for us. I mean, I feel like it's hard to beat the real deal, but hey, you never know if you don't try. Keep my face away from this, this stuff smells so bad. All right, oh God. I don't really know the proper way to hook this stuff up if I'm being honest guys. Maybe you just like ball it up and stick it on this thing. I could be totally wrong. Let me know down in the comments below if this is how you fish with this. This should be a good spot. Time to let that thing soak. Looks like we're on the bottom. That should be good. Let's leave that bail open. Yo. Something's taking off. Oh yeah, something's definitely got that I think.  Oh my gosh. Yeah. Dude, I don't feel nothing, but this bait was right there. Well, that's our first bite of the day guys. So set that back down there.  That line, see how it's tangled? Yeah, I just looked over and I haven't seen it straight out and I'm like, hold on, there's something going on. Oh, oh, just take it in, take it in, take it in. Oh dude, he's taking all of the flesh. He's taking all of the flesh. He's back, he's back. Do you want it? You can go for it bro, this is you. Well, I'm gonna let him eat for a second. I'm gonna let him eat for a second. Dude, that fish was taking off. So just remember too, we don't have regular circle hooks. Like just like the octopus ones, you know? So you might need to give it a little more.   At least you know the shad works. We got bit like, I mean, that was quick. I would just drop it back down, you might get hit again. Maybe we should change the hook location. I'm gonna try changing the hook location. I think mine's getting hit. You have one? I don't know. Oh, there's a dragonfly hitting your line. I was like, it's bouncing. I was like, what's it bouncing for? Oh, you get stripped? I guess, you ate whatever I had. All right boys, what we're gonna do in the time being, I had a fish eat that.  Since he had that bite on the shad, two bites really. I'm just rigged this up. There's something else down there. I might actually move it over here by Fletcher because that was two quick bites. Let's go ahead and put that in there like that. Just move in the hook location, maybe have some luck with a better hookup ratio there. I'm just not really sure. Not my specialty. Probably put mine over here, Fletcher. Yeah, but I'd say right up against. I mean, clearly, they're on this side. Oh, no way. You're not getting bit, oh dude, that's insane. Do you see the line pop? Yeah, I don't know if it's just, because I put it like a little bit over there, I just don't know if that just strained out a little bit.  Yeah, it definitely could have been a turtle. We've had that happen before. We've had turtles just totally fool us thinking it's a big catfish. It's definitely a possibility. Hold on, boys. I don't know what Fletcher has going on over here. Oh, dude, there's a fish on it. Yes, let's go, baby. Oh, let's go. Oh, I got him. He's got me all stuck in something. Oh, can we get him to come up? Can we get him to come up?  Oh my goodness. Wow. Out of all the effort that we have done to put in the live bait, to put into the cut bait, to put into everything for the freaking catfish bait that we bought pre-packed, it smells like horse crack to get bit first or to get the first initial fish, it's just beyond me. It's funny because I've been checking all the rods and I haven't even thought to check this one. I was like, oh yeah, forgot Fletcher put this on and this is the bait that's getting bit, that's insane. Put up a pretty good fight. He had the drag very loose. I didn't even remember to tighten that down, but I mean, that was a decent fish, whatever it was. I'm assuming he just had it drop all the way down here. Oh, Fletch. Take it off? Yeah, it is like crazy, bro. No, dude, no, dude. It's like they're not having it. It's like they're just grabbing the tail and you really gotta just let him eat it. It's probably the hooks. I mean, we don't have any regular circle hooks. Dude, he took, I think, all the way under the dock. At least we're good bites. Geez, man. My GoPro battery just died and Fletch went to go grab something. I was like, tell him, get the camera quick. Look how far he ran. He went from here all the way out there. You ready? Get him. Yo, let's go. Yes, sir. Let's go. Let's go. Oh, no, look at him running. She feel good? I can't even tell, dude. There's definitely a cat. Yeah, it's a little cat. Okay. Listen, we gotta get started somewhere. That fish was so aggressive, bro. Okay, spit out the shad. I have never seen a fish just run so far in my life, bro. Like when you went out over there to the golf cart. Just darted. Dude, my whole line, I thought my whole spool was about to be gone. Like he was just going, but listen, not a big fish by any means, but that is a start. And Fletcher, you know, I think we could stay here, but I think we should go to the spot. We're back in the day. Back in the day. We caught some monsters. What do you think about that? Yeah, I'm definitely saying we make that move. It's like just around the corner over here, guys. There's like a back corner of this marina and we have caught some lunkers. Yeah, we've caught some monsters. So I'm going to go ahead and get this guy off the hook and then we're going to move spots. Boys, this catfish choked this line so bad that sadly I just got to cut it, but I want to get him back in the water. I just, it was honestly the worst timing for anything and everything that happened. The big camera died. Fletcher took his GoPro off. Fletcher was over there. The moment it starts taking my line, my GoPro overheats. It was the worst timing for anything. Literally there's been a camera rolling nonstop for the last like four hours. The whole day. Yeah. All right, boys. Well, let's go ahead and move to this other spot. Some took your bait, but some might have that catfish. Oh yeah, it looks like it's out a little more. It looks like it might be stripping that line, Bubba. Uh, took it. He took it. That bait's actually working pretty well, man. Did you put the stinky on this? That was the stinky. Let's go ahead and grab this little shad. See how he's doing. If he's still kicking, I'll be surprised. Oh yeah. She done. She done. We're just going to go ahead and take her right off. Dude, what did this fish do with your line? What? Oh my God. How is that? How did you, look how lively he is, bro. He's just laughing at us. How did he even manage? I've never seen nothing like that in my life. That is, that is something. That's one of those things that you might just need to leave and cast a bag out. I'm actually going to go ahead and walk this over here, Fletch. Okay. I guess you could just toss him in like that. I'm going to, man. I ain't even trying to mess with that, boys. I'm going to cast him right out there. We're going to go over here for the cut bait. So I'm just going to cast him out, just kind of let him roam, do his thing. We'll get the rest of the supplies. I think the dead shad is the move. Yeah, you want to give that a shot? You get in there, boy. These are all slimy bingers. Oh yeah. Nose manicured hands. Come on now.  I don't even know how you managed to catch these, to be honest, but I'm not asking questions. Hey, as long as the fish like them. Dude, that cut's so easy. Boom by the bay. That cut's ridiculously easy, boy. Where did you put the hook on that last bait that we actually caught a fish on? I'm probably going to go right through the eyes, bro. See that? I think that's the best way to rig it right through the eyes is fire. And I'm going to actually toss it right there on the corner slip. I've caught a lot of massive catfish there in the past. And I would not surprise me if we get a bit. So it's just a waiting game now, boys. And we're just going to drop this one right here where all our stuff is. And just as long as it's up against the dock should be a pretty saucy spot. Dude, I had one the whole time without not even knowing. And something even grosser is I officially got both of the fish's eyeballs on my hook and they just fell somewhere. They just slipped right off. I guess we're going to need another piece of cut bait. I mean, we caught that one on the tail. Might as well not let it go to waste and use this one as well. Just like that, man. Pretty simple. Whatever's down there is hungry, guys. That was two bites very quick. See if we can get us another one, another bite. I didn't even know we bit it that second time. Right there, money hole. So if that bait sink, let it do its work. I'm just going to sit here and kind of baby it. I actually had to get bit pretty quick with how fast he was biting it both those times. I mean, that was intense. Dude, he's already got it. That quick, Fletch. That quick. I mean, look at him taking my line. Whatever's down there is hungry. See him taking my line? She got it? Oh yeah, she going off to the right. I think I got a hook set with this hook, man. Yeah, no. She didn't get my bait, but she tore it up, bro. Look at that. I got to re-rig that. Straight up just messed her up. Definitely the hook set got a little bit of it, but I mean, it's just crazy how quick they're biting this corner. It's a great sign. Got guts on that hook. Oh, that's money. Rigor through the body piece. That's prime. Let's drop her back down there, man. Hoping these aren't baby catfish just playing with me. I'm hoping we can get just some megalodon of a catfish. I'm thinking we pop a squat right here. This looks like the ideal spot to rig up some leader. He's taking it now, guys. He's messing with it. I can see my line tightening up. Yeah, I just watched him eat it. Here we go. I'm losing my sanity by the second.  Hold on. We got something over here, Noah. We got a runner. I got a runner, too. Dude, mine is cooking. I think I'm gonna set. Yeah, we're on. We're on. Let's go. Fish number two. Oh, she got some handshake. I'm on two, Fletch. I'm on two. Come on, come on. You on two? Yeah, it's a baby, but I got her. Let's go, man. Oh, what do we got? She got some weight. Oh, yeah. Oh, yes, we got a good one, boys. We got a good one. You wanna walk her all the way down and bring her up on one of these ramps? I don't know. I think I can flip her with this. I got 20 pound on this. We're gonna go ahead and do the flip, guys.  All right, hold on, one more 10. Boom! Let's go! That is what I'm talking about. Dude, that's a good one, Fletch. Good job. Let's bring her over here. We all can see her a little bit better.  Got my line fully coated. What did you catch this one on? A headpiece? A tail. A tail. I think we got this one on a tail. Dude, that's a good one, Fletch. Yes, sir. I mean, look at the difference. She got some size. I said at least six times the size of this. Yeah. All right, boys, this ain't nothing to show compared to Fletchers, but second fish. There he goes. Wow, those little ones are feisty. Let's see if we can get this hook out of here for you, pretty lady. Here, actually not that pretty. Kind of ugly. There we go. Got that hook out and she's free at last. Let's check this fish out, guys. I'm gonna try to get a good grip on her. These guys can be a little bit floppy. There we go, guys. That is a really good-looking cat, if you ask me. Probably about four pounds or so. We definitely know there's some bigger ones out there, but I'm stoked to get this one. Finally getting some size on the board, and I think it's time to go ahead and release this bad girl back into the water. Here you go. We're just gonna launch ya. Choo!


Hey YouTube, it's Dabbin Doug and we have an amazing review to do today. We're gonna be reviewing the Pax 2 vaporizer It's an amazing new vaporizer sleek new design all cool stuff. Let's get straight into it The sleek new package here that Pax 2 has got brought us today. We're gonna open this up right here nice little pedal design they have here and Voila beautiful sleek design looks something like an iPod here Beautiful aluminum here nice deep red here. We have the mouthpiece right here pops right off and The oven where you put in your dryer right in there Well first we're gonna take a look at the inside of the kit that it comes with it's a complete kit comes with everything You need we're gonna look at the charging features of the Pax first now It's an interesting magnetic charging system at the Pax features On the bottom of the Pax there are these two magnetic posts there Thought those magnetic posts Lie on this one here. They just clip on right there as you can see It's completely magnetized to the Pax 2 vaporizer then That charging station plugs into the USB cord that's attached to it Plugged on the computer it starts charging Fully charged battery for the Pax 2 it runs around three hours. It's a really long battery life It's gonna last you a really long time Really cool stuff next let's check out this cleaning kit here It comes literally with everything you need Inside the cleaning kit it comes with a hundred percent isopropyl alcohol in a nice little container and Some pipe cleaners you get the pipe cleaners a little bit wet with the isopropyl alcohol You pop off the mouthpiece and Right in there you can see where the vapor comes out You stick the wet pipe cleaner inside of that push it down in there and then Just pull it out and you can see the residue that's come off You'll clearly be able to see if it's been being cleaned, you know You just pop that back on there good as new It also comes with the different types of mouthpiece The mouthpiece that it comes with is a flat mouthpiece You you can see on the top there. There's a little opening right there You can get closer looking right there and Then you put to use it you put your mouth on the top here Don't you don't not even have to and encompass the entire lid but only this little slit right here and you take a small Cigar like pool on the vaporizer just a on that good as new You can pop off that me out mouthpiece and trade it for this lifted mouthpiece that they have works a bit differently similar to other vaporizers like the Titan out there and Just like that you put your mouth right over there and pull just normally Works just fine pops right off made out of rubber Has a little baggy for But yeah, those are the the different mouthpieces that it comes with They're really easy easy to change remove everything But everything Is easily contained inside this kit, but let's get back into the features of the packs If you can see a little bit on the mouthpiece There's a little circle right there in the center and that's the power button You press it once To turn on as you can see it's turning on there They they feature a pedal design right on the front of the packs with the LED lights And that's pretty much how you control the packs via the power button and these lights Right when it turns on it starts heating up It has about a 30 second to one minute heat up time as it's pulsing like that As you can see when it's changing in colors like that as when you don't hit it In this time you could be putting your herbs in the convection of them right there This is a true vaporizer because it has a convection oven by the way There is no coils that you're using or anything because it's it's a heating up these Dry herbs in this oven. It's truly vaporizing the herbs and that's makes it a true vaporizer once it turns green like that That means that it's ready to be used And you can pop that lid on back there. There's also a magnet on this lid to the oven right there just slides right on Once it's green like I said, it's ready to use and you can hit it normally like I said earlier once again like a cigar like pool There are a lot of different features while it's on they can use as well starting with Changing the temperature of it. You just hold down The power button right there and then when it changes to these different color pedals here You can see what stage of temperature it's in on this stage. It's in the second highest stage You press the power button one more time as you can see it's in the third highest stage And let's say that we want to add that stage. All you have to do is shake it. You shake it and We're back to normal. It's we're back to it's charging Heating up back to that point once it turns green again. It's ready to use Well, and it has so many different features like that things that you'll just never need to use like for instance Let's say you're just fiddling with the packs flipping it around a couple times and then all of a sudden It starts changing colors It has so many features even a rainbow dance party. It does so many different things You can even flick it a little bit and then it'll tell you how much battery it has left when it has four pedals lit up That means it's got full battery. You'll also see that when you're charging it It's got so many crazy features so many cool things to do with the packs, too We've got we really hope that you guys enjoyed this you guys can check it out at smokeartel.com Thanks for stopping by dab and dug out


Welcome back to the breakfast here in plus TV Africa on this day in history I'm going back to the year 1930 to tell you about the Hoover Dam It began of course a construction. Well, the United States began construction of the Hoover Dam on this day this Dam that eventually took about five years to construct and 21,000 workers that were part of the construction of the Hoover Dam. It of course had an average of 3,500 men working every single day on the dam Author Powell Davis an engineer from the Bureau of Reclamation originally had a vision for the Hoover Dam back in 1902 and his engineering report on the topic became the guiding document which was eventually put to use when the Construction eventually began in 1930 for many many years Actually, the dam was the construction of the dam finished on this day For many many years the water had been a source of contention among Western states that had claims on the Colorado River and so they of course began construction It took like I said 21,000 men I'll prove off from Congress and of course a little bit of controversy here and there before it eventually was fully constructed a few facts Also concerning the Hoover Dam initially it was meant to be called the Boulder Canyon Dam and You know that was mostly because of where it was located and and the place that they decided that they were going to You know create the dam just right next to the Grand Canyon, but eventually after construction it was renamed by the guy in the head engineer after President Herbert Hoover in 1930 that of course eventually was changed after Hoover left Presidency Franklin Roosevelt who wasn't a huge fan of Herbert Hoover Then renamed the dam and said it was going to be turned back to the Boulder Canyon Dam But later on after Franklin Roosevelt President Truman eventually then approved it to be called the Hoover Dam much later in 1947 I believe Also an entire city was created for the people working on the dam at that time And that's you know, some of the people who currently still live in those areas Descendants of the people who built the Hoover Dam and it reminds me of their job at your quota still complex You know, I've heard of stories from their Dakota still complex and now The original plan for you know all of it also included a small city Which I think they probably achieved in to some extent, but there was also a plan for a small city for workers and Yeah, but not to that I don't think it did to the extent that they had planned You know because of the failure of the dam of the still complex to fully kick off and be fully functional I don't think they eventually did but some of those complexes that you see around their Dakota still complex now where constructed way back then While the dam was still while the I keep saying damn while the my Dakota still complex was being constructed I think many interesting facts about the Hoover Dam about how, you know one of the world's largest refrigerators cooled all the concretes that were to be used for the dam and That all the concrete used in the dam can you know, fill the entire US and About how nobody actually once said that they are to be called to be named after, you know President Hoover. He's it was stated that he's one of the least beloved president. So there was lots of Oppositions of that, but it stood and now it's called the Hoover Dam. Yeah, so moving on from 1930 I believe to the year 2005 on this same history There was a terror attack in the US actually four different terror attacks in the US and this Was at London's systems transit London's transit system Okay, in the UK in London. All right, so it was on the morning of July 7th, 2005 Bombs were detonated in three crowded London subways and one boss during the peak of the city's rush hour So there it was basically a synchronized Suicide bombings, which you know people had believed to be the work of Al Qaeda It killed about 56 people including the bombers and injured another 700 people it said to be the largest attack on Great Britain since World War two There was no warning for this these three guys, you know, they had planned this they picked up the fourth guy on their way They all wore backpacks with the you know, these explosives they entered different You know trains heading in different directions They detonated the bombs and basically, you know, killed themselves and other people now these attacks took place as World leaders including British Prime Minister Tony Blair were meeting in the G7 G8 summit in nearby Scotland now despite You know thought that this probably was a walk of Al Qaeda foreign powers You know was found that these people were actually British and maybe even one was Jamaican Well, it was thought to be doing funny enough This attack happened on July 7th by July 16th The police had investigated about six hours six thousand hours worth of footage and found the men July 16 I mean look at that look at the short time. Yeah, even though they were dead they were able to trace CCTV footage and found people who carried out this attack and Much later that month July 21st another group of four men tried to replicate the attack Well, luckily the devices did not detonate now guess what on the same day of this four bombings attack in London There was a consulting company that ran mock exercise Entirely on paper in preparation for a possible terrorist attack in London similar to one that actually happened So they were preparing for a mock exercise to say if such a thing happens in the UK How can we deal with it and such a thing happened that same day in reality? So this basically prompted initial conspiracy that you know the company the government had something to do with it But you found that this exercise was common. It was unaffiliated and these these men would let go So about two other people were in basically fingered in this attack, but they were cleared two years later So it was in this day in history that four bombers attacked transit station in London in 2005 to coordinate the suicide bomb on London's transit system With explosions that thought through the transit trains in London on the ground. I remember, you know this day very well This was a couple of years after the World Trade Center bombing in 2001 and it was kind of in the peak of Al Qaeda and its attacks on the United States and on the West generally in Europe Later on there was you know others in France, you know, I think there was one in Australia or Switzerland I'm not sure now, but it was in in that era when there was a lot of those terrorist attacks, you know But this also showed The importance of intelligence gathering for for them to be able to prevent, you know later further bomb attacks in the future I remember that there's also, you know, sit at times when there are certain raids that you hear about That's you know, eventually bossed Terrorists that we're going to you know attacks on time later in the future The importance of intelligence gathering the importance of funding security networks Properly that we lack very well here in Nigeria and also the perspective of Or the idea, you know, that Al Qaeda doesn't necessarily Al Qaeda Members or attackers don't necessarily have to be from Afghanistan or Iraq Because this was at the same time that the war in Iraq was going on, you know, the Saturday in 2003 the war in Iraq was going on There were still of course bombing left and right in Iraq and Afghanistan while this happened in the UK And so it's just it was just a reminder that terrorism always have to come from those areas, you know They could be British citizens. They could be American citizens They could be you know from any from your own country And there's always a presence of sales terrorist cells in every country I mean that were possibly a presence of a terrorist cells in every country in the world And loan I think they're called loan strikers or loan rangers who also carry out these type of attacks But don't you think that when we look at how other countries foreign powers like the US the UK How they handle, you know attacks like this An attack was was, you know happened July 7th Almost a week later They apprehended or actually the people died but they identified people who perpetrated an attack and many instances throughout history Why is it that we're not looking in that direction? Let's get like Training from people who do it better. That's really why do we have, you know, you can send up We used to have things like this, you know back then Yes, well, we had people get authentic training about how to actually prosecute Anti-insurgency war why why don't we have things like that because we are in dire need of military intervention in Nigeria How can you know our security situation being such shambles that people are Abducted and parents are protesting and they are threatened with arrest and bullets, you know So our do you agree with me that we could use Some training by foreign powers who know how to do it better So I don't think we necessarily need foreign powers to train I think what we have lacked for a long time. We're speaking about the Hoover Dam now from 1930 Over time at that time it was the biggest or the highest, you know, dam over time They've been able to develop, you know other dams that have, you know, be more than 700 meters higher than the Hoover Dam So but that's where they're coming from The UK the US Germany France Russia They have a massive budget every year for security and that includes not just equipment No, no, we don't have peanuts compared to what they compared to what they would not exactly compare our economies are different I wouldn't exactly compare them with that level But I am saying that the need for training is something it's something you cannot So so this is where I'm this is where I'm headed. They have a massive budget, you know, they do not You know means words or you know Shortchain themselves when they come to spending on security They spend billions and billions of dollars every year on security and pounds every year on security We don't and it's not I don't think you will blame it on, you know having two different, you know economies We I don't think we prioritize these things well enough for us to be able to spend, you know, as much as we should on them We started talking about Boko Haram in what 2009 or so Since then we've had enough time to increase our budget for security But because we've continued to do the same things every year. I don't know what's on the miracle that we expect to happen We keep bringing in the army when we instead should be spending more on intelligence gathering spending more on Local security. We've had this conversation of state police for a very very long time Intelligence government you hear how they basically rummage through Thousand and thousands because they spend enough because they spend enough to have cameras They spend enough to keep those cameras running every single minute of the day Since time we had stories about contracts, you know to a Chinese company to install cameras in our highways Where is that? Yes, the Abuja was a thing called it was some CCTV, you know scam or something in Abuja When we're obviously not serious about these things because they spend money for those cameras to be running every single day Every single minute every second it costs money and they do not you know in any day anytime feel like okay Maybe would have money for cameras today Maybe would have money for fingerprints, but there's so much that they do in order to ensure that their cities are safe And for anybody who wants to carry such an attack in the UK You will be worried that you you definitely will be scared that you will be caught or in the United States You you have those fears because the the chances and the loopholes that exist are so little You know for you not to be caught But here in Nigeria you can have a serial killer walk around for ten years and commit murders in In in hundreds and nobody would ever figure it out Community policing and all of that, you know They have their system of the 911 where once they see anything suspicious They inform and alert the police and they get right on it, but our police funding our police people tell you They don't all of this is still funding. Yeah things like that. So it's still funding You know they invest enough and that's why they have what they have we don't We haven't in any way been able to be To open up more avenues with which we can make more money and invest in education investing security investing infrastructure We keep giving the same excuse and every day where we are currently when Economically in such a bad state that there's not even a 2,000-hour extra that can be put into security If we're serious about it because we don't even have money to fully fund our budget We have to keep borrowing every year to fund budget to pay back loans that we've taken And so it's just of it's these these are the effects of a country that has not been serious With these things for a long time. I keep speaking about blocking leakages in government reducing the cost of running government in Nigeria There's so much and so many ways that we can save money and we can make more money But we don't seem to be taking it seriously if we have a You know ways that we can create systems create Better auditing and save 500 million save a billion save 2 billion every year save 200 billion every year From different sources from customs from immigration from there We make so much money in Nigeria that just disappears into thin air And so if we have better auditing and we're able to save all that money and put it into infrastructure or put it Into security some of all these things would not happen. How how can you imagine that a hundred and fifty kids will be kidnapped from a School and just be taken away How I mean you hear these things that you can't just imagine that this is real life It can even in movies this kind of things won't happen. So how is it possible that in 2021? This is a country that has been dealing with insurgency for more than a decade in 2021 We still have a situation where students in the hundreds can just be taken away from a school And there's no checkpoint. There's nothing that stops them from being taken away We then start to you know negotiate or find ways to do you know have a rescue efforts You know to bring these kids back it makes absolutely no sense But imagine this happened imagine a hundred kids were kidnapped in the UK in the US in 1930 today you would not be able to try it will be almost impossible because they would have improved with regard to intelligence and security and You know better ways to protect school children, you know by one year later The minutes of sports Sunday there I hope that we can bring this up and one bring this up is just to show that over time we always fail to Look for excellence with the people that we put into into office look at what is happening with Nigeria sports perfect example men and female Athletics Teams are not getting into the Olympics We have that and we have a minister of sports at the end of his time as minister of sports I don't think there's anybody who's going to question what he did and what happened to Nigerian sports while he was there Nobody's gonna ask all those questions. They will simply say oh is not is in on Nigeria would laugh over it and we'll move on But there's the person whose responsibility it is that Nigeria sports should be seamless and we should be able to break boundaries You we should get the best out of our athletes, but nobody's going to ask him why we have failed to make it to the Olympics Nobody's gonna ask why don't have the best athletes in the Olympics and that's the same thing with every other MDA in Nigeria today There would only we're even speaking about Ibrahim Magu being given appointment again Well, all right, so that's that wraps it up for today in history talked about 2005 attack in the UK and the 1930 construction of the Loverdam. Let's take a break here We'll return to talk more about security


Can you hear me all right? Fantastic, okay. Thanks very much for the introduction and thanks for the invitation also. It's really a great pleasure to be able to speak here. So this course will be a short course, so there's only three lectures. And the main purpose is to give an introduction to the foundations of the theory of rigid co-cycles. And it's used to give an attempted construction of singular moduli for real quadratic fields. Now, this theory hasn't been around for that long, only a couple of years. And what I'll do here is I'll give a sort of friendly introduction to the foundations of the theory, which can really be made very elementary and very computational as well. Now, this is in sharp contrast with the increasing number of theoretical results that are emerging also in this area, which in contrast have kind of formidable prerequisites to appearing in their proofs, and in particular the proofs of special cases of some of the observations that we're about to make in this course. So instead, we'll take a very kind of elementary and computational approach in this course. The main idea of course being to embody as much as possible the spirit of this summer school, which is number theory informed by computation. And I very much hope to make it sufficiently concrete and computational for you to apply everything that you've learned so far in this summer school, all your skills with your favorite computer algebra system and really engage with this theory in a very concrete and examples-based way. To help you do that, there's a series of exercises which you can find online and we have a problem session, first problem session later today. So the official TA for that is James Rickards. I don't know if he's around, if you can raise your hand, yeah, there he is. He'll be leading those sessions. There's also my student, Horvak Damjohnson, who knows a fair deal about these computations that go into it. Is he here? I don't know. Oh, there he is, yeah. So those two people will be around and you can ask them lots of questions. And I'll be around also and encourage you to do so. Okay, I think that should cover the practical side of things, if there are any questions, please interrupt me. So today will be mostly motivational and including also some very classical background. And on Thursday and Friday, I will start in earnest with the rigid co-cycles. And I would like to begin today by discussing the first two words here in the subtitle, which is singular moduli. These boards really vibrate a lot. Okay, I'll do my best. Okay, so the values of the modulo j function, which has appeared many times already, I think in other courses, which are called clients j invariant, which is defined by a very explicit Q expansion. So this is n cubed Q to the n divided by one minus Q to the n, where n is greater than or equal to one. You'll recognize the Eisenstein's using weight four here and I'll cube it and divide it out by the following infinite product, which defines the Ramanujan delta function. If you expand this out as a Q series, you obtain one over Q plus 744 plus its magical number 196884Q, et cetera. And I can view this as a function on the Poircari upper half-plane in the variable Q in e to the two pi i tau. The sentence starts with the values and I'll be interested in very specific values, namely those values of tau in the upper half-plane, which satisfy a quadratic equation with integer coefficients. These are called CM points. So these are points in the Poircari upper half-plane, which I'll denote by age sub infinity, which is just a set of all complex numbers whose imaginary part is positive. So what about these values? I'll claim that they are arithmetically rich, which is not a mathematically very precise term. So let me try and point out a number of ways in which they are arithmetically rich, perhaps guided by a few basic examples, most of which, I mean the most standard ones. Or if you start by evaluating the j function at i square root of minus one, there's a unique such square root in the upper half-plane, you get the number 1728, which appeared in many talks already, as well as the j invariant of a cube root of unity, which is zero, perhaps a slightly more interesting, randomly chosen example, is the j function evaluated at the square root of minus 15, and I'll have to look at my notes, which I computed for you as minus five squared times three cubed times this magnificent number, 637 plus 283 square root of five, divided out by twice the square root of five, okay? Minus 15, sorry, this is not very clear. Thank you, minus 15, exactly. Now there's two things I'll note about this number. It's an algebraic number. It's an algebraic number that's defined over a different field than the argument that we fed into the j function. What that field is, I'll tell you in a second, and you probably already know, but I would like to point out two things. The first is that we can compute its norm, it's an algebraic number, and we can factorize that norm. What we get is minus three to the power of six, five cubed, 11 cubed, this is a cube. So we obtain an integer, that's an algebraic integer, if we take its norm we get an honest, rational integer, and if you factorize it, we find that it's extremely smooth. It's divisible only by very small prime numbers to some large exponents. I'll also mention, because it'll come back potentially later, that it also has a trace, and that trace, well you can kind of read it off from this, I guess it's minus three cubed, five squared times 283, which happens to be a prime number. I'll get back to these two observations and why I bother telling you this about this particular j-value. Now, classically, the reason people were very interested in these j-variants, oh yeah, absolutely. I have a tendency to write very small, so please don't hesitate to remind me if I exhibit some recidivism. So classically, and I'm talking now sort of late 19th century, they were notable because of their ability to generate ring class fields of imaginary quadratic fields. And class field theory was in full development in those days. Putting the singular module, I really center stage around the turn of that century. In this particular example, the relevant ring class field is just the Hilbert class field for the imaginary quadratic field q join the square root of minus 15, which is a field of definition of the arguments that we plugged into the j function and out came this number, which is contained in the Hilbert class field, which in this case is a genus field, it's a bicodratic field obtained by joining the square root of minus three and the square root of five. Indeed, it generates that field over the imaginary quadratic field. This singular modulus does. So this is classically why people were very interested in the singular moduli. And for a long time, I think, certainly up to and including World War II, people had the feeling that this theory was really satisfactory concluded. Now, huge renaissance for singular moduli happened much later. And this came with the very celebrated work of Gross and Zagier. Their first paper together dates from 1985. And what they did is something that looks very strange at first sight. They say, take one singular modulus, J tau one, that gives you such an interesting algebraic number that is a generator for a ring class field. And now let's take another one of a discriminant that has nothing to do with the first one. So J tau one minus J tau two, where the discriminant of tau one is less than zero and the discriminant of tau two is also less than zero. And in the paper of Gross and Zagier, the original paper, they required that these were co-prime and fundamental. What they did is this is an algebraic number, an algebraic integer. So you can take its norm and that gives you an honest integer. And what they do in their paper is they give an explicit formula for what this integer is. Looks a little bit bizarre at first sight, but what is so fantastic about this discovery is that it led to really deep and very important developments and the foundations of these discoveries. I mean, if there's anything that screams the theme of this conference, I think it's very much the origins of this work of Gross and Zagier. And fortunately, Zagier had some significant birthday, I forget which one, a couple of years ago. And Dick Gross gave a talk on the occasion of this birthday and he made public also the letter that he received from Zagier in 1983, announcing some of these first discoveries. And this is such a wonderful document. So Gross says the following. He says, singular moduli were studied intensively by the leading number theorist of the 19th century, as we remarked. They're algebraic integers, which generate certain abelian extensions of the imaginary quadratic fields. The theory was believed to have been brought to a very satisfying completion in the early 20th century. That was before Don got his hands on it. In early 1983, Don sent me an amazing letter from Japan containing a proof of a factorization formula for the integer, which is the norm of the difference of two singular moduli of relatively prime discriminants. This was a completely new aspect of the theory which Don had discovered by extensive numerical experimentation. So that's very much in the spirit of this conference. And here you can see this letter. It's very nice. I recommend anyone who hasn't seen it to look at it. It's really fun to read. And I'll read sort of the first page of it. And you'll see in the exercises, the first exercise really is to try and recreate for yourself this moment of doing these numerical calculations and to actually spot some of the patterns without looking at the letter first and try and relive this moment, which seems like a very frivolous exercise. It's very ill-defined, trying to find patterns and see if they match up with patterns that someone else found 40 years ago. But it's very instructive because later on in the third lecture, we'll do precisely that, but in the setting of real quadratic fields. We'll try and mimic as much as possible that initial process of being guided by computations. So this is what Zagir writes. Dick, he says. I've been in Japan for two weeks now and I'm enjoying it tremendously, both for sightseeing and mathematics. However, telling you about the trip can wait till you get to Germany. I'm writing now for mathematical reasons only. Yes, as you may remember, I had asked you whether our results might be generalized to results on the norm of the difference of singular moduli for arbitrary CM points, tau one and tau two, with unrelated discriminants. You poo-pooed the idea, explaining why your method only applies to elliptic curves having CM by the same order. Not daunted, actually I was, I didn't do the calculations till now, I calculated the difference of singular moduli for many different examples of class number one. A somewhat tricky business since my HP has only 10 places. And I found the values and then here's some big table. Now already, I am just mesmerized by this letter because I don't know, I wasn't born, but in 1983, if you were on a trip to Japan, I don't think people had laptops, so this HP with the 10 places he's talking about, I could only surmise he's talking about a calculator with sort of 10, I don't know, but this three years, this is amazing, that he had a calculator with 10 places and worked out all of these tables. You have one? Fantastic, okay. So if you really wanna be hardcore, you should do the first exercise for this course with only that HP. That's for the really hardcore people. Other people can use a laptop and that also makes the exercise, I think, quite a bit more palatable because it doesn't sound like the most fun obstruction to put yourself under. Okay, now I wanna highlight one table. So Zagyyei computes many things in this letter and I'll just pick out sort of one example that we can stare at and try and explore a little bit together. It's the J value at one plus squared of minus seven over two and I'll subtract the J invariant of one plus the squared of minus 43 over two. Now, see, M theory tells us that these are both integers. So the norm is not necessary, it's already an integer and that integer, when you factorize it, is three to the six plus five cubed, so times five cubed rather, times seven times 19 times 73, okay. Again, very smooth number, which is quite remarkable because the J invariants themselves, as we already noted here, maybe this norm, they tend to be very smooth themselves so we have the difference of two smooth integers which is again very smooth. The ABC conjecture tells us that that sort of stuff shouldn't happen very often. Luckily, there's only a finite number of examples here so we can still believe in ABC if we want to. Now, this is what Zagyyei does, is he makes a little table and here, I mean, of course, he had some foresight so this is the part that will seem a little bit strange but he had good reasons to be interested in those particular expressions. So what he's going to do is he's going to take X to be one, three, five, seven, nine, that's probably enough and then let's make another one here of X going this time 11, 13, 15 and 17, okay. Now the expressions that he's going to compute for each of those X is the following. So it's seven times 43 which is the product of the two discriminants of the CM points that we plug in, minus seven and minus 43 but that's the same, you subtract X squared and you divide it by four and he's going to list all of the positive integers that are of that form, right. So this can only happen for odd Xs in this range so those are the only examples where this expression is a positive integer. So he computes those. So for instance here we get seven times 43 which is 301 minus one squared which is 300 divided by four which is 75, okay. So 75 is three times five squared. I'll just fill in the table and you look at the numbers and see if you notice anything suspicious. So this is a table, okay. Who's willing to make a conjecture? Observation zero. Every prime factor that arises that divides the difference of these two singular moduli seems to arise somewhere in this table, correct? That seems to be true. But conversely that's not such a hot statement it seems. Some of the prime factors that appear in this table do not appear on the left. Oh, there's something wrong because I probably made a horrendous mistake. Three squared five, thank you. Yeah, this is a problem with a board token for this reason I want to show you a huge amount of numbers later on in the third lecture. There'll be slides. There's something else wrong. 13, I wrote, yes, this is a three. I can claim that I meant to write a three but this is just a mistake. This is a three. Thank you. Let me scan again while you observe. Now what Zagir goes on to do in his letter and he describes his thought process which is very interesting. He's trying to figure out which primes in this table actually do arise and to what multiplicity they arise. He kind of walks you through his thought process. He comes up with a formula and then he proves it in this letter. He proves the formula exactly and this first exercise is to try and do the same because by doing it I think you'll have really the right reflexes when we get to the real quadratic, okay? So I highly recommend doing that particular exercise. So that's what happens in that letter. Now the rest is kind of history because Zagir challenges Gross in the letter to find a proof that's different from his proof because his proof uses, it's a very analytic proof. It uses families of Eisenstein series. It revolves around the families of Eisenstein series that was written down by Hecker about 100 years ago and he challenges Gross to find an algebraic proof which Gross does. So they give two proofs in their first paper and later on they combine the two into the contributions of the Archimedean and the non-Archimedean primes to a height pairing of Hegner points and these results they led. Let me just say, and I'll say more about this in the final lecture, to progress on the Bach and Swinnett of Diaconjecture. Relating the heights of Hegner points to the first derivative of the central value of the L function of an elliptic. So this is Gross Zagir, 86, and then there's also Gross Kronen Zagir, 87. Yes, some question. Bigger, yes, thank you. Okay, so the goal, this course, will try to do a very similar kind of experiment and to try and construct also by analytic means for the periodic analytic means. In variants that look very similar to the differences of singular moduli at two CM points where we replace these two CM points by a pair of RM points. So real quadratic fields instead of imaginary quadratic fields. And so this course will aim to discuss the foundations of this theory. The real quadratic analogs. And this is the subject of rigid co-cycles and everything that I'll mention is joint work with Harry Darmel. All right, before we do that, it's important to also mention that we're certainly not the first ones to try this. And there's very, there's alternative approaches that I should mention that you may have heard of, but they yield a very different set of invariance. So other approaches have been explored. Of course, very famously,


Have you had a chance to watch that ground ball that you went home with in the ninth? Yeah, watch it. I don't know. I mean we're in there to stop the the corners are in in order to stop the runs from scoring. The only ball that really that I was going to go to second base with at that moment was a player like a one hop ball at me that had enough steam with knowing how well he runs. It's going to have to be a perfect throw to second, perfect transfer, perfect throw to first base in order to get him just for the respect that we and that we know how well he runs that. And you know we were willing to take the out at home right there. That's why we were playing in to get the out at home. Overall, cocking was it being out on that field for the way everything unfolded in the ninth?  I mean we've been very successful for, you know, a large part of the season. Obviously the last few games is not indicative of who we are. It happens. It's baseball. When it was Garrett, you're staying right there at third base. How dominant was he? Yeah, I mean he made it look easy tonight for sure.  That's the second great game losing streak of the season. How unusual does that feel? I mean for this season it definitely feels a little bit unusual which I think that's a good thing. It just tells you how well that we have played. You know not to take anything away from our opponent. They battled in the ninth inning and you know we're able to put them in their selves in a position to win the ball game and they were able to come through.