Amen. So Deuteronomy chapter number four, we're in American heresy tonight. Before we get into the topic for this evening's sermon, Deuteronomy, this is Moses speaking here. So Deuteronomy is basically Moses giving, he's been speaking for several chapters and he's going to continue speaking to the children of Israel. It's basically, Deuteronomy means the law again. So he's basically giving a final warning to the children of Israel before they go into the promised land. So he's not, as he says in Deuteronomy chapter four, he's not going with them. We know why that is, but he's not, you know, God said that he's not going to go over the Jordan. That's when Joshua took over and we see the book of Joshua detail, kind of the handoff there and Joshua taking the children of Israel into the promised land. But I mean, there's about 50 sermons just in this chapter alone, but Moses is giving some warnings here to the children of Israel before they go over into the promised land, even though Moses is not going with them. So he's kind of giving his final warning to the children of Israel. Look down at verse number 26 before we start the sermon tonight. I mean, just a couple things here. I mean, I don't know how people can miss that, you know, the children of Israel being in the promised land was a strict condition on them following the word of God, the statutes that Moses is literally giving them again right here. Okay, look at verse 26.      All right, because what was the thing that allowed them to get the victory when they went into the land? It's that the Lord fought for them. All right, whether the Lord was on their side was the variable that meant victory or defeat. All right. So look at verse number 27.  He says, if the Lord shall scatter you among the nations, he shall be left few in number among the heathen and whether the Lord shall lead you. So you're going to be amongst these powerful people. He says they're more powerful than you are and you're going to be there few compared to them. And then it says, and you shall serve God's lowercase g. It's like, look at this curse that's going to come upon them. The work of men's hands, wood and stone, which neither see nor hear nor eat nor smell.  You're going to become like these people that just worship rocks, you know, and just, you know, that's one of the, the, the ironies of mankind right there. Or that's the, the funny things about mankind is left alone.  So tonight's topic of the American heresy series is made the main topic of tonight is going to be the religion of Hinduism. All right, Hinduism. Now, Hinduism, you're going to see, I'm going to talk about Sikhism and Hinduism together. But and I'll kind of point out the differences. And I was actually going to do a separate sermon on Buddhism. But it's so much the same that I think it would just be a huge waste of time and kind of get boring. So I'll try to just touch on that a little bit tonight as well. But one of the claims about Hinduism, and I think that it's interesting that the Bible brings this up is one of the claims about Hinduism and their scriptures is that it's one of the oldest religions on the face of the earth. Agree. Agree. Because it says old as idol worship itself. So what you can see here from this curse that this curse that Moses is warning them about, it's saying what is he saying? If you abandoned the Lord, the one true God, you are going to end up worshiping rocks and trees and moons and stars and all this. You're going to end up worshiping the creation, not the creator. It's a curse. It's a curse. So we could just deduce from that curse right there that as long as men have been abandoning the Lord, this type of thing existed. So you can see why Hinduism and this this religion that has so many different gods and just lower case G was what we would call those gods that are that are not gods. They're just works of men's hands. They're just paintings. They're just things that men have invented themselves. It's literally a curse from the Lord God Almighty because they've abandoned him and they've turned to these things. So Hinduism's old. Agree. It's as old as man. You know, it's as old as Cain. It's as old as man abandoning the truth, the truth of the one true God. So we're going to talk about Hinduism tonight. There's a lot of Hinduism in our area here, especially in West Fresno. There's a lot of Hinduism. More so in this area, it's Sikhs, not so much Hinduism, but I'm going to kind of explain to you the difference. So Hinduism is kind of that you can kind of look at it like Hinduism is is the Catholic Church and the Sikhs are like the Protestants. You can they came out of you know, the Hindu religion. So that's why you'll see so much similarities between the two. Alright, so look, Hinduism is generally this I just kind of break it down for you. They believe they believe in Hinduism can believe in many gods. Most Hindus believe in many gods. And but there's one big God called Brahmin where that this God is not he's not everywhere and all powerful. He is everything kind of thing. So they kind of create this one main God that is the creation. Whereas the Bible is very clear that God is not the creation. He is the creator. He made the creation. Alright, the scriptures of Hinduism are called the Vedas. That's their their their scriptures that are going their claim to be so old. And look, I don't have any problem with that that they're old. They're thousands of years old. Another concept in Hinduism that's kind of a main construct is Dharma. I don't know if maybe you've heard that in pop culture this word Dharma before but that's kind of the the ethics of the Hindu culture.  You should perform the moral law. It's basically this duty the Dharma to Hinduism is this duty to do the right thing. And you're going to see look I'm just going to you already know but I'm just going to give it away. It's just super works based. It's like that's all it is. It's just works based but there's two things that I want to point out that are unique about Hinduism that flow into Sikhism and Buddhism that we have not talked about before. So they do believe the Hindus believe in the immortal soul. Okay, they believe in the immortal soul and you're going to maybe I should have done this one before Scientology but they believe in the immortal soul in the sense that your soul is immortal and just reincarnated into another being or another creature. Okay, then they believe in the moksha which is basically the the the ultimate immortal state. You know this is if you've ever heard of Nirvana not the not the rock band okay not the wicked evil you know band from that was that the 80s or 90s that people still wear t-shirts today. I mean like the guy committed suicide and like was devil worshiper and it's like what in the world but anyway not to but Nirvana is the Buddhist state of moksha. It's like you've you've done so good in your life that you finally get to this state where you are one with God and you know it's the ultimate state like you got you know if I do bad as you know as a as a Buddhist you know I'll come back as a bug or something bad right and you know if I do good I'll come back as something better and then I just get better and better and better and better and then Nirvana. Alright so Hinduism is the same thing but Hinduism is kind of the original gangster of this theology Hinduism was kind of the the start of it and Buddhism came out of Hinduism I think it was like the fifth century Buddhism came out of Hinduism so that you know the guy that became Buddha I can't remember his name but the guy that became the Buddha you know he of course he got to the he got to that level right it's kind of like it's kind of like Elron Hubbard right I mean are you noticing a pattern here is like you get to level eight in Scientology and they're like oh yeah you know you're still all messed up and you're like alright well where's level nine I'm here to write another check and they're like it doesn't exist but Elron Hubbard got to 15 like what so you know you got these religious leaders in these you know false religions that achieve these high states and that's the idea so they do they will say they believe in the immortal soul but they believe in reincarnation all that to say that we'll talk about that a little bit more and the idea of reincarnation is to reach this top state where you become one with the the Brahma God whatever you want to whatever you want to call it all right now Hinduism originated in you know northern India and it you know thousands of years ago I guess and Sikhism in the 15th century Sikhism what we see around here that came out of Hinduism in 1500 and something okay so that's where you know Sikhism came from now Sikhism has a little bit of a couple of twists on Hinduism the main difference between Sikhism and Hinduism is that Sikhism will claim to be monotheistic so they're they're going to be like no there's not all these different gods there's only one and that's it but they're still very like all inclusive and universalist if that makes sense so there's standing from the Bible looking at both of these there doesn't seem to be a lot of difference from a biblical standpoint but I will say this just just having soul winning experience with Hindus and with Sikhs Hindus are more likely to let you preach the gospel to them because because of the fact that they believe and I'm not saying that they're more receptive to get saved I mean I've seen I've seen Hindus saved we have seen Hindus say but I'm not saying they're more likely to get saved they're just they're more likely to listen because they believe in this plethora of gods they kind of believe this idea and they'll even tell you at the door they'll be like oh we believe that all you know it's it they kind of believe this philosophy that we all call God a different name you can have your own gods and I can have my own gods and like Hindus some will have six gods and they say that there's up to like 33 million different gods that you could have I don't know you know where they come up with that number but the point is there's no set amount of gods so that's why you'll see Hindus that are just like open to hearing about Jesus and the gospel and hey if somebody is open here in the gospel I'm going to tell it to them all right so but there are some things that we need to kind of check you know some boxes we need to check when we know somebody comes from this type of background all right all right so 15th century we see Sikhism come out of you know it's kind of the the reformation of of Hinduism right and there was a man named Guru Nanak so if you've ever heard the term Guru you know Guru's comes from this idea of this religious expert or this religious teacher in Sikhism or Hinduism and this one named Nanak started the Sikh religion all right the Sikh religion has throughout history said 10 main Gurus I guess but basically Sikhism is this the fundamental beliefs I'm just going to read this to you of Sikhism are articulated in the sacred scripture you know Guru Grant Sahib include faith and meditation on the name of one creator so it's monotheistic all right but then it teaches this and you compare this with Christianity unity of all humankind engaging in a selfless service striving for social justice and for the benefit and prosperity of all an honest conduct and light and livelihood while living a householder's life so this Guru is worshipped as the Supreme authority of Sikhism and considered to be the final and perpetual Guru of Sikhism and as the first Guru of Sikhism this Nanak contributed a total of 974 of these Proverbs or hymns or whatever they want to call it but anyway the point is it was invented in the 15th century by this guy and the main difference is that it's monotheistic versus polytheistic or you know whatever however many they want but let me just say this and many of you probably thought this when I talked about you know the Scientology last week where it was invented like in the 50s you know a religion that is invented in the 1500s you have to ask yourself and I don't understand why more people don't ask themselves well what about the people before this all right this is why it is so important that you understand that Baptists are not Protestants Baptist we did not we did not if you read any Wikipedia article or anything on Baptists and most things on the internet about Baptists they're going to tell you that Baptists came out you know of the Reformation in 1600 something or whatever the point is Baptists are not Protestants and this again is why that you know it's really why doctrine is so important as well I mean there's many reasons doctrine is important ultimately because it's the truth but doctrine is so important because we could never see the trail of Baptists if it wasn't for doctrine because Baptists you know we call ourselves fundamental Baptists today but Baptists have had many names you know throughout history and down to you know Jesus Christ you can definitely trace the doctrines that we hold to in the Bible back to groups of people back all the way back to Jesus Christ and this again doctrine is important and it's also why the trail of blood is so important because these trails these people these wall denzians and all these different names of people that that you know held to these doctrines that we hold to today that many times their doctrines and their history was written down in terms of their persecution and that's why the trail of blood and doctrine is so important because look if I'm following something that came out of a truth that came out of the 16th century my question is well what about before that if this isn't something we're not following something that was invented by some guy that was invented by Martin Luther you know a few hundred years ago we are following the doctrine of John the Baptist we are following the doctrine of Jesus Christ we are following the doctrine of the apostles as they're written in the Bible and the trail of blood and those doctrines that those people in that trail of blood were persecuted for they document the proof of that all right they document that history but that's why it's so important you know that's one of the reasons that you know once we start giving up doctrine think of it this way once we start giving up doctrine we think about the trail of blood and you think about all the different martyrs throughout history all the way back to Jesus Christ and you think about how we are connected to all of them then you think about if we would start giving up doctrine today and guess what you know what would you become unsaved if you just like put your head down and just like you know compromised on all your doctrine no you've not become unsafe but you know what you would be unrecognizable to history we as a church as a Bible believing ministry would not be able to be recognized by the people two hundred years down the line when they look back and say where were the Christians then oh well I guess they just had all their their heads under their their desk or they were just hiding at that time but that is how and that's why Hebrews eleven is in the Bible that's why Hebrews 12 verse number one is in the Bible so we can look back at these martyrs and at these persecutions and see you know what there was the doctrine you know the martyrs mirror is all people being bad or being being persecuted basically for being Baptist for refusing to add even baptism to the gospel to refusing to add anything to the gospel but that's why it's so important so anyway all that to say this Sikhism was invented in the 15th century now let's talk about the aspects these two main aspects I want to talk about tonight about Hinduism and Sikhism that shares okay so the soul is immortal but it's only immortal through reincarnation alright so being reborn you know you you live a life and you live this works-based life and if you do good you're reborn into something that is closer to God closer to divinity or you reborn into something that is further away alright Sikhs believe that heaven and hell are also both in this world where everyone reaps the fruit of karma they refuse to refer to good and evil stages of life respectively and can be lived now they refer to good and evil stages of life respectively and can be lived now and here during our life on earth alright so look you according to karma karma is this idea that you reap what you sow and we'll get there in just a minute but first of all you're just reincarnated again and again Hindu deities are the gods and goddesses in Hinduism deities in Hinduism are as diverse as traditions and a Hindu can choose to be polytheistic pan theistic monotheistic you can about tell where a lot of this stuff goes to alright this is why you see a lot of people that are in the reprobate culture today that are you know just like they just hate the God of the Bible you see that a lot of these people like embrace things like Buddhism embrace things like Hinduism so like I said in Hinduism there's at least six gods but you could max out at 33 million alright but basically if you go down to Deuteronomy chapter 27 or Deuteronomy chapter 4 verse number 27 we see that there's nothing new under the Sun it says in the Lord shall scatter you among the nations and he should be left few in number among the heathen whether the Lord shall lead you and there he shall serve gods the works of men's hands woodstone which neither see nor hear nor eat nor smell so the idea here folks is that all these idols and all these gods they're not gods I mean you know this but they're not gods they're they're dumb idols meaning they can't hear anything they're just a piece of wood just a rock and you're gonna end up you know a curse from God for not following him is to become an idol worshiper I mean that's when you just think of the logic of it it's if it wasn't a curse from God it would be really hard to understand I mean could you not I mean it would be really hard to understand how you know you just imagine God looking down here and seeing you know people abandon him and then just on their own worshiping the dirt or worshiping the rock or you know a cloud or the Sun or whatever I mean it's a very strange thing it's one of the strangest things about man but really we see here kind of it's explained it's a curse from God all right so now let's talk about this reincarnation this immortal soul of the Hindu of the Sikh and really of the Buddhist as well and it's really based on this idea of karma okay I'm gonna read for you a definition of karma out of you know some Hindu explanation website I can't remember what the site was but it says basically karma you have to think about it this way so here's their words karma is a concept of Hinduism which describes a system in which the beneficial effects are derived from past beneficial actions and harmful effects from past harmful actions creating a system of actions and reactions throughout a soul throughout a soul's reincarnated lives forming a cycle of rebirth so again you live a bad life and you get reincarnated into something bad I couldn't find like a scale of things I was really looking forward to seeing something like that I couldn't find anything like that but works-based you know works-based religions never have scales like that right if the Bible was truly works-based that's what we would have we would have scales we would have sin rankings we would have all these things because God doesn't want us just to wonder if we're going to go to heaven he wants us to know what it takes all right that you may know that you have eternal life so you're reincarnated according to how good you are now in pop culture you'll hear people use the word karma all the time what do they what do they use the word karma you see something bad happens to somebody that everybody thinks is you know a big jerk and they're just like yeah see that's karma well what they're doing is they're quoting some Hindu belief basically karma you know what it means it means you get what you deserve that's kind of the idea that it portrays all right but turn to Galatians chapter 6 karma is kind of a ripoff of biblical reality all right look at Galatians chapter 6 look at verse number 7 I mean karma is basically saying you get what you deserve the only difference is is that you know they add this reincarnation and say you'll you'll pay in another body or another life or another lower life form or whatever that is Galatians chapter 6 of verse number 7 the Bible says be not deceived God is not mocked for whatsoever a man soweth that shall he also reap in second Corinthians chapter 9 go ahead and turn to second Corinthians chapter 9 second Corinthians chapter 9 this is more of a application of this to the Christian as Paul is talking to the the church at Corinth he says but I this I say he which soweth sparingly shall reap also sparingly and he which soweth bountifully shall reap also bountifully so this is kind of this concept applied to the soul winner right here all right so this concept is like hey you never go soul winning you never get by saved hey if you don't you know live the Christian life you know you're not gonna get bear fruit in the Christian life but if you go out there you just so like all the time is like you are going to see fruit that's a nice little guarantee for the Christian right there you go and you do what God wants you to do you do what you're told in the Bible and you're gonna be what you're gonna be extra save no you're saved just the same you're gonna bear a lot more fruit you're gonna reap a lot more you're gonna you're gonna if you sow bountifully you will reap bountifully but the whole point is this whether in this life or the next in Galatians chapter 6 7 is saying you know whatsoever man a man so if that also shall he reap in this life for the next the Bible says that everybody's gonna get what they have come into him the Bible is very clear about that but the problem is is that the Bible teaches that everybody's going to be judged everyone's going to get what they have coming to him the saved is going to get what they have coming to him on this earth God's gonna punish you on this earth you're gonna you're gonna reap that punishment you know in your life here on earth if you don't do what God says but like everybody else they just get off the hook you know the Bible is very clear and a lot of people have this question you know they're just like especially when you tell them that salvation is free you know you have to do is trust in Jesus and you'll be saved for eternity you're sealed you can never lose that salvation a lot of people have a hard time with that because they're just like well why does God just let all these bad people get away with all these bad things it doesn't seem fair well yeah it is true that the Bible or that the people in this life that are not saved wicked people that hate the Lord they're gonna get away with a lot of stuff here on earth but everyone will get what's coming to them the difference between what the Bible says and what the Hindus and the Buddhists and the Sikhs teach is that everyone that you know deserves punishment that is not saved is going to pay in hell not in the life of a puppy or whatever all right you're gonna pay and look you're gonna pay the reality of the Bible is much much worse than anything that this religion could possibly eat you because look it's called it's literally called ever lasting destruction and first Thessalonians it's destruction that will last forever there's no you're there's no second chance at it you know I mean basically reincarnation it the problem is is it I mean the problem is it's not true but I mean it gives people this idea that no matter what maybe I don't need to seek truth because if I'm wrong I can just try again there is no try again there is no try again that's why the in Luke chapter 16 that story is in the Bible with us I mean if the guy the rich man in Luke 16 he's like hey I want to get out of here it's like sorry you're still you that's it you cannot pass from there to us or us to you we can't come get you and you can't get out go tell my family I already gave him the Bible they wouldn't listen even if you came back so look there is no second chance it's it's it's extra wicked in that case that it yes it's a works-based religion like many all other religions other than true Christianity are but it teaches this idea that you're gonna get another shot at it and another shot at it and another shot at it and that is a super wicked seed to plant in people's heads because you know what they won't be seeking the truth you know that you know what that is that Satan trying to take away the urgency it's Satan trying to take away the fear of the wrath of God and guess what the fear of the wrath of God is hardwired in but religions like this will cover it and they will they will block that fear that God wrote in your heart turn to Hosea chapter 8 turn to Hosea chapter number 8 Hosea chapter 8 the Bible says look at verse number 7 Hosea chapter 8 Hosea chapter 8 look at verse actually go to verse number 4 Hosea chapter 8 and verse number 4 the Bible says they have set up kings but not not by me they've made princes and I knew it not of their silver and their gold have they made them idols that they may be cut off thy calf osomeria has cast the off mine anger is kindled against them how long will it air would be ere they attain to innocence see far from Israel was it also that the workman made it therefore it is not God but the calf of Samaria shall be broken in pieces and then here we see in verse number 7 it says for they have sown the wind and they shall reap the whirlwind it at no stock but the bud shall yield no meal so if it be yield the strangers shall swallow it up talking about a nation that turns against the Lord here they've sown the wind they're going to reap the whirlwind nations are judged on this earth it's interesting though that it's a calf they're always worshiping a cow why is that now I actually looked up and Hindus don't actually because a lot of people say oh Hindus you know they don't eat beef I mean most of the Hindus that I've worked with they they literally they don't eat they don't eat beef at all but I guess they don't really you know actually worship the cow it's just this sacred animal to them I'll read you a little quote here it says Hindus see the cow as a particular genus generous generous docile creature one that gives more to human beings than she takes from them the cow they say produces five things and they're really big on these five things milk cheese butter so those are the good things and then urine and dung those are the bad things right those are bad things the first three are eaten and used in worship of the Hindu gods well the last two can be used in real religious devotion and in penance or burned for fuel so it's talking about how they will use so that this is why they they reverence the cow so much I just think it's interesting that people end up worshiping cows in the Bible all the time all these different gods bail and all these other gods are there always there was a calf there was a cow it's almost like the same person is behind it you know what I mean so here we've got this idea that cows are sacred they produce these things I remember during I don't if you all remember this but during the COVID-19 thing there was a big thing in India and of course it says here that you know all of these things are used by Hindus even the last two the last two can be used in devotion or penance ceremonies I remember there was some of these gurus I don't remember if they were seeker Hindu but there is these gurus in India and there was this big like like thing where they're all worried about the health of people in India because they were smearing themselves with cow dung to protect against you know COVID-19 right and I'm like okay you know so there was like there was all these articles about these people in India doing this but look I mean that the Hindu majority people in India use cow dung and urine for their wellness and cure of illness since ancient times it's a normal thing that they've done because this cow is a sacred thing I mean any time I've been covered in cow dung it's been on accident all right it's it's been not a you know not something that I wanted at the time all right see the sermon from this morning all right but that's the whole thing on cows right they don't technically worship the cow but you know it's this high thing for them and most of them do not eat beef all right so the first thing that I really wanted to point out about Hinduism was this idea of they'll say oh we believe the soul is immoral but it's this idea of reincarnation that is driven by this machine karma so they believe they're like karma is kind of like this is my words not any any Hindus words but karma is like this machine in motion so it's kind of this thing that just automatically handles you know the reincarnation cycle all right but the next thing and you will notice this soul winning the next thing about Hinduism especially is this idea of universalism and you could even see that in the the description even though Sikhism Sikhism was theoretically monotheistic you could see that it was very open to things and wanting to bring all humankind together and all these different things so it's this idea of universalism that you can just have whatever God you want and we all just call God a different name this all this idea came from Hinduism there's a lot of people who are not even religious that have this type of belief like all the Muslims because they grew up there and the Hindus because they grew up there and the you know the Chinese are into Buddhism because that's where they grew up and then you have the Christians over here in Europe and that's how they grew up they're all worshiping the same God they just call him different names okay that is one of the ideas of Hinduism and this is why you will see Hindus that are open to hearing they're doing it for two reasons when you walk up to a Hindu's door and they're open to hearing the gospel there's there's two possible reasons they're doing it number one they want to just add a belief they want to just add a God to what they believe and number two they're they're being nice they're doing a good work in in their in their mind okay they're being open to any form of God okay it's this idea that we all worship the same God he's just got different ways of revealing himself to different people and different cultures in different parts of the world turned to Luke chapter 12 but the Bible is the opposite and Christians need to remember this today this is what Christianity is losing today you know other than you know the true gospel even Christians that have the true gospel churches that are preaching the right gospel they're losing this divisiveness today they're losing this idea they're like oh we just all need to come together and stop you know we all need to forget our differences we all need to forget our differences and quit you know you know nitpicking on all these things like God the gospel and like repenting your sins aren't you kind of like you know splitting hairs on that one it's like no look what Jesus says in Luke chapter 12 I'll read for you John 14 6 which is everyone probably has memorized it says I'm the way the truth and the life and no man coming to the father but by me in Luke 1251 Jesus himself listen to this Jesus himself says suppose that I'm come together to get come to give peace on earth I tell you nay but rather division this is Jesus talking saying that to be a Christian to follow me you must divide Christians need to relearn this today Jesus divides people there is only we just studied in John chapter 10 there is only one way there's many people that are going to try to crawl up the building and go through the windows it's like no Jesus is the door and the only way is through the door Jesus is the only way so all these religions that we talk about all these religions whether they're inclusive or not inclusive of other religions all these religions that we talk about if they do not hold to the gospel of the Bible and someone dies in that religion they are going to spend in eternity in hell that is divisive that if you don't trust in Jesus Christ that's why you have to say when you're out and you're out preaching the gospel you're out talking to people and they say well faith in God how do you how do you know even heaven just have to have faith in God hey what is God's name does God have a name you get all kinds of different answers once you ask that question well it doesn't matter what God's name is yes it does matter the door is Jesus Christ there is no other name there's no other name all other ways lead to destruction that is a divisive thing but guess what we're not here to lie to people we're not here to tell people that oh it's great and and just you can believe whatever you want and you know you can believe in works and you can believe in repenting your sins and you can believe in worship salvation and you can believe that you know you can believe in tulip and you can believe in all these different things it's fine those are all you know whatever as long as you say Jesus in there somewhere you're fine no we're not here to lie to people we will divide people as we need to for them to know the truth because how loving is it to not tell somebody the truth and have them not be saved that's why Jesus said that I came to divide because you tell people that no it's not one of these 33 million gods there's one God and one mediator between God and man the man Christ Jesus that's it that's the only way that is divisive and people there's gonna be people that do not like that turn to first Corinthians chapter number two so look I mean it's not complicated it's I'm sure it is a lot more complicated if you want to dig more into it but it's a workspace religion it's based on reincarnation and the idea of karma Sikhism Buddhism and and Hinduism all the same because Sikhism you know came out of Hinduism in 15th century Buddhism came out of Hinduism way earlier I think like the fifth century so Buddhism is actually older than Sikhism but it's all the same thing it's do good and you're gonna get somewhere better it's all the same idea all right now look turn to first Corinthians chapter number one and I want to kind of show you something a little bit different here you hear in this sermon series one of the things that I wanted you to see in this sermon series as we go through a lot of these different heresies as many people involved first of all they're ridiculous you will listen to these descriptions of these different religions these pagan religions these cults whatever it is and it's just it's full of ridiculousness and but guess what you are not the only one that thinks that it's ridiculous it is not only the same Christian that thinks that that Scientology story that I told you last week is ridiculous but I want to kind of show you tonight how ridiculousness is part of the design of false religion it's baked into the cake by Satan the inventor of these false religions that's why they kind of all look the same not only are they all workspace religions every single one I preach on you're like oh it always comes back to that it always comes back to worst bakes work space salvation it's always the same thing but then you see certain things like the heathen the pagans worship in the cows worship in the calves all these different things their sacred in this one it's all because you know all you know airplanes look the same because they were designed by the same type of people it's it's just all designed by Satan but ridiculousness is baked into the cake on purpose say who would believe these wicked ridiculous things that you could be reincarnated into something else we even thought of this stuff but look here's what we have to understand folks a lot of people that are Hindus and Sikhs and all these different religions Mormons Jehovah's Witnesses whatever it is all these different ridiculous false religions a lot of these people are born into these religions just like I was born into the Lutheran religion so we always need to remember that like look we love these people we love these people that were born into these religions and we want to reach them with the truth of the Bible but the point is is that they have ridiculousness baked into their cake on purpose I mean just think about the reincarnation you think about the Dianetics book a psychological book written by a science fiction or author you think about that whole story you think about the the book of Mormon and the ridiculous things in the book of Mormon just like that there were submarines after the Tower of Babel and like just crazy stuff just ridiculous things you think about the Koran where it says the earth is flat Muhammad split the moon in two and then put it back together I mean just like all this crazy ridiculous stuff meteors or if you see a meteor it's because it's a star that was shot by Satan or whatever I mean it's something like that maybe I misquoted it but the point is they're ridiculous books with ridiculous teachings you say why is that it's the same reason for multiple Bible versions follow me here you say why are they all so ridiculous because Satan he doesn't want to just corrupt the people that were born into these religions he wants to corrupt people that were not in those religions from finding the truth so what did God do God gave every man no matter who this man is no matter what this person is when they're born they had this law written in their heart they had the proof of creation all around them in Romans 2 15 God wrote the law in their heart they have this desire to seek the Lord you don't believe me I just talk into the girls that went out soul winning today and they had this child following them around this eight-year-old child can you read me the Bible again can you read me some more of the Bible can you read me the Bible they just inherently know that the Bible is truth no one showed her anything out of the Bible before they just inherently know why because God wrote that in her heart guess what she's not saying can you show me the Book of Mormon can you show me the Shiva can you show me the the Dianetics book that Eloran Hubbard wrote no she's saying can you show me the Bible she doesn't know what's in the Bible her heart's telling her that the Bible is true and she wants to hear the Bible that has to be broken someone has to go out look and people are breaking it today but children are a perfect example of this so Satan makes all these religions ridiculous so people that aren't part of any religion that have that law written in them to seek truth in their life they see this ridiculousness all around them you know what they say then somebody comes up to them this is the Jehovah's Witnesses that are out knocking doors right before we get there they see this ridiculousness ridiculousness and then when the truth comes up there is like that's just more ridiculousness it's like spiritual wolf crying it's like the boy that cried wolf he cried wolf so many times when there actually was a wolf nobody wanted to listen it's the same theory these people the Jehovah's Witnesses they come up the Jehovah's Witnesses they don't have any answers you want to talk about ridiculous you read the history of the Jehovah's Witnesses it's one of the most ridiculous things you'll ever read leader of the Jehovah's Witnesses that that they deny Jesus Christ leader of the Jehovah's Witnesses builds a mansion and remember this one builds a mansion in Los Angeles and they've they've predicted the end times I think more than any other false religion out there like they've picked a date and been wrong so many times you're just like who's a Jehovah's Witness at this point who's following this point remember the guy in 1914 or whatever it knows 1920s he goes and he builds this huge mansion Rutherford I think his name was the leader of the Jehovah's Witnesses builds this huge mansion in LA it's on like a hundred acres and he's like this is for Abraham and Jacob and the patriarchs that are going to come back they're coming back soon we have to build this huge mansion for them imagine me standing up here and telling you this like what we do this huge mansion well I will live there until they they get here that's what happened I mean it's the it's the epitome people that aren't saved that are looking at this from the outside they're like you people are all nuts for following these people you must be out of your mind it's as bad as Xenu the alien dictator I mean it's just like people look at this and they're just like what are you nuts and then somebody knocks on their door that's when they put signs on their door that says don't bring any religion here because they're sick of the ridiculousness of it because they've seen so many ridiculous things and ridiculous people that are involved in all these ridiculous baked into the cake on purpose so somebody shows up at the truth and they're like man it's all crazy get away from me that's why that's why even people that the but the devil's like but there's still the Bible I still got these eight-year-old kids who want to hear the Bible what should I do I'll make a hundred and fifty Bible versions that's what I'll do and then every single logical person with a brain in their head will take a book a that's a Bible and book b that's a Bible and set them next to each other and see that they say different things and be like how could this be true if a doesn't equal be how could one of them equal and they're both the same how could they be true that's why there's a hundred and two hundred whatever how many hundreds of Bible versions there are because he still got this problem of the conscience that starts out with every man so he's got to corrupt the very Bible that people are going to be seeking and then you see what's corrupted in the Bible who Jesus was what's corrupted in the Bible I don't know the gospel itself what's corrupted in the Bible the divinity of Christ you look at all the things that are corrupted is all the things that would cause people to be saved and now on top of that you say well but there's more but there's more you see any person here's another thing any person can open up one of these religious books the Book of Mormon the Quran whatever it is and you could read the ridiculous things that I just shot off to you you could look at it you could read it yourself be like this is ridiculous and you just you read two pages of any one of them and they sound they sound you can just see someone wrote it it's I don't even know you just open it up yourself and you'll see you're like this is not a spiritual book but here's another problem well they can open those books and they can see the ridiculousness of those books look at 1 Corinthians chapter 2 in verse number 14 they cannot do the same with the Bible because the natural man can not understand what's in the Bible so if there is somebody that is telling them a false thing about what's in the Bible is very likely that they're going to believe it look at 1 Corinthians chapter 2 and verse 14 it says the natural man receive it not the things of the Spirit of God for they are foolishness unto him neither can he know them because they are spiritually discerned you say who could who could read the Bible and think that it's foolishness I have sat down with people who've explained that they can't understand Ephesians 2 8 through 9 to me they're like gift of God that's so confusing they're unsaved they're trying to understand it and then you sit down you have to explain the gospel to them that's how God designed it he designed it were someone who is saved would come down and use the Holy Spirit that is in them to preach the gospel to someone and then they get the Holy Spirit and now they can understand the Bible so it's not like they can just pick up the Bible like they could pick up any ridiculous book and see that it's ridiculous but if a false prophet is telling them that it's ridiculous that's why I mean that's why we are needed that's why we are needed to go out there and spiritually discern these things with the Holy Spirit in us it's really we're kind of like just a suitcase that rolls up to the door you know kind of carrying the Holy Spirit he's kind of the important one in this situation it's kind of like you know Jesus is the important one the Word of God and the Holy Spirit that is with us we're kind of like the suitcase that just kind of rolls up to the door but we're necessary that's how God designed it say why God designed it that way you ask him when you get to heaven but that's how it is so I mean look these ridiculous books they're providing cover including all the false Bible versions they're providing cover of the truth and that's a wicked thing about how ridiculous some so we laugh about it how ridiculous it is but it's really a wicked thing because it's spiritually crying wolf to the world and there really is the wrath of God out there it's a real thing and people need to know the truth all these things have no answers all these ridiculous things there's no answers there's no answers Hinduism has no answers JWs they don't have any answers so look one thing that all that to wrap this up let's take it back to Hinduism I kind of went on a tangent there but the reality of the Sikhs and Hindus to the soul winner is this because of this inclusivity of the Sikhs and because of this polytheistic universalism of the Hindus you may be allowed to preach to them you may be allowed to give the gospel to these people but what you need to be sure of is that they understand that it is only through Jesus that this cannot be a plus Jesus situation it needs to be you must repent of Hinduism and turn to Jesus Christ it needs to be a decision that has been made look and I've seen it happen typically with what younger people that I've seen younger people who are raised in a Hindu home save but they must understand that they can't just add this hopefully you mentioned that to everybody but they can't just add this to all their different beliefs but you know a lot of the time you're going to find out that they're just doing a good work listening to you they're just being you know that it's just another workspace religion and part of their good works is to include all ideas of God okay and look that being said if somebody is going to let me preach the gospel to them I'm still going to do it I'm going to preach the gospel to them but there will probably be a point where you kind of have to what you have to do that you have to divide so you preach the gospel and then you just make sure you give them that dividing line and then look it's up to them what they believe you're not going to be able to force anyone to believe anything so that's Hinduism it's not a huge mystery you know it's another it's a it's a workspace religion it's full of idolatry and it really pushes this same philosophies that the Buddhist and the Sikh push because Hinduism was kind of that the Catholic Church of it and then you had Buddhism split off and then you had Sikhism split off and of course each one of their their head you know it's kind of like you know John Calvin and Martin Luther and each one of their their heads I mean it can't you just see the same things and all this stuff each one of their head gurus or whatever it was achieved that high state that you know for some reason no one else is going to be able to to reach right because it's fake it's not real there is nothing reaching there there's there's Jesus Christ the door and everything else is damnation what's power heads and have a word


And thanks to Mike and Philip and Alina for putting the series together. It's a great pleasure to speak in the series. What I'd like to talk about today is some methods for determining rational points on curves and I'd like to illustrate each of the methods with A prototypical curve that might want to consider. And so all these techniques sharing common some chaotic methods and I'll kind of start from the ground up and build our way to some interesting curves that we'll look at. So the first question comes to us from Question about rational triangles. So we say that a rational triangle is one with slides that are all rational. So we can ask then if there exists a rational right triangle and a rationalized sauce least triangle that have the same perimeter and the same area. And this might seem like a question that the ancient Greeks looked at maybe one that We've known the answer to for quite a long time, but it turns out that this was just looked at very recently by two graduate students in Japan, my KO university here, Kawa and Matsumura in 2018. So this will lead us to a question about a particular genius to curve. I thought we'd take a look at their solution. So we'll assume that there is such a pair of triangles a rational right triangle and a rationalized sauce least triangle. And so we rescale and we can assume that their lengths look like the following. So for the rational right triangle to KT K times one minus T squared K times one plus T squared Where K is a positive rational and T is a rational between zero and one and for the rationalized sauce least triangle one plus you squared one plus you squared and for you for you between zero and one rational. Alright, so we've set up our two triangles. And now the question was, can they have the same area and the same perimeter. So we compute And we compare their parameters and areas and now we have two equations and three unknowns and KT and you and then we do a small change of coordinates and find ourselves looking at a genius to curve. So the question now about our rational triangles is about rational points on this genius to curve given by the following equation. Why squared equals this cubic squared minus 8x to the sixth. So here is our first curve that we'd like to look at. So what can we say about rational points on this curve. All right. So something very special happens and the shadow tea Coleman bound kicks in and it tells us that this particular genius to curve has at most 10 rational points. So this is incredible and I'll tell you where this comes from and why it applies in a moment. But we'll hold on to that for now. And since we know that this curve has at most 10 rational points. Let's try to find some rational points in this curve. So, you know, we search in a box and maybe write a little script to try to find some points of small height. And when we do that. Well, there are a number of rational points and in fact there are 10 rational points. So this is great. We've determined the set of rational points on this curve. So from the point of view of this question we've now found what we can say about our triangles. And if we trace through the different coordinates. We find that the one point with x coordinate 12 over 11 and positive y coordinate gives rise to a pair of triangles. So, we've answered this question about triangles using the shadow tea Coleman bound on this genius to curve. And this is the theorem of here a column at some of the that up to similitude, there exists a unique pair of a rational right triangle and a rationalized sauce least triangle that have the same parameter and the same area. All right. So, you might be wondering about the shadow tea Coleman method, which is what allowed us to compute the set of rational points in the previous example. So that's what I'd like to talk about next. So it implied for this curve that there were at most 10 rational points. And behind the scenes crucially, we satisfied an inequality between the genius of the curve, which is to, and the rank of the mortal V group of its Jacobian, which we were able to compute and find out that it was one. All right. And since that applied, we were able to apply some work of shadow tea that was later be interpreted by Coleman to get an effective bound that turned out to be sharp. And that's a bit of luck that's involved for this particular curve. So, there's some choice of prime. This is a periodic method and there's some choice of prime that that you're using behind the scenes. I'll tell you more about that a little bit. All right. So maybe to just give a little bit more context about why we care about questions like this. So, faultings proved more dels conjecture and so we know that we have a smooth project of curve X over the rationals of genius at least to find out that the set of rational points is finite, but faultings proof doesn't give us an algorithm to determine the set of rational points. Now, there's another proof of finiteness due to Voida, but it also doesn't give us an algorithm for finding set of rational points. There is recent work of Lawrence and bank attached that also gives another proof of finiteness. And here remains to be seen that this can be made algorithmic. But what we do have that works in some cases is the method of shabushian Coleman. So it only applies to curve satisfying certain restrictive rank bound. But then when it does apply, it works very well in practice for determining rational points possibly combined with other methods. So that's our motivating problem to look at the method of shabushy Coleman or possibly other refinements of this to give an explicit version of faultings theorem. So if I give you a curve to find over the rationals. I know that it's genius is the least to compute explicitly the set of its rational points. All right. So here is one more example one more curve. Smooth plain quartic reasonably nice coefficients. So maybe this is some random curve that we'd like to look at, but maybe not. So it's not so random, but we would like to look at it. This is the split carton modular curve of level 13. And there are some interesting consequences for knowing its rational points. And I'll tell you more about that a little bit. This will be the third curve that we look at. What we do know about this is that Galbraith did some searching and found seven rational points. And of course, that then leads to the natural question. If that's everything the set of the seven points precisely the set of rational points or were there more just kind of beyond the range that he considered, and we'll answer this at the very end. So maybe just a little bit of setup for working with higher genus curves. So for curves over the rationals of genius at least to the set of rational points is just a set. So we need to have a little bit more structure to say something about rational points. And so what we do is we associate to our curve other geometric objects that do have extra structure. So with all of this, I'll assume that our set of rational points is not empty, and I'll fix one point I'll fix the base point and use that to embed the curve into its Jacobian. I'll do that using this obel Jacobi map Iota that sends a point P to the class of the divisor P minus B. So the more del V theorem tells us that the group of rational points and the Jacobian, it's finally generated.  And here, the rank this is very important for us often will need to know what it is before we can even start, but I just want to emphasize that this is also something that's very hard to compute. Typically we get it through some Oracle, like magma. But in general, this is also something that's really difficult to compute and we don't have an algorithm that will compute this in general, but in all of what we do we kind of assume that we're handed the rank and that we can proceed. All right. So the idea, the main idea behind all the methods that I'll talk about is to associate either one or maybe a collection of other geometric objects to our curve. And then use that to explicitly compute a slightly larger, but very importantly, finite set of chaotic points containing the set of rational points. And then from that finite set of chaotic points, use that to extract the set of rational points. And this is what the Shabbat-T Coleman method does. And in subsequent years, there are various refinements of this method that allow us to consider other curves of slightly higher rank. So by associating covering collections or maybe maps elliptic curves over higher degree number field something that is elliptic Shabbat-T. There's also a very large generalization of all this. This is a program initiated by Min Young Kim, where instead of just considering the Jacobian and Abelian geometric objects, he considers non Abelian geometric objects associated to the curve. And this leads to a non Abelian version of the Shabbat-T method and non Abelian Shabbat-T program. What I hope is that this will allow us to understand rational points on curves whose Jacobians have higher rank. So beyond the frontier what's possible with the Shabbat-T Coleman method. All right. So let's start with Shabbat-T's theorem. So Shabbat-T proved the following special case of Mordell's conjecture in the 1940s. So that if X is a curve of genus at least two over the rationals, and the Mordell Bay rank is less than G, then set of rational points is finite. And in the 1980s Coleman gave a theory of periodic line integration and a number of spectacular applications of this theory. And one application that he gave was a reinterpretation of the method of Shabbat-T. He gave an effective version of Shabbat-T's result by reinterpreting everything in terms of periodic line integrals. He constructed a certain annihilating regular one form and computed its integral and show that this really annihilated rational points. So the rational points had to occur as solutions to this integral of an annihilating regular one form set equal to zero. He did more than that. He was able to count the number of zeros, this periodic integral. And in doing so, he gave this bound that for a good prime greater than 2G, the number of rational points is bounded above by the number of Fp rational points plus 2G minus 2. So this bound comes from studying the number of zeros of a periodic power series, but also using some global information about this differential that he constructed. All right, so here's a little bit more about the method of Shabbat-T and Coleman. So we'll fix the prime of good reduction for our curve. We'll assume that is greater than two. And then remember we have the obel Jacobi map, iota that embedded our curve x into its Jacobian j. And so that induces a map between the spaces of regular one forms in the Jacobian and the curve. And in fact, this is an isomorphism of g-dimensional piatic vector spaces. So we'll suppose that we have a regular one form on the Jacobian omega j and that it restricts to regular one form omega on the curve. All right, now we have a piatic integral on the Jacobian. The Jacobian over Qp is a piatic lead group, and so naturally carries piatic integrals. And we'll use that to define a piatic integral on the curve. So I want to define the piatic integral on the curve between points Q and Q prime on the curve in terms of this piatic integral on the Jacobian. So we'll define it to be the integral between zero and the class of the divisor Q prime minus Q. And then if the rank of the Jacobian is less than g, the claim is that there exists a regular one form such that the integral between our base point and any rational point of this differential omega vanishes on all rational points. All right, so why, why should this be true? Well, the Jacobian, we said this is the genus G curve. And so the space of regular one forms is g-dimensional. And if the Jacobian has rank r, essentially we have r independent linear conditions. And so if g is greater than r, then we have some positive co-dimensions in the space that we're trying to cut out using conditions coming from the rank. And so if r is less than g, then we can construct at least one annihilating differential. And so just using the linear algebra, using the values of the Coleman integrals coming from each piece of rank, we can then compute the kernel of the corresponding matrix and then compute a basis for the space of annihilating differential. So this really is now linear algebra problem if we're able to compute values of these Coleman integrals on each of the independent points in the Jacobian. So if we're able to do that, then we can construct a basis for the space of annihilating differentials and then study the corresponding zeros of those integrals and then find a finite set of chaotic points containing the set of rational points on the curve. All right, so just to kind of recap what's happened so far. So we're going to have our curve of genus at least two. We're going to embed it inside its Jacobian. And if the rank of the Jacobian is less than g, then we're going to use a Shabu-T Coleman method to compute a regular one form whose Coleman integral vanishes on rational points. Coleman computed an upper bound for the number of zeros of this chaotic integral and showed that if the prime is greater than 2g and if the prime is good, then the number of rational points is bounded above by the number of fp points plus 2g minus 2. So this is first a local computation in each residue disk and then globally using lemur rock, we get this bound. Now, in practice, this can be sharp. So in the triangle example, the genus was two, the rank was one, and behind the scenes we picked five, and that gave us eight f5 rational points and putting that all together we got the separate bound of 10. But you know that there's some dependence here on the prime p, right? And we kind of understand how the number of fp rational points on our curve grows as p grows, is linear in p. So we want to pick this p to be as small as possible, essentially, so that we have the best shot at getting a good upper bound. In practice, it isn't always going to be a sharp upper bound. So you compute whatever you can on the right hand side and you do a search on the left hand side for the points of bounded height, but you might have some discrepancy, right? And it might just be because genuinely there is some difference between the size of the set of chaotic points cut out by these intervals and the number of rational points you have. Yeah. So it's a question in the audience, from the audience. Rasima Kumar, please and ask away. No, in the second point, the number of points, the fp rational points and x, is it for a specific p or is it for all p? What is the p? So this is the p that you choose. p has to be a prime of good reduction. And also it has to be larger than two times the genus of the curve. But then this is true for any such p. You get to pick. Okay, thank you. Are there any other questions? No, I don't think so. You can continue. Okay, thanks. All right. So when there is some discrepancy between an upper bound that you compute and kind of the size of the known rational points that you have, it might seem that all hope is lost, but in practice, there is a good method of sitting so you kind of do a little bit more work and then throw all this into the more divisive. So you compute Shabu-T Coleman data at a number of primes, you carry out the Shabu-T Coleman method for a collection of favorable primes, and then use conditions on the Jacobian at these various primes to then sieve out the fake points, the points that are not honestly rational. And in practice, this works quite well to then give you the set of rational points. All right, so what we're trying to compute in the Shabu-T Coleman method is the following. So we construct an annihilating differential omega. And then we want to compute all chaotic points in the curve Z for which the integral vanishes. And we're going to call that set of chaotic points X QP one. So we want to do two things. We want to compute this annihilating differential omega, or maybe a collection of annihilating differentials, and then the finite set of chaotic points X QP one. So I thought I'd try to make this as explicit as possible to kind of convince you that we can compute these objects. So this is the case of genus two and rank one. And we'll assume that we have at least one rational point on our curve and we'll fix the base point V. So genus two, we have two dimensional space of regular one forms, call the basis elements omega naught and omega one. And since the rank is one, which is less than two by setup, we know that we can compute this set X QP one is the zero set of the periodic integral. So the next point isn't essential, but it just kind of helps simplify some of the computations. So if we know one more rational point P for which the divisor P minus B is not torsion and the Jacobian said some gives us a point of infinite order in the Jacobian. Then we can compute these two integrals, the Coleman integral between P and B of omega naught and the Coleman integral between P and B of omega one that gives us two periodic numbers and B, and then that gives us the constraints that we need to construct our annihilating differential. So taking, for instance, B omega naught minus a omega one gives us an annihilating differential. And then that is what we want to integrate and find all points Z for which that vanishes. And so maybe I should say a little bit about computing these intervals. So for hyperliptic curves, this is something that you can use sage to compute these numbers and B. And for more general curves. There is a package that young Topman and I put on GitHub to compute these values of Coleman integrals. So if you need to compute Coleman integrals either for this or for other purposes I encourage you to check out those two software libraries and let me know, especially if you find any bugs. So, in any case if you're able to compute these numbers, then you're able to compute the set of chaotic points for which the integral vanishes. And then you're able to compute the sex QP one. So, from that you might have to do a little bit more work to compute the set of rational points, but hopefully things like the more divisive will kick in and then you can determine the set of rational points. All right, so we've talked a little bit about finding rational points on curves. Maybe there's another question of which curves we actually want to do the sort of thing for. And, while sometimes we like to look at curves that have been mysterious for a number of years. And so one particular curve that was difficult for many years was a curve that appeared in the Earth Medical. So, DiFantis posed the following question. So, to find three squares, which when added to give a square and such that the first one is the side or the square root of the second, and the second is the side of the third. So, if we put this all together, DiFantis is asking if we can find positive rational x and y, such that the equation y squared equals x to the eighth plus x to the fourth plus x squared is satisfied. And DiFantis found a solution. So, he gave x is one half and y is nine sixteenths. So, from DiFantis's point of view, he was done, he solved this problem. But from our point of view, we'd like to see if we can find all the rational points on this curve. So, let's remove the singularity of the curve at zero zero. So, this amounts to determining the set of all rational points on our next curve now this genus two curve with affine model y squared equals x to the sixth plus x squared plus one. The reason this curve is very nice is because it's the only higher genus curve that's considered in the 10 known books of the Arithmetica. Another reason that this is nice is that it's just beyond the frontier of what we can do using shepherds Coleman strictly speaking. Why is that? Well, this is the genus two curve, but it's Jacobian has more del V rank two. So, just one more than what we can use shepherds Coleman for. Nevertheless, Joe weather all in his thesis show that the set of rational points in this curve is precisely the following. And let me tell you how weather all did this. So, this curve is a little special so it's a genius to curve, but you'll notice if you look at the equation here so y squared equals x to the sixth plus x squared plus one. There is an extra automorphism, right so at least one. So you look at x going to minus x in addition to the hyperliptic evolution, you have more symmetry than you would in a generic hyperliptic curve. Indeed, this is a bi elliptic genus to curve, and the Jacobian of this genus to curve is isogenous to a product of two elliptic curves. So what weather all did was to consider a collection of covering curves and apply which he Coleman on the covers, and then deduced the result about his curve about di fantasies curve. Alright, so by a cover of a curve I mean a surjective map from a curve F on to see. And whether all idea was to look at a cover or really a collection of covering curves, such that every rational point on his curve came from a rational point upstairs or on one of the curves in the covering collection. Then we can compute the rational points on our curve by computing the various sets of rational points in the covering collection. Now because the Jacobian is isogenous to the product of two elliptic curves in this case, they're very nice covers that you can construct this genius to curve. And whether all found some genius five curves G1 and G2 covering this genius to curve. And then he was able to further quotient out by some automorphisms, which then gave genius three curves that were hyperliptic. And moreover, those hyperliptic Jacobians had ranks zero and one. And so then Shabbatee Coleman kicks in and he carried out Shabbatee Coleman on those genius three curves and use that to determine the set of rational points on his genius to curve. So Shabbatee Coleman plus a little bit extra geometry looking at this collection of covering curves, and then carrying out Shabbatee Coleman upstairs is the idea that allowed whether all to tackle the second curve in our trilogy. All right, so taking this question a bit further for which curves do we care about and do we want to actually compute rational points on. So there are a number of interesting questions that come from modular curves. And so maybe one very nice example of this is this beautiful theorem of major that tells us if we have an elliptic curve over the rationals and P rational point of finite order and then N has to be one of the following values one through 10 or 12. And the idea behind this is to find the rational points on the modular curve X one event. So non hospital rational points and X one event correspond to elliptic curves over the rationals with a rational point P of order n. So Mazer's theorem is equivalent to the fact that the set of rational points on X one event consists only of cus if n is 11 or n is greater than equal to 13. So here is another modular problem. So we'll look at elliptic curves of the rationals and L of prime number. And then we have a natural Galois action on the L torsion points. And if we fix the basis of our L torsion, this is isomorphic to Z model C model. We have the following residual Galois representation row bar EL. And we can try to understand its image and Sarah proved that if he does not have complex multiplication, then row bar EL is surjective for L sufficiently large. So here went on to ask if this can be made uniform. Regardless of E, does there exist an absolute constant L not such that row bar EL is surjective for every non same elliptic curve E over the rationals and every prime L greater than L not. And the folklore conjecture is that taking L not to be 37 should work. Alright. If we're trying to understand the image of this model gal representation. Well, to show that it's surjective we want to show that its image is not contained in a maximal subgroup of GL to FL. So what are these maximal subgroups, we have the borrel subgroups that were triangular ones, exceptional subgroups, those with projective image a four s four or a five. These are the normalizers of the split carton subgroups. So these are the ones that can be diagonalized over FL. And then the normalizers of the non split carton subgroups. And the idea and doing this analysis is that for a maximal subgroup G. There is a corresponding modular curve XG over the rationals, for which the non hospital rational points correspond to elliptic curves with the image of this gal representation containing G. So now we've translated this question about looking at these images into, well, we've translated this question about looking at the possible subgroups into question about the possible rational points on these modular curves. So what do we know about Sarah uniformity. So in the borrel case. This is handled by major his work on rationalisogenes of prime degree, the exceptional subgroups were handled by Sarah. The normalizers of the split carton subgroups. This was work of below and parent. And the non split carton subgroups are much harder and essentially, not much is known. So this is pretty wide open. What I'd like to focus on is the work of billy parent and then billy parent rebella. So they showed that the split carton curves, their rational points are costs and CM points for L greater than or equal to 11, but they couldn't say much about 13. Alright, so what went wrong at 13. So a number of things went wrong at 13. In fact, they refer to 13 as a cursed level. And one thing that they use is Mazers method for integrality of noncustital rational points, which uses the following decomposition the Jacobian. So the Jacobian of the split carton curve of level L is isomorphic to the Jacobian of X not of all squared plus, which in turn is isomorphic to the Jacobian of X not of L cross the Jacobian of X non split of L. And Mazers method applies whenever the Jacobian of X not of L. This is what J not of L is whenever this is not trivial. And this is true for L equals 11 and L greater than or equal to 17. And this is the first thing that goes wrong for 13 J not of 13 is trivial. So the first curse of the cursed curve, and there are quite a few is that the Jacobian of X split of 13 is actually isogenous to the Jacobian of X non split of 13. And there are not any nice factors that we can take out and analyze the Jacobian of X not X split of 13 is absolutely simple. All right, so we said earlier, the Jacobian or the non split carton curves are kind of more difficult to deal with and so the fact the Jacobian of the split carton curve is isogenous to the Jacobian of the non split carton curve level 13 doesn't bow ball for us kind of suggest that maybe things are somehow more difficult level 13, but it gets worse. So, virtually on did some very nice work on the non split carton curve level 13 and the split carton curve level 13 she computed explicit smooth playing quartic models and showed that actually the curves are isomorphic over Q. So, the curve at level 13 that split carton curve level 13 actually turns out to be isomorphic to the non split carton curve level 13 not just that their Jacobians are isogenous, but the curves themselves are isomorphic over Q. And there's no real good reason for this there's no modular interpretation of this exceptional isomorphism. And again we said that the non split curves are much harder and so now we find ourselves dealing with the non split curve of level 13. So, here is Farron's model for X split of 13. So this is that curve that I showed you earlier and so now we can maybe try to analyze this curve and see what we can say about its rational points. So of course the first thing that might want to try is the Shabbat C Coleman method. So, what can we say about the rank of this, first of all, that's our third curse. So the, the rank of this Jacobian is at least three. Beyond Shabbat C Coleman. And there aren't any nice covers that we construct, at least not obviously to kind of use the second idea of whether all. So we're really out of luck here, and we need to do something else. But this is exactly what Kim's method proposes. This is kind of the right sort of example to consider in Kim's method. One can possibly hope to carry out Shabbat C Coleman ideas when the rank is greater than equal to the genus using some non-Abelian geometric objects and corresponding iterated Coleman integrals. So, in this situation, or kind of beyond the frontier of Shabbat C Coleman, we would like to construct some non-Abelian geometric objects associated to our curve. So this is what's often referred to as looking at Selma varieties. So these are cut out using homology and some local conditions. And so this should kind of feel like the construction of Selma groups and setting elliptic curves. And these summer varieties give rise to a sequence of sets. So XQP1 was the first Shabbat C Coleman set that we were looking at. But there is a depth 2 set XQP2 that's cut out by these double Coleman integrals and further endfold iterated integrals give rise to further refined sets XQPN. So the hope is that maybe we can compute the equations that these iterated Coleman integrals have to satisfy, and then we can use these now to cut out these finite sets of piatic points and then extract rational points from these sets. That's the idea. Alright, so Kim has conjectured that for n sufficiently large, the set XQPN is finite. And this is implied by standard conjectures like block HATO. So what do we know now what's what's known in practice. Well, one of the first nice results in this area to do coats and Kim for curves whose Jacobians have CM Jacobian. So for n sufficiently large the set XQPN is finite. And Ellenberg and Haast showed that that can be extended to give a new proof of faulting theorem again. So for n sufficiently large we have some finiteness and then that applies to curves whose curves that can be written a solvable Galois covers of P1. So both of the first two results are showing this asymptotic finiteness that there's some controlled dimension of Selma varieties as n gets large, and that gives you the finite result that you need. In a different direction together with Dogra, we showed that for curves X over the rationals with genus at least two and the rank less than G plus the rank of Neuron severity of the Jacobian minus one, then the set XQP2 is finite. And this XQP2 now is kind of this next set cut out by double integrals and computing XQP2 is what we call quadratic shepherty. So we would like to compute the set XQP2 because it turns out that the curse curve satisfies this bound that the rank is less than G plus the rank of Neuron severity minus one. And so if we can compute the set XQP2, then maybe run business and we can compute the set of rational points. All right, so more broadly, we can hope to use quadratic shepherty to compute XQP2. In the case when the rank is equal to the genus and the rank of Neuron severity is greater than one. But there's some difficulty making all this effective and implementing this on a computer to get some sensible answers at the end. So the functions, you know, where do they come from? Well, it turns out that they can be expressed in terms of piatic heights. And the idea is to move from the linear relations among these abelian integrals and the shepherty Coleman method to bilinear relations among these integrals. And these piatic heights have a natural interpretation using piatic differential equations and the constants can be computed using a collection of known rational points. So there is also still a bit of luck here involved as well that you kind of need enough rational points on your curve to start off with. So not only do you need to know the rank of the Jacobian and to have some explicit work on the Jacobian as well, but your curve naturally to have a collection of rational points and sufficiently many to be able to start this process. Right, so just a little dictionary between the two methods classical and quadratic shepherty. So the Jacobian now is replaced by this non abelian object the Selma variety. The integrals are now replaced by some iterated integrals, and there's some bilinear algebra of piatic heights replacing linear algebra computing kernels. So to reframe what happened in classical shepherty Coleman, instead of just using the linear relations, we're going to try to use some bilinear relations as well. So let me just remind you about the linear relations so these are among the functionals integral from B to X for omega regular one form in the case when the rank is less than the genus. So this gives us some obel Jacobi map with base point AJB. And what we like to do now is to generalize this to consider now the bilinear relations when the rank is equal to the genus. So this brings us to the definition of a quadratic shepherty function. So this is a function theta from QP points on the curve to QP, such that on each residue disc the map obel Jacobi base point and the quadratic shepherty function this is given by pietic power series. And moreover, we have an amorphism E, a functional C and a bilinear form B, such that for all rational points, the quadratic shepherty function minus the bilinear form on obel Jacobi base point and then slightly translated version of this, the and we can show that a quadratic shepherty function induces a function that vanishes on rational points and has finally many zeros. So this is how we'll compute the sex QP to I'm sorry to interrupt. There is a question please from Joel Rosenberg. Joel, will you please ask your question. Sure. So, earlier in the talk, there was an obstruction to using classical shabbatee in G equals three and R equals two. Does what what is that obstruction does quadratic shabbatee solve this case and is there a corresponding obstruction to quadratic shabbatee. I might have misheard the question. So you were saying there is an obstruction to genus three and rank two. I thought that was the problem in diaphantesis. I thought that was the setting of diaphantesis problem. Oh, okay. So yeah, and in diaphantesis problem is actually a genus two curve with rank two and so indeed there is an obstruction to using classical but you're absolutely right there is an obstruction to using classical shabbatee Coleman and genus to rank to whether all didn't use quadratic shabbatee. He had this covering collection, but you're totally right. You can apply quadratic shabbatee in that scenario as well. So this condition about now and so very ranks works in that case as well. And in fact, my student Francesca Bianchi worked out an example using quadratic shabbatee to solve diaphantesis problem again, and to show that it has just the rational points that whether all found. Thank you, Jennifer. Are there any other questions. Okay, great. All right, so in quadratic shabbatee we need to construct this function so we need to have some way of having this endomorphism is functional, and we need to solve for the bilinear relationship. Let me just say for this last thing solving for the bilinear relationship is similar to solving for the linear relations and shabbatee Coleman using linear algebra. And one natural source of bilinear functions for us comes really through the theory of chaotic heights. So the fact that this is quadratic shabbatee suggests that maybe there should be some quadratic functions, maybe taking patic values and patic heights are precisely the sort of tool that does that for you. So that's that's how we're going to find some of these quadratic functions. Okay, so to warm up. Before we get to the curse curve I thought I just say a little bit about how this can be used to find integral points on hyperliptic curves. In the case when the rank is equal to the genius and then how we can extend this then to rational points on more general curves. So the little bit of theory that you need now here is the Coleman gross patic height, which is a symmetric bilinear pairing on degree zero divisors on the curve. This takes values in QP and the height on D with a principal divisor this is zero so this is a well defined pairing on the Jacobian. So global height has a decomposition is a finite sum of local heights over the finite primes fee. And then how do you compute each of these local heights HP. Well, there's some distinguished crime P that you're picking this is a patic height so at P, there is something honestly patic that's happening. So there's a normalized differential that you construct with respect to splitting the hodge filtration on each one drop. And then you compute the Coleman integral of this differential the third kind away from P. This is more common tutorial, and you compute intersection multiple cities of your device is extended to a regular model. Roughly speaking, what is happening in quadratic Chevy T is to use this decomposition of the global height h is a sum of local heights. So, local height at P is something really honestly patic it's the solution to a patic differential equation. So local heights away from P, take on finitely many values on rational points, but in certain favorable cases, these are all trivial, and there are no local heights away from P to consider. And that's kind of a situation that we'll want to look at. So in the case of looking at integral points on curves, the main idea is to first compute this local height HP is a double Coleman integral, and then to control the finite number of values that the local heights take on integral points. All right, once we have those two pieces well the other thing that's going on is on the left hand side, the global height, we said was a quadratic form. So let's take down a basis for the space of quadratic forms using again Coleman integration so we can rewrite the global height h in terms of a basis for space of quadratic forms, and then everything on the left is a Coleman integral, and most of what's on the right is a Coleman integral so we move things around so that we have a Coleman integral set equal to finitely many values. And then, just like shepherty Coleman, it mounts to this pietic calculation of looking at pietic power series and understanding when they can take on each of these values. Right, so that's the theorem with Bessar and Mueller, then the case of a hyperliptic curve this can be made very explicit. When the rank is equal to the genus. And you know that these linear functionals f by the intervals of the regular one forms a mega I are linearly independent. So we have an explicitly computable finite set of values s and explicitly computable constants alpha ij such that the quadratic shepherty function theta, which is a double Coleman integral the integral of omega i omega i bar, where omega i bar is the dual undercut product pairing such that theta minus this combination of products fi fj this takes values in this finite set s on integral points. So this theta minus alpha ij fi fj. This gives us the analog of that function that we looked at the integral the nightly differential on the shepherty Coleman method. And then we just systematically set it equal to each of the values in this finite set s, look out all the possible pietic solutions and then take this across the whole set us. So this now gives us the sex QP to the case of integral points.  But there's one main problem here is that controlling the local heights away from P is quite difficult when X is rational but not integral. So in the Coleman growth setting these local heights away from P are essentially measuring denominators. And so if you want to study your rational points well you need to have some sense about the denominators look like, but this is, this is the whole thing right this is the entire question we don't know some absolute So this is tricky. How do we control the local heights away from P. So the workaround is to look at a quadratic shepherty function that comes to us from a slightly different construction of pietic heights one due to neck of our, and here instead of pairing divisors we're going to look at the pietic height on an associated gal representation. For every rational point we can construct a pietic gal representation a of X. This depends on having this data of a correspondence, a nice correspondence Z. And we have such a correspondence when the rank of Neuron Severe the Jacobian is at least two. And then we're going to make the following extra assumption that our curve has everywhere potential good reduction, because in this case, the neck of our height has the following very nice property that the local height each V of this representation is trivial. So then there are no local heights that we need to compute. And so we're essentially done here we just need to compute the local height at P. And this we can do using some pietic hot theory. So just like with the Coleman gross decomposition the neck of our pietic height has a local decomposition as a sum local height of local height at P of the representation plus the local heights away from P. The local height at P is a locally analytic function. This will give us our quadratic quadratic shepherty function theta. And by assumption since we're now going to put ourselves in the position that our curve has everywhere potential good reduction, the local heights away from peer trivial. And this will give us a quadratic shepherty function whose bilinear pairing is now this global height H and the anamorphism is induced by this correspondence Z. So with all the assumptions now that we've kind of stacked on. So if the rank of the Jacobian is equal to the genus, the rank of Neurons of very is greater than one. The pietic closure of the Jacobian has finite index and jqp access everywhere potential good reduction, and that we know enough rational points on our curve. So if we can solve the following two problems, we're in business and we can compute the quadratic shepherty set. So we need to be able to compute the local pietic height as a pietic power series, and then evaluate the global height. And so we can compute the known rational points, but really the second problem, computing the global height on each of the known rational points is essentially the first problem since we have everywhere potential good reduction, computing the global height is just computing the local height at P. Right. And so the cursed curve. There are many things that go wrong with the curse curve but some nice things about the curse curve well it satisfies all these conditions. So we can check and compute the rank of Neurons of very and so on, we can compute the rank of the mortal V group the Jacobian. And we can show the rank, the Jacobian is three. It turns out to have enough rational points so Galbraith found seven rational points and we can do a small change of coordinates to work with this model of the curve so that we can put five rational points in each of two affine patches covering the curve. Since the rank of Neurons of various three, there are two independent non trivial nice correspondences that actually give us to 17 at a kite so we work with 17, but there are two height functions that we can write down. And so it's a little bit better than just having one independent function so having two functions now we want to look at the intersection of the solution sets, and then to see what happens if this is precisely on the seven known rational points, or rational points. All right, so the theorem. This is joint work with Doug Romulo Taubman and Bonk is that the split carton curve level 13 has just the seven rational points that Galbraith found earlier. So this completes the classification of rational points on the split carton curves that was started by Billy Pajón Rebolleto. And then thanks to the work of Bertrand Barron. Since we know that the split carton curve level 13 is isomorphic to the non split carton curve level 13. We get for free that we know the rational points and the non split carton curve of level 13 as well. There's seven rational points there. And I think I'll stop there. Thank you very much for your attention.