Questions tagged [arithmetic-geometry]

A subject that lies at the intersection of algebraic geometry and number theory dealing with varieties, the Mordell conjecture, Arakelov theory, and elliptic curves.

469 questions
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Can someone suggest a path to study Mordell-Weil theorem for someone studying on their own?

So, I want to read the proof of Mordell-Weil theorem and so, I picked up the book 'Arithmetic of Elliptic Curves' by J. Silverman and J.S. Milne's Elliptic Curves book. But after going through both ...
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Riemann Roch theorem for surfaces

Hi am a student of Maths at university; I am studying the theorem of Riemann-Roch for curves. I am interested in understanding what happens in the case of surfaces. I do not want to look for the whole ...
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Prerequisites/Background for Arithmetic of Shimura Curves

A proposed graduate course for next fall on the "Arithmetic of Shimura Curves" caught my eye for various reasons. I would like to know what background one would need to have to start learning about ...
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Ring of Definition of Morphism

Let $\phi \colon \mathbb{P}^n_\mathbb{Q} \to \mathbb{P}^m_\mathbb{Q}$ be a morphism over $\mathbb{Q}$. Is there a way in general to find the smallest positive integer $N$ such that $\phi$ is defined ...
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Pointless, non-singular, absolutely irreducible affine plane curves over finite fields

Crossposted from MO because of contradicting comments there. We think the following is true and have partial results: For all sufficiently large primes $p$ and all natural $g \ge 1$, there exists ...
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Existence of a $\mathbb{Q}$-rational point from the existence of an $\mathbb{F}_p$-rational point for all $p$?

Suppose I have some $f\in \mathbb{Z}[x]$ and some $a\in \mathbb{Z}$ such that $f(a)\equiv 0 \mod p$ for all primes $p$. Then of course $f(a)=0$. Suppose now I have a scheme $X$ over $\mathbb{Z}$ and ...
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$\mu_p$ as a $Z/p$ torsor

Let us fix the base field $k=\mathbb Q$. Then, say on the etale site, $\mu_p$ is a $\mathbb Z/p$ torsor since locally (after base change to $Q(\mu_p)$), we can pick a root of unity, aka a section ...
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Riemann-Roch theorem and divisor of differential forms

I am studying Riemann-Roch theorem but I have some difficulty understanding the concept of divisor of a differential form and the link between differential forms, their divisors and the RRT. Could you ...
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The $abc$ conjecture as a special case of Vojta's height inequality

From Quanta I've learned that Peter Scholze and Jakob Stix rejected Shinichi Mochizuki's proof of the $abc$ conjecture in September 2018. As a non-expert one stumbles a little earlier than necessary ...
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Local height function on affine variety

Let $K$ be a global field (e.g., a number field) and let $M_K$ be its set of absolute values. Further, let $\overline{K}$ be the algebraic closure of $K$ and denote by $M$ the set of absolute values ...
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Quotients and exact sequences of algebraic group schemes

Now I study group schemes to understand fundamental properties about semi abelian varieties and generalized jacobian varieties. Let $G$ be an algebraic group scheme over a field $k$ ($=$ $k$ group ...
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Why is the kernel of reduction contained in the image of this isogeny? (From a paper of Cassels)

$\newcommand{\fp}{\mathfrak{p}} \renewcommand{\phi}{\varphi}$ I'm currently trying to understand a paper of Cassels [see page 189 for the relevant content, esp. (4.9)] and I've hit a little snag. The ...
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slightly different from AGM

$a_{n+1}=\frac{a_n+b_n}{2}$ $b_{n+1}=\sqrt{a_{n+1}b_{n}}$ also, $a_1=a$ and $b_1=b$ evaluate $$\lim_{n\to\infty}a_{n}$$ This question is from guillaume musso's novel La Jeune Fille et la Nuit . ...
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What is a supersingular elliptic curve over arbitrary rings

I read in Katz Mazur's book on moduli spaces of elliptic curves that an elliptic curve over an $\mathbb{F}_p$-algebra $R$ is called ordinary if its geometric points are all ordinary. Now the question ...
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Moduli interpretation of the integral anticanonical tower

This question is related to my reading of On torsion in the cohomology of locally symmetric varieties by Scholze. In chapter $3$, using the theory of canonical subgroup, he produces Frobenius maps ...
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Are there counterexamples of isogeny elliptic curves with non-isomorphic integral Tate modules?

Let $K$ be a field and $G_K$ be its absolute Galois group. Let $E_1,E_2$ be two elliptic curves over $K$. Assume that there exists an isogeny $f:E_1\rightarrow E_2$. Let $p$ be a prime number. Then $f$...
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How to compute the $p$-adic period of $\mathbb G_m$?

I am reading one paper by Colmez about periods of abelian varieties with complex multiplication. As a motivation, he says we can compute the periods of $\mathbb G_m$: I know the example for the ...
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I am interested in the study of p-adic geometry. Unfortunately what I know is basic Algebraic geometry and basic number theory. To get an idea of the amount of material to study what could be a "...
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Finding elliptic curves achieving the upper and lower bounds of Hasse's Interval

I always thought that Hasse's bound is sharp (at least for elliptic curves). In other words I always thought that given a prime number $p$, I can find two elliptic curves $E_1,E_2$ over $\mathbb F_p$ ...
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Is Hom scheme between projective curves of large genus finite etale?

Let $K$ be a number field, $T$ be a finite set containing some finite places of $K$, and $S=\operatorname{Spec} O_{K,T}$. If $X,Y$ are two projective smooth curves over $S$ with genus large than $1$ ...
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An abelian variety not isogenous to a Jacobian

In the L-functions and Modular Forms Database is an isogeny class of an abelian variety of dimension $2$ over $\mathbb F_5$. They claim that it is principally polarizable but not the Jacobian of a ...
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Modular parametrization from equality of $L$-functions

In the literature, an elliptic curve $E/\mathbb{Q}$ is defined to be modular in two different ways 1) if there exists a nonconstant morphism $X_0(N) \to E$, 2) if there exists a modular form $f$ ...
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decomposition group and inertia group, the minimal polynomial,surjectivity of the map $D_{M/P}\rightarrow Gal$

Can anyone explain the underlined sentence? For notation, A:Dedekind domain, K=Frac(A), L/K:Galois extension, B:The integral closure of A in L, M:A maximal ideal of B, P:The intersection of M and A (...
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Finding the integral closure of $k[x]$ in $k(x)(\sqrt{f})$, where $f(x)=x^6+tx^5+t^2x^3+t\in k[x]$.

This is an exercise (1.15) of Dino Lorenzini' s An Invitation to Arithmetic Geometry. Let $F$ be a field of characteristic $2$. Let $k:=F(t)$, the field of rational functions in the variable $t$. Let ...
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Bad reduction at prime numbers for the elliptic curve $y^2+y=x^3-x^2+2x-2$

Consider the elliptic curve $$E:y^2+y=x^3-x^2+2x-2.$$ My goal is to compute the conductor of the elliptic curve, the example is from https://planetmath.org/conductorofanellipticcurve. My problem isn'...
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Prerequisites for reading the paper “Birational Calabi–Yau n-folds have equal Betti numbers" by Victor V. Batyrev?

As the title suggests, I'm trying to read this paper. But it demands a lot of prerequisite knowledge that I'm not just unaware of, but I'm having trouble narrowing down what books and notes I can take ...
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Why é​t​a​l​e​?

Background: The notion of an étale morphism has proved itself to be ubiquitous within the realm of algebraic geometry. Apart from carrying a rich intuitive idea, it is the first ingredient in notions ...