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.

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In search for methods for finding a closed form for the roots of a non-homogenous diophantine equation.

I'm trying to solve a special case of a number theoretical problem, and it relies on finding a closed form for the roots of a non-homogenous diophantine equation with four variables, but I could only ...
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What is the idea behind the proof of the Isogeny theorem and Theorem III.7.9 (Serre) in Silverman's book?

Let $E_1$ and $E_2$ be Elliptic curves over the field $K$ and $\ell\neq\mathrm{char}(K)$ be a prime number. Let $T_\ell(E_i)$ is the Tate module of $E_i$, $i=1,2$ and $\mathrm{Hom}_K(T_\ell(E_1),T_\...
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Equality of divisors under Galois action in Silverman

I am trying to understand a formula in Silverman's The Arithmetic of Elliptic Curves. Let $K$ be a field, $\overline{K}$ an algebraic closure of $K$, and $C$ be a curve defined over $K$, and $\sigma \...
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Singularities of arithmetic surfaces

I have a problem understanding the discussion of example 8.3.54 in Liu's Algebraic Geometry and Arithmetic Curves. The setting is the following: We have a DVR with uniformiser $t$, characteristic of ...
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If E is non-singular then it has non-singular weierstrass equation

Let $E$ be a non-singular projective curve of genus one. There exist regular functions $x,y$ on $E$ satisfying a Weiestrass equation $$ y^2 = x^3 + ax + b$$ Is this equation necessarily non-singular? ...
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Existence of a model for $(X,L)$

Recently I was reading Moriwaki's work about the heights on arithmetic varieties, and I came across the following doubt about the setting. Let $(X,L)$ be a couple where $X$ is a projective nonsingular ...
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Torsionness of the kernel of the pullback map of Picard groups of a normalization map

Let $X$ be a projective variety over a number field $k$, $\pi: \tilde X \to X$ be its normalization, and $\pi^{*}: \mathrm{Pic}(X) \to \mathrm{Pic}(\tilde X)$ be the corresponding map of Picard groups....
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Meaning of Torsion in Chow groups

I am reading the paper "Remarks on Correspondences and Algebraic Cycles" of S. Bloch and V. Srinivas. In the proof of proposition 1 in chapter 1 says: $Ker(CH_0(U_L))\rightarrow CH_0(U_{\...
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Global sections are finite product of global sections of irreducible comoonents

I would like to extend the question here. Assume that $X\rightarrow Spec(K)$ is proper and X is reduced. Since is proper and $Spec(K)$ is Noetherian, $X$ is Noetherian and hence it will be finite ...
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$4$-torsion points of elliptic curve in field extension of odd degree

Let $E$ be an elliptic curve over some field $k$ of characteristic not $2$. Let $m$ be the maximal number of $4$-torsion points of $E(L)$ where $L$ ranges over all finite field extensions of $k$ of ...
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Split semi-abelian varieties

I am looking for a reference on semi-abelian varieties (there are many for abelian varieties), as I have some basic questions. For example Let $G$ be a semi-abelian variety over a field $k$ of ...
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What is a shtuka?

In the Berkeley lecture by Scholze, an $X$-shtuka over $S$ (usually $\operatorname{Spec}\bar{\mathbb{F}_p}$) is defined to be a rank $n$ vector bundle $E$ on $S\times_{\mathbb{F}_p}X$ together with a ...
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Global functions on arithmetic varieties

Let $f:X\to\operatorname{Spec} O_K$ be an arithmetic variety where $K$ is a number field and $O_K$ is its rings of integers. We assume that $X$ is integral, projective, regular and $f$ is flat. If ...
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prime index coefficients of $L$-functions of Elliptic Curves over $\mathbb{Q}$ and analytical rank [closed]

I have searched without success if there was any known relationships or conjectures between the rank and the prime index coefficients of the $L$-functions of a modular elliptic curves. Anyone has a ...
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Quotient of algebraic curve

Say we have an algebraic curve $C:s^6 = t^6 + 1$ defined over $\mathbb{F}_p$, and the automorphism $\phi :(s,t) \mapsto (s,\xi_6t)$ (we assume this is defined over the base field). I am trying to ...
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Relation between Petersson inner product and cup product

I was wondering how to define Petersson inner product for Hilbert modular forms. I had a discussion with my supervisor who vaguely suggested that in the case of elliptic modular forms the Petersson ...
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CM points on Shimura curves

I am trying to understand CM points on Shimura curves and I got confused. Before I get the point that I got confused and stuck I need to introduce some notations. $F$: a number field, $\mathbb{A}_f$: ...
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Induced map on jacobians of curves of genus one

Let $K$ be a field and consider two genus-one curves $C,C^{\prime}$ (probably without $K$-points). Let $f:C\to C^{\prime} $ be a morphism defined over $K$. Do I understand correctly that if we take ...
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Isogenous elliptic curves with the same j-invariant are isomorphic

Let $K$ be a field. Consider two elliptic curves $E,E^{\prime}$ over $K$ isomorphic over $K^{\textrm{sep}}$ or equivalently having the same $j$-invariant. Suppose there exists a non-zero $K$-isogeny $...
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What is the needed background to study Euler system?

I'm interested in Euler system.I heated this theory was used to prove the finiteness of Tate-Shafarevich group of certain elliptic curves by rubin. To read Karl Rubin's 'Euler system', what is the ...
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Decomposition of the regular representation of $S_3$ into irreducible ones.

The regular representation of $S_3 = \langle e, a, a^2, b, ab, a^2b \rangle$ is realised into the $6 \times 6$ matrices acting on the six dimensional vector space spanned by each element of $S_3$. OK, ...
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Injection from $WC(E/K)$ to $H^1(G_K,E)$ from Silverman's 'the arithmetic of elliptic curves'

This is question regarding Silverman's 'the arithmetic of elliptic curves'(AEC), p325, theorem 3.6. I want prove there is bijection between $WC(E/K)$ and $H^1(G_K,E)$. The bijection is given by {$C/K$}...
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Morphism between algebraic (smooth) curves of degree 1 is an isomorphism. Does the converse holds?

Morphism between algebraic (smooth) curves of degree 1 is an isomorphism. Does the converse holds ? In other words, isomorphism between algebraic smooth curves has always degree 1 ? Degree of morphism ...
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Let $C$ and $C'$ be algebraic (smooth) curves.If morphism $φ:C→C'$ satisfies the condition that $#φ^{-1}(Q)=1$ for all $Q∈C'$, why $φ$ is isomorphism?

Let $C$ and $C'$ be algebraic (smooth) curves. If morphism $φ:C→C'$ satisfies the condition that $#φ^{-1}(Q)=1$ for all $Q∈C'$, why $φ$ is isomorphism ? If I could prove $φ$ is a degree one morephism, ...
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Elliptic surfaces: translation by a torsion section preserves intersection numbers?

Let $S$ be a rational elliptic surface with elliptic fibration $f:S\to\Bbb{P}^1$ (with a section) over an algebraically closed field. Let $E(K)$ be the Mordell-Weil group of $S$. In this text, Example ...
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Why does transforming an Elliptic curve into Weierstrass form not affect the group of rational points?

There are various steps involved like realising the curve in projective space, multiple coordinate transformations, homogenising/dehomogenising the curve and converting a curve with rational ...
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The affine scheme Spec $\mathcal{O}_K$ is a curve over F1 too?

I read that we can considerate the affine scheme $\mathrm{Spec}(\mathbb{Z})$ like a curve over $\mathrm{Spec}(\mathbb{F}_1)$ where $\mathbb{F}_1$ is the field with one element. So, can we also ...
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$p$-adic valuation on algebraic numbers

Is there a way to define in a canonical way a $p$-adic valuation on complex number or at least on algebraic number, that extend the $p$-adic valuation on $\mathbb{Q}$? That is, is there a canonical ...
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What is an isomorphism between $L:aX+bY=Z$ and $\Bbb P^1_K$?

It is known that genus $0$ smooth curve over field $K$ with base point is isomorphic to $\Bbb P^1_K$ over $K$. I want to understand this with a lot of examples. For example, let $a,b\in K^\times$ and ...
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Translation of Loic Merel's Paper on Bounds for Elliptic Curves

I have been reading elliptic curves and had a discussion with my professor regarding the torsion subgroup of the Mordell-Weil group. He advised me to read Loïc Merel's "Bornes pour la torsion ...
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Closure of finite type generic fibre

Let $S$ be an integral noetherian scheme and let $X\to S$ be a locally finite type morphism. Suppose that the generic fibre $X_K$ is of finite type over the function field $K=K(S)$ of $S$. Let $Z\...
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Properness and compactness for rational points

I am reading Rational points on varieties by Poonen and I'm not sure about this proposition. Prop. 2.6.1 Let $k$ be a local field. Let $X$ be a $k$-variety. Il $X$ is proper over $k$, then $X(k)$ is ...
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Why does Weil cohomology have to be on $\mathbb{Q}_l$ for $l\neq p$?

Specifically, I saw a brief explanation: The construction of Weil cohomology isn't easy. Here's an example by Serre: Consider the endomorphism ring $\operatorname{End}(C)$ of an elliptic curve $C$, ...
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2 votes
1 answer
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Dense rational points of an elliptic curve

$\newcommand\Q{\mathbb Q}$Could anyone please provide me the following two examples of elliptic curve defined over $\Q$ (if they exist) It has infinite $\Q$-rational points and these points are (in ...
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Isomorphism of schemes over a DVR is determined by the isomorphism over its generic fiber?

Let $R$ be a discrete valuation ring. Let $X\to \operatorname{Spec} R$ and $Y\to \operatorname{Spec} R$ be separated (not necessarily proper) $R$-schemes which are flat over $\operatorname{Spec} R$. ...
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Finitely many elliptic curves isogenous to a given one (over number fields)

Let $K$ be a number field and $E/K$ be an elliptic curve (or an abelian variety). Is there an "easy" proof that there are only finitely many isomorphism classes of elliptic curves $E' / K$ ...
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Understanding Shioda-Tate's isomorphism $E(K)\simeq \text{NS}(X)/T$.

Let $\pi:X\to\Bbb{P}^1$ be a rational elliptic over an algebraically closed field $k$. If $K:=k(t)$, we have the Mordell-Weil group $E(K)$, which is an elliptic curve over $K$ (which is also the ...
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Why $\tau$-pure sheaves can be twisted to be a absolute pure one

I'm now working on Kiehl and Weissauer's book Weil Conjectures, Perverse Sheaves and l’adic Fourier Transform. All sheaves mensioned are on the small étale site. Consider an algebraic scheme $X_0 $ ...
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Does an Elliptic Curve has to have a rational point by definition?

This is how an Elliptic Curve is defined by my professor, which also appears in Silverman and Tate's book Rational Points on Elliptic Curves. The set of solutions to the equation (over a field) $y^2 = ...
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How does the geometric Frobenius act on stalk?

How does the geometric Frobenius act on stalk? The first image is from http://virtualmath1.stanford.edu/~conrad/Weil2seminar/Notes/L19.pdf page 9. The second image is from Kiehl-Weissauer page 7.
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The textbooks of number theory including p-adic cyclotomic character and Teichmuller lift

I would like to learn number theory, and it seems like necessity in number thoery that ingredients which is p-adic cyclotomic character and Teichmuller lift. Therefore, I wish to learn these, but in ...
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Understanding "invariant differentials" of singular Weierstrass curves

$\newcommand\spec{\operatorname{spec}}\newcommand\P{\mathbb P}\newcommand\msO{\mathscr O}\newcommand\push[1]{{#1}_*}$ Say I have some relative Weierstrass curve $f:W\to S$ (so $f$ is flat and proper ...
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1 answer
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Left versus Right regular representations.

Let $G$ be a finite group. $G$ can bear the so-called regular representation. Let $\chi_g(h) \colon= \delta_{g,h} ~ {\mathrm{for}} ~ h \not= g$. Let $X \colon= {\mathrm{the\, vector\,space\,of\,...
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Relation between $S$-ideal class group and usual ideal class group

For $\mathcal{O}_{K}$, the integer ring of a global field, we denote $S$ to be any set of primes of a global field $K.$ Let $$\mathcal{O}_{K,S}:=\{x\in K\mid v_{\mathfrak{p}}\geq 0\text{ for }\...
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What is a Galois group $Gal( \Bbb Q(E[4])/ \Bbb Q)$?

Let $E$ be an elliptic curve defined by $E:y^2=x^3-x$. What is a Galois group $Gal( \Bbb Q(E[4])/ \Bbb Q)$ ? we can easily find $E[2]=\{(-1, 0), (0, 0), (1, 0), \infty\}$ by finding a point at which ...
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Let $E$ be an elliptic curve defined by $E:y^2=x^3-x$. I want to figure out all points $P∈E(\bar{\Bbb Q })$ which satisfies $[2]P∈E({\Bbb Q })$.

Let $E$ be an elliptic curve defined by $E:y^2=x^3-x$. I want to figure out all points $P∈E(\bar{\Bbb Q })$ which satisfies $[2]P∈E({\Bbb Q })$.In other word, I would like to explicitly write down the ...
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Textbook of p-adic hodge theory

It seems that p-adic hodge theory is essential ingredient in arithmetic geometry. I would like to learn p-adic hodge theory, so I have searched for textbooks of p-adic hodge theory, but I wasn't able ...
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3 votes
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How does the Frobenius really act on Weil sheaves in $\ell$-adic cohomology?

Let $X_0$ be a connected scheme defined over $\mathbb F_{p}$ and let $X$ be the product $X_0 \times_{\mathbb {F}_p} \overline{\mathbb F_p}$, as usual, with the natural map $\pi:X \to X_0$. Given an ...
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3 votes
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What is the meaning of integral structure and generic fibre?

In Peter Scholze's perfectoid spaces, Almost mathematics, I have difficulty understanding the sequence of localization functors $$ K^{\circ}-\textrm{mod}\rightarrow K^{\circ}-\textrm{mod}/(\mathfrak{m}...
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Why 13 is special for the Mazur's theorem on torsion of elliptic curves?

I'm studying the proof of Mazur's theorem on the torsion of elliptic curves. He first show that, for a prime $N > 7$ and $N\neq 13$, there's no elliptic curve over $\mathbb{Q}$ with a rational ...
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