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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6 views

reference request: From hypergeometric functions to hypergeometric motives

I am interested in the hypergeometric motive and its application in arithmetic geometry. As a beginner, I see a lot of works in this area refer the readers to the book of N. M. Katz, "Exponential ...
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138 views

Infinitely many solutions leads to existence of a polynomial

Let $P$ and $Q$ be monic polynomials with integer coefficients and degrees $n$ and $d$ respectively, where $d\mid n$. Suppose there are infinitely many pairs of positive integers $(a,b)$ for which $P(...
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Explicit elliptic fibration for $x_0^4+x_1^4+x_2^4+x_3^4=0$ (Fermat's quartic)

In Huybrecht's Lectures on K3 surfaces, there's an explicit description of an elliptic fibration for the K3 surface $X:=\{(x_0:x_1:x_2:x_3)\in\Bbb{P}_\Bbb{C}^3\mid x_0^4+x_1^4+x_2^4+x_3^4=0\}$ (Fermat ...
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How many ways can this hexagon be tiled by 11 rhombuses of unit side length?

I came across a question in an exercise booklet for Mathematic Olympiad for primary school students in Australia. The question is shown in the following picture: I barely have any clue how this sort ...
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Textbook about the proof of Serre's modularity conjecture

I want to understand serre's modularity theorem(Serre's modularity conjecture).However, there are many conjectures named serre's conjecture and it seems to be difficult to find textbooks. Is there a ...
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26 views

Product formula for zeta function over finite field

I'm trying to understand the Euler product form for a Zeta function over a finite field. So let $k = \mathbb{F}_p$ and $X = V(\mathfrak{a})$ be an algebraic set with coordinate ring $k[X]$. I believe ...
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A twisted version of the modular curve $X_1(N)$

The modular curve $Y_1(N)$ is known to parametrize pairs $(E,P)$ where $E$ is an elliptic curve and $P$ is a point in $E$ of exact order $N$ (at least if $N$ is invertible in the base; if not, one can ...
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Smoothness of diagonal hypersurfaces

I am working on concrete examples of projective varieties over finite fields where one can verify Weil's conjectures. Following the ideas in Ireland-Rosen "A classical introduction to modern number ...
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148 views

An intersection calculation over a finite field

Question: I made a calculation that must be wrong, but am having trouble spotting the error. Which steps below are invalid? Thank you in advance for your attention! Setup: Let $p$ and $\ell \neq p$...
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If $E$ has additive reduction at $v$ then $H^0(I_v,E[p^\infty])$ is finite

Let $E$ is an elliptic curve defined over a number field $F$, $p$ a prime of $\mathbb{Z}$ and $v$ a valuation of $F$ that does not lie over $p$. Call $F_v$ the completion of $F$ with respect to $v$ ...
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39 views

Relations between good ordinary reduction for $E/\mathbb{Q}$ and for $E/K$ with $K$ number field [duplicate]

Reading some articles and Silvermann's "The arithmetic of elliptic curves" I found these two different definitions for good ordinary reduction. Silvermann, with Theorem V.3.1, says that an elliptic ...
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Kolyvagin's Euler systems and their properties

I am confused with the different definitions of the Euler system I understand from the book "Cyclotomic Fields and Zeta Values" that an Euler system is a map $φ: W\smallsetminus S \to Q$ (algebraic ...
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Are there upper bounds for the genus of an extension of function fields, can you have unbounded genus growth?

If $F$ is a function field with constant field $K$, and $E$ is a finite extensions of $F$, then Riemann-Hurwitz gives a way to compute the genus, $g_E$, of $E$ from the genus, $g_F$, of $F$ so long as ...
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Frobenius between $\mathbb{Z}_p[p^{1/p^m}]$-modules etale

The questions are motivated by P. Scholze's answer in MO question https://mathoverflow.net/questions/132438/why-is-faltings-almost-purity-theorem-a-purity-theorem. Consider for every $m \in \mathbb{N}$...
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Change of variables in the Weierstrass form $y^2=x^3+a_2x^2+a_4x+a_6$ for the generic fiber

I'm reading Schütt-Shioda's survey paper on elliptic surfaces, section "Tate's algorithm". My question doesn't concern the algorithm itself, but only the initial manipulation the authors do on the ...
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Singularities of the quotient minimal surface with an involution

I am reading Guletskii Paper "Bloch Conjecture for surfaces with involution and of p_g=0" and I would like to have any advice to prove the following affirmation (see page 13 of this paper): If S is a ...
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32 views

Divisibility properties of point counts for elliptic curves over finite fields

Let $E$ be an elliptic curve over the finite field $\mathbb{F}_q$ (of characteristic $p$). My questions are about the following ideal $I_E \subset \mathbb{Z}$: $$I_E=\langle \#E(\mathbb{F}_q), \#E(\...
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Status of Deligne's Weight-Monodromy Conjecture?

I'm interested in Deligne's weight-monodromy conjecture, and was wondering if anyone could provide any insight on its current status... What is completely resolved? What is being worked on and is ...
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Is Bhargava multidimensional matrices and elliptic curves methods applying also to finite fields?

There are a bunch of papers of Bhargava et al. 1 and 2 where they use multidimensional matrices and hyperdetermimant to derive theorems about the average ranks of genus 1 or 2 curves over rational ...
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Motivation and Interpretation of Classical and Geometric Satake Equivalence

I am struggling with understanding the motivation for both the classical and geometric forms of Satake equivalence. What prompted the development of this equivalence? Any open problems that it was ...
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Arithmetic Satake Equivalence?

I was wondering if there is an arithmetical analogue for Satake equivalence? I'm looking for a topic to focus on for my thesis, and although I haven't been able to find any arithmetic analogue, I want ...
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102 views

A curve over a finite field with apparently no points

I am probably making a really silly mistake but I can't figure it out: Let $\mathbb F_q$ be a finite field and $a$ an element in it that is not a square. Let $E$ be the elliptic curve corresponding ...
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Elliptic curves over extension fields with Galois group the Monster

I was watching Mazur's 2019 Einstein lecture at the AMS joint sectional meeting, and he proposed the following question following a discussion, which I paraphrase here: Given an elliptic curve $E/\...
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Reduction of elliptic curves defines over general field

let $(K,v)$ be a field with a non-archimedean valuation and let $E/K$ an elliptic curve. The field $K$ is not supposed to be local. We denote by $\mathcal{O}_v$ the valuation ring (not a DVR) and $...
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Relation between the selmer group and an etale cohomology group.

Let $E/\mathbb Q$ be an elliptic curve with good reduction away from a finite set of primes in $S$. Let $\mathscr E$ be a model for $E$ over $\mathbb Z[1/S]$. Then I know two ways to prove the weak ...
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Geometrically irreducible schemes

$\newcommand\spec{\operatorname{Spec}}$Hartshorne Exercise II.3.15 a) states the following: Let $X$ be a scheme of finite type over a field $k$ then the following are equivalent (in which case we ...
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Places and schemes

In my algebraic number theory, we're currently studying local fields, and we've noted that given a number field $K$, its normalized discrete valuations correspond exactly to primes of $O_K$, and we ...
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54 views

Lorenzini's “Invitation to arithmetic geometry”, 2nd exercise

I have some trouble trying to prove that \begin{equation} \mathbb{Z}\left[\frac{2+i}{5}\right]\cap \mathbb{Q}=\mathbb{Z} \end{equation} which is the second exercise of Dino Lorenzini's "An ...
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92 views

Dieudonné module associated to the dual of a $p$-divisible group

Let $k$ be a perfect field of characteristic $p>0$, and consider $X=(X_m,i_m)$ a $p$-divisible group of height $h$ over $\operatorname{Spec}(k)$: it is an inductive system where $X_m$ is a finite ...
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Étale fundamental group acts by similitude on Tate modules of abelian variety

Note: the below question doesn't make sense. I have mistakenly mixed two different $\pi_1$ and came up with a false claim. See my last comment. Let $X$ be an abelian variety over an algebraically ...
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33 views

Generalisation of the Rank of an Abelian Variety

If $A$ is an abelian variety over a number field $K$, then the set of $K$-rational points $A(K)$ is a finitely generated group by the Mordell-Weil Theorem. By the classification of finitely generated ...
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25 views

(In)dependence of the conductor of a Galois representation and the choice of l

In the work of A. P. Ogg, Elliptic Curves and Wild Ramification, he proves that the conductor of an elliptic curve is independent of the choice of $\ell$. That is, for example, if $E$ is an elliptic ...
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33 views

Geometric interpretation of $0$-cycles

I am learning about cycles on schemes, and I am wondering how I can think about a $0$-cycle, $1$-cycle (or more generally, $r$-cycle) geometrically?
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Some questions regarding the computation of the Mordell-Weil group

I was reading the theory relevant to Selmer and Shafarevich-Tate groups from Silverman's AEC. And I have a lot of doubts related to these topics: First, I don't understand the reasoning behind the ...
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Confusion about Lang's Conjecture

I've been reading the following Paper (https://arxiv.org/abs/1809.06818) on arithmetic hyperbolicity. We say that a scheme $X/\overline{\mathbb{Q}}$ is arithmetically hyperbolic over $\overline{\...
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Understanding partial derivative of a global section of a sheaf

This question comes from a paper of Vojta "A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing": https://www.jstor.org/stable/2152710?seq=23#metadata_info_tab_contents Here is the ...
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70 views

What fields of arithmetic/algebraic geometry benefit from infinity/derived techniques?

I'd like to get a better picture of how infinity/derived techniques become more important in algebraic/arithmetic geometry. I'd therefore like to know: What questions/subfields of algebraic/...
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Riemann Roch Space for an algebraic surface and degree of rational function.

I am not familiar with the language of cohomology and definitely not fluent with the modern language of sheaves. I had learnt about Riemann Roch spaces and the associated theorem in the context of ...
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Serre's paper on non-existence of singular cohomology analogue for schemes

I read an argument - attributed to Serre - as to why one cannot construct a cohomology theory for schemes, analogous to singular cohomology, providing $\mathbb Q$-vector spaces. My understanding is ...
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67 views

norm of invertible element in Banach ring

Let $A$ be a commutative ring with a norm $∥·∥\geq0$ such that $∥ab∥\leq ∥a∥∥b∥$, $∥a+b∥\leq \text{max}\{∥a∥,∥b∥\}$, $∥1∥=1$ and $∥a∥=0$ if and only if $a=0$. We assume $A$ is complete for this norm. ...
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What does “generic fibre” mean in elliptic fibrations?

I've just started to read about elliptic surfaces in algebraic geometry. Here's a quote from wikipedia: An elliptic surface is a surface that has an elliptic fibration [...] such that almost all ...
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Comparison between height of polynomial and height of coefficients of polynomial

Let $K$ be a number field, and let $P\in K[x_0,...,x_n],\;P=\sum_{\gamma}a_{\gamma}x^{\gamma}.$ There is a notion of the height of the polynomial, denoted $h(P)$ which can be described in one of the ...
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minimal distance between algebraic numbers of given height and degree

Let us start with an example. If $p,q$ are rational numbers of height $H$ (the maximum among the absolute value of their nominator and denominator in reduced form). Then we can bound the distance ...
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The image of cyclotomic character is open?

Let $K$ be a finite extension of $\mathbb{Q}_p$, and define $t=(1,\varepsilon_1,\varepsilon_2,\varepsilon_3...)\in \bar{K}_\mathbb{N}$ where $\varepsilon_i^{p^i}=1$ and $\varepsilon_{i+1}^p=\...
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$1-\phi$ is surjective for a $\phi$-module?

Let $A$ be a ring which is complete for the $p$-adic topology, in another word, there is an isomorphism between the two topological rings $A\rightarrow \lim_{\longleftarrow_{n \in \mathbb N_+}} A/P^...
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Height of system of equations in terms of height of a variety

Say $X\subset \mathbb{P}^{n}$ be a $\mathbb{Q}$ variety of degree $\beta$. It is well known and not hard to show that if $X=Z(f_1,...,f_{k})$ where $f\in\mathbb Z[x_0,...,x_n]$ are polynomials of ...
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Silverman's AEC, X.4: How to construct local version of fundamental exact sequence?

I need some clarification in the second last paragraph from Silverman's Arithmetic of Elliptic Curves. I am not sure what 'For each $v \in M_K$ we fix an extension of $v$ to $\bar{K}$, which serves to ...
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30 views

Discriminant of $X=Z(f)$ in terms of join, intersection and projection.

Say we have $X\subset \mathbb{CP}^{n}$ a projective variety, we may even assume it is a hypersurface for convenience. Let $\pi:\mathbb{CP}^{n}\to\mathbb{CP}^{n-1}$ be the standard projection. I am ...
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72 views

$p$ adic Galois group and $p$ adic logarithm

In the book "Theory of p-adic Galois Representations" of Fontaine, I have two questions in Prop3.45 on page60. Let $K\subset L$ be a finte extension of local fields of characteristic $0$. And $H_L:=\...
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84 views

Image of $G_K\rightarrow GL_n(\mathbb{C}_p)$ for a closed subgroup?

Given a continuous map $G_K\xrightarrow{f} GL_n(\mathbb{C}_p)$, where $K$ is a finite extension of $\mathbb{Q}_p$,$G_K:=G_{\bar{K}/K}$ and $\mathbb{C}_p$ is the $p$ adic complex number field. Fixed ...

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