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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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Is $p$ an anomalous prime?

Let $E$ be an elliptic curve defined over the number field $K$ and $p$ be a rational prime such that for all $v\mid p$, $E$ has good ordinary reduction. If $E[p]\subseteq E(K)$, can we conclude $p$ is ...
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$p$-th coefficient of weight $2$ new form with $p | N$ must be $1$ or $0$ or $-1$?

Let $f= \sum_{n=1}^{\infty}a_nq^n \in S_2^{new}(N)$ be a normalized new form of weight $2$ with respect to $\Gamma_0(N)$ and assume $p|N$ is a prime. Then must $a_p=0$ if $p^2|N$ and belongs to $\{-1,...
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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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65 views

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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30 views

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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39 views

Roadmap to rigid cohomology

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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2answers
58 views

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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43 views

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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20 views

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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68 views

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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1answer
124 views

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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1answer
44 views

$\mathbb{C}$-valued points and flatness

Let $X$ be an integral scheme over $\mathrm{Spec}(\mathbb{Z})$, we denote $X(\mathbb{C})$ as the set of $\mathbb{C}$-valued points in $X$. Then $Y\rightarrow \mathrm{Spec}(\mathbb{Z})$ is flat if $Y(\...
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1answer
40 views

Torsion points of an elliptic curve (example in Silverman)

Let $E$ be the elliptic curve $$y^2=x(x-2)(x-10)$$ Silverman claims (Example. X. 1.5 p.315 Arithmetic of Elliptic Curves) that $E(\mathbb{Q})_{tors}$ injects into the reduction $\widetilde{E}(\mathbb{...
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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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1answer
226 views

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 ...
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42 views

Frobenius for modular curves.

Actually, I am a bit confused about the notation and what is called Frobenius for modular curves. Let $N\geq 4$ be an integer, let $R$ be an $\mathbb{F}_p$-algebra, where $p$ is a prime not dividing $...
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Pullback of a scheme along the Frobenius morphism

Let $S$ be an $\mathbb{F}_p$-scheme for a given prime $p$. Let $X$ be a scheme over $S$. Then, we can consider the Frobenius morphism of $X$ relative to $S$, defined by taking the unique morphism ...
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Base change of horizontal divisor in semi-stable curves

Let $X\rightarrow Spec(O_K)$ be a semi-stable curve, where $K$ is an algebraic number field. Let $D$ be a horizontal divisor and finite extension $K'\supset K$ contains the splitting field of the ...
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Algebraic 1-cocycles and Galois gerbs

We have the following set up: $K/F$ is Galois, $D$ is an algebraic group of mult. type and $E$ is an extension of groups: $$1\to D(K)\to E\to Gal(K/F)\to 1$$ Now take a linear algebraic group G over F....
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Base change to make singular points on singular fibers split

Let $C$ be a semi-stable curve over $Spec(O_K)$ where $K$ is an algebraic number field. Assume that $X_s$ is singular for $s\in Spec(O_K)$. How do we know that there exists a finite separable ...
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1answer
45 views

Is an integer extension of a ring the integral closure of this ring in some extension?

This is a question that I just came up with, so it may be completely stupid and I sincerely apologize if that's the case. The question is; Take $A$ to be an integral domain. Let $B$ be an integral ...
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22 views

What would a arithmetic surface look like

By arithmetic surface we mean a projective, regular and flat scheme of dimension 1 over $Spec(O_K)$ for some algebraic number field $K$. So we can view such a scheme $X$ as a closed subscheme of $...
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Global sections of a Vojta divisor: a lemma for Faltings' theorem and Vojta's inequality

This is E.5 of Hindry, Silverman's Deophantine Geometry. I want a global section of a Vojta divisor from some polynomials. Let $K$ be a number field, $C$ a smooth projective $K$-curve of genus $g \ge ...
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42 views

Degree of variety over $\mathbb{Q}$ versus over $\mathbb{F}_p$

Let $V$ be a projective variety (possibly reducible) in $\mathbb{P}^n$ defined over $\mathbb{Z}$. What is the relation between the degree of $V$ seen as a variety over $\overline{\mathbb{Q}}$ and ...
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30 views

Is rational connectedness a constructible property?

Let $f:X\rightarrow S$ be a morphism of varieties (say, over an algebraically closed field $k$). Is the locus of points above which the fibers of $f$ are rationally connected $\{s\in S:X_s \text{ is ...
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Finiteness of the Brauer group and torsion-freeness of Picard group of a rationally connected variety over an algebraically closed field

Let $X$ be a rationally connected variety over an algebraically closed field $k$. Then (1) is the Brauer group $\text{Br}(X)$ finite? (2) is the Picard group $\text{Pic}(X)$ torsion-free?
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1answer
27 views

Rational points on elliptic curve over a local field correspond to integral points on minimal Weierstrass model

Let $R$ be a complete discrete valuation ring, with fraction field $L$. Let $E$ be an elliptic curve over $L$, and $W$ a minimal Weierstrass model over $R$. Why is $W(R) \simeq E(L)$? We have a map $...
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41 views

Formal completion of modular curves

Let $N\geq 4$ be an integer coprime with $p$, where $p$ is a fixed prime number. Then we know that there exists a scheme $Y_N$ over $\text{Spec}(\mathbb{Z}_p)$ whose $\text{Spec}(R)$-points are ...
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1answer
85 views

topology on the ring of Witt vectors in the theory of period rings of Fontaine

For a $p$-adic field $K$ with perfect residue field $k$, we know the standard construction of the ring $R$. I will recall it briefly. It is $\varprojlim_{x \rightsquigarrow x^p} O_{C_K}/pO_{C_K}$, ...
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1answer
83 views

Points of Scholze's Anticanonical Tower

In his paper "On torsion of locally symmetric spaces", Schole introduces a perfectoid space, which is the anticanonical tower. He claims that points coming from $\text{Spa}(C,C^+)$, where $C$ is a ...
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1answer
46 views

Endomorphism of the elliptic curve $y^2 = x^3 - ax$

For the elliptic curve $$E:y^2 = x^3 - ax$$I know that $E\left(\mathbb{C}\right)\cong\mathbb{C}/\left(\mathbb{Z}i + \mathbb{Z}\right)$ and hence $\text{End}_{\mathbb{C}}\left(E\right)\cong\mathbb{Z}\...
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Geometrical proof of the impossibility of angle trisection by straightedge and compass

There's a fascinating (almost) geometrical proof of the impossibility of angle trisection by straightedge and compass by Terrence Tao which is somehow more comprehensible (and more directly to the ...
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Infinite subgroups of elliptic curves and quotients

Let $E_1$ be an elliptic curve over $\Bbb Q$ and $S \subset E_1(\Bbb Q)$ be a subgroup. Is there an elliptic curve $E_2$ with an algebraic group morphism $E_1 \to E_2$, and such that $E_2(\Bbb Q) \...
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33 views

Smooth rational points of affine curves under birational maps

Suppose $K$ is an infinite field of char. $0$. Let $C$ be an $K$-irreducible affine algebraic $K$-curve and suppose that $P_1,\ldots, P_n$ are non-singular $K$-points of $C$. Can one always find a ...
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Tate module of an Abelian Scheme

It is well known that if $A/k$ is an Abelian Variety over (the spectrum of) a field, a very important object to consider is its Tate module $T(A):=\underset{\underset{n}{\longleftarrow}}{\lim}A[p^n](\...
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2answers
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Construction of arbitrary regular polygons with ruler and compass

A lot of arithmetic geometry has to do and took off with constructions or proofs of non-constructibility of specific figures only with ruler and compass. See e.g. the non-constructibility of the ...
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1answer
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Analogies between Hodge conjecture and Tate conjecture

I hear sometimes that there is many analogies between the Hodge conjecture and the Tate conjecture. If we take a look at the statements of this two conjectures, we have the followings : The Tate ...
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Why do vertical divisor not contribute to the “intersection pairing”?

Let $X\to S=\operatorname{Spec}(O_K)$ be an arithmetic surface. We denote with $X_s$ the fiber over $s\in S$ and let $\operatorname{Div}_s(X)$ be the set of divisors on $X$ with support contained in $...
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Solving equations in $\mathbb{Z}/p\mathbb{Z}$ versus $\mathbb{Q}_p$.

According to Silverman: In order to show that an algebraic set $V/\mathbb{Q}$ has no $\mathbb{Q}$-rational points, it suffices to show that the corresponding homogeneous polynomial equations ...
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266 views

Logarithmic height of algebraic numbers

Let $a$ and $b$ algebraic numbers over $\mathbb{Q}$. Do you know (or recall) if there are simple uppper bounds relating the logarithmic height of $ab$ (or $a/b$) with the logarithmic height of $a$ ...
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69 views

motives in current research

Are motives still studied in current research? Searching various papers I can see that a lot of these works about motives are quite old. I'm interested in arithmetic geometry and i'm also really ...
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2answers
40 views

(Reference request) How to show elliptic curve has positive Mordell-Weil rank

I know there must be a lot of ways to show an elliptic curve has positive Mordell-Weil rank if it really does. And I guess that I am supposed to collect them by myself. But since I am not working in ...
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Checking equality of maps over a reduced scheme enough to check on fibers

Let $E \to S$ be elliptic curves over a reduced scheme $S$. Let $f:E \to E’$ be a non constant homomorphism of elliptic curves over $S$. Let $N=\deg(f)$. Let $f^t:E’ \to E$ be the dual morphism. To ...
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height function over function fields

I want a good modern reference for basic definitions and facts about function fields. lang wrote some books and articles about height functions but they are in old language and I cant understand them.
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Involution On elliptic curve

In an elliptic curve E given by $y^2=x^3 + ax^2 +bx+ c$ and origin at the point of infinity, why does the map $i$ sending $(x, y)$ to $(x, -y)$ send $i(P)=-P$, where $-P$ denotes the inverse of $P$ ...
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1answer
20 views

Calculate x/y coordinates of an overlayed image when the underlying image is resized. [closed]

I can't post images until I have 10 reputation points. I'll try to explain without them. I have an image that is 35x29 pixels in size and overlayed on an image that is 168x46 pixels in size. The ...
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1answer
34 views

Weierstrass equation for a family of elliptic curves, pushforward of sheaf

Let $f:E \to S$ be a (family of) elliptic curve. Let $[0]:S \to E$ be the zero section and $I:=I([0])$ its ideal sheaf. Why is $f_*(I^{-n})$ locally free, as claimed by Hida in the following excerpt ...
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Pushforward of sheaf of relative differentials in family of elliptic curves [duplicate]

Update: never mind, This question has been asked before here Let $f:E \to S$ be an elliptic curve (precise definition is given below from Hida’s book Geometric Modular Forms and Elliptic Curves). Is $...