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Area of an ellipse, Grid points, Approximation

Let $f(x,y)=ax^2 + bxy + cy^2$ with $a,b,c \in \mathbb{Z}$ a primitive and positiv definite quadratic form with discriminante $D < 0$. Then $ax^2 + bxy + cy^2 = N$ with $N \in \mathbb{N}$ describes ...
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27 views

Lifting points via étale morphism of adic spaces

This question was suggested to me during the reading of Huber's book about Etale Cohomology of Adic Spaces. I formulate this question here in the context of adic spaces, but I think, since a morphism ...
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19 views

Lifting morphisms of $p$-divisible groups using Grothendieck Messing theory

During my reading of Peter Scholze and Jared Weinstein's paper ``Moduli of $p$-divisible groups'' I found this assertion in the proof of Proposition 6.1.3. Consider the following situation. Let $k$ be ...
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44 views

Is there an universal term for difference of transformations?

Tl;dr: What is the universal name for $C$ when comparing two transformations in $C = B * A^{-1}$, assuming C is a transformation itself? I'm looking into comparing two transformations and extract ...
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31 views

Closed point in the generic fiber of an arithmetic surface

Let $S$ an irreducible Dedekind scheme of dimension $1$, and let $\pi:X\to S$ be a regular, integral fibered surface. We assume that $\pi$ is a flat morphism and that $X_\xi$ is the generic fiber over ...
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28 views

Weil conjectures - If two varieties have the same of Fq^d - valued points for all d >> 0, then they have the same Hasse - Weil function

I was working on the following exercise for fun, and I haven't really gotten anywhere with it. Let Z( X; t) be defined as exp ( $\sum_{r= 1}^{\infty} N_r t^r/r$), where $N_r$ is the size of X($\...
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84 views

Intersection of algebraic curves at a point with given multiplicity

I don't know if this question is too basic for MO, so I put it here, but if you think I should migrate the question to MathOverflow please suggest me. Let $C/k$ be a smooth curve over a perfect ...
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1answer
28 views

Morphism smooth over the function field. What does it mean?

Look at the following lines I found in the book "Moriwaki - Arakelov geometry" (beginning of chap. 4): Let $S$ be a connected Dedekind scheme with function field $K$. Let $\pi:X\to S$ be a ...
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1answer
25 views

The cardinal of the Mordell-Weil group is prime for certain elliptic curves over $\mathbb{F}_{q}$ for certain $q$.

Let $p\in\{2,3\}$ and $r\in\mathbb{Z}_{\geq 2}$. I would like to find if there exists an elliptic curve defined over $\mathbb{F}_{p}$ such that $|E(\mathbb{F}_{p^{r}})|$ is a prime number. If $p>3$ ...
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54 views

Elliptic curves with trivial Mordell–Weil group over certain fields.

I am looking for elliptic curves $E,E'$ defined over $\mathbb{F}_{3}$ and $\mathbb{F}_{4}$ respectively and given by a Weierstrass equation such that their Mordell-Weil group is trivial, i.e. such ...
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43 views

How to compute explicitly the covering map in the modularity theorem?

The modularity theorem (original Shimura-Taniyama-Weil conjecture) asserts the existence of a covering (uniformization) map $\pi:X_0(N) \to E$ for every $E$, an elliptic curve defined over $\mathbb{Q}$...
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46 views

Is torsion of a topological module closed?

I was asking to myself the following question. Consider a $p$-adically complete and separated topological algebra $R$ over $\mathbb{Z}_{p}$. As $\mathbb{Z}_{p}$ is a domain, we know that the $\mathbb{...
4
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73 views

Structure Sheaves of Rigid Analytic Spaces: What is the right Value Category?

Let $K$ be a complete non-archimedean field and let $X$ be an affinoid $K$-space. Then the structure sheaf $\mathcal O_X$ of $X$ is defined first on the "weak Grothendieck topology", ie on affinoid ...
3
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54 views

Infinite primes (places) of a number field geometrically

Given a (global) number field $K$, thinking of the affine scheme $\mathrm{Spec}\mathcal{O}_K$ can gige an insight into (at least) some kf the number-theoretic terminology, e.g. ramification or local ...
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66 views

What constitutes a good reading course in $p$-adic number theory?

I have had a course in number theory where I studied Marcus and also a course in differential geometry. I have read Koblitz's introductory book on $p$-adic numbers. I am roughly interested in both $p$-...
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21 views

Reference for Cohomology of Arithmetic Groups

Does anyone know a good lecture notes that explains arithmetic groups and their cohomology from basics? Thank you
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87 views

Kummer map and cohomology group for an elliptic curve

Let $E=E_q$ be the Tate ellipitc curve over a finite extension $K$ of $\mathbb{Q}_p$ for a $q$. Let $T$ be its p-adic Tate module. Let $\mathfrak m$ be the maximal ideal in $K$. I saw in this paper (...
4
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2answers
73 views

${\mathbb{Q}_p^*}^2$ is open in $\mathbb{Q}_p^*$

Show that the set of squares in $\mathbb{Q}_p^*$ is open in $\mathbb{Q}_p^*$. Here $\mathbb{Q}_p$ is the $p$-adic numbers and $\mathbb{Q}_p^*$ is the set of units in $\mathbb{Q}_p$. I know that $\...
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44 views

Finding a finite Galois cover trivializing a lisse sheaf

Given a lisse $\mathbb{F}_{\ell^r}$-sheaf on a smooth curve $U$ defined ofer $\bar{\mathbb{F}_p}$, Katz says here, in p. 33, that $\mathcal{F}$ "becomes constant on some connected finite etale galois ...
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1answer
73 views

“Lifting” fibres of morphism of arithmetic schemes to get rid of “nongeometric” ramification

This is a soft question and really a request for pointers towards a certain rigorous formulation of geometric intuition I've had for some "arithmetic schemes". I'm looking for ideas and key references ...
2
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81 views

What are some topics of advanced number theory every young geometers should know? (soft question)

By "advanced number theory", I mean topics like arithmetic/Diophantine geometry, modular/automorphic forms and Shimura varieties. I'm interested in derived/non-commutative algebraic geometry, some ...
3
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1answer
83 views

What is the significance of Coleman maps arising in Iwasawa thoery?

I have come across two instances of "Coleman map" Let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $k_\infty$ be the unique $\mathbb{Z}_p$ extension of $\mathbb{Q}_p$ contained in $\...
2
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1answer
46 views

What's so special about hyperbolic curves?

This is really a two-part question, but I would be happy to get an answer for either bit. By a hyperbolic curve as defined by e.g. Szamuely in Galois Groups and Fundamental Groups (p.137) I mean an ...
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2answers
47 views

A fundamental tool used in the study of Diophantine equations…

Notations: $K$ is a perfect field, $\overline K$ an algebraic closure and $V\subseteq \mathbb P^n(\overline K)$ is a projective variety on $\overline K$. If $V$ is defined over $K$, in symbols $V/K$,...
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1answer
51 views

Is every integral point of an arithmetic scheme contained in an affine open set?

The schemes $$X = Proj \mathbb{Z}[s,t] = \mathbb{P}^1_{\mathbb{Z}}$$ and $$Y = Proj \mathbb{Z}[x,y,z]/(x^2 + y^2 - z^2)$$ both have isomorphic generic fibers as schemes over $\mathbb{Z}$, and there is ...
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1answer
63 views

Will someone kindly explain Kato's dual exponential map?

I am reading this article by Rubin. Will somebody how to derive the formula given in equation 2 of section 5? It states thus: $z$ corresponds to the map $$x\mapsto Tr_{\mathbb{Q}_{n,p}/{\mathbb{...
3
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1answer
63 views

Multiplication by $m$ isogenies of elliptic curves in characteristic $p$

I've been attempting to prove some comments I've read on MO by myself for my undergrad thesis regarding étale morphisms of elliptic curves. My definition of an étale morphism is taken from Milne's ...
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1answer
65 views

Geometrically ireducible curve

I know that curve with coefficients in $k$ is geometrically ireducible if it does not factor over algebraic closure of $k$. I have this curve, for example, $$2x^2+2x^2y+2y^2+2xy+3xy^2=1.$$ It's ...
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107 views

J. Silverman exercise 3.12 “The arithmetic of Elliptic curves”

I have question regarding exercise 3.12 of J. Silverman "The arithmetic of Elliptic curves". It states the following: Let $m \geq 2$ be an integer, prime to $\text{char}(K) > 0$. Prove that the ...
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37 views

Generalized elliptic curves over cusps and orbits of $\mathbb{Q}\cup\infty$

In the post http://mathoverflow.net/questions/51147/what-objects-do-the-cusps-of-modular-curve-classify, it says that the fibers over the cusps of a modular curve are n-gons. Wikipedia (https://en.m....
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70 views

Tate curve and cusps

I know this is a naive question, but what is the relation between the Tate curve and cusps on a modular curve? Naive googling seems to suggest that level structures on the Tate curve (up to ...
3
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1answer
56 views

Let $K = \mathbb{F}_p(t)$, $X$ be the affine curve $(t^5 + 1)y^2 = x^5 + 1$, what is $p_g(X)$? Is $X(K)$ finite? [closed]

Let $p$ be a prime not equal to $2$ or $5$. Let $K = \mathbb{F}_p(t)$. Let $X$ be the affine curve $(t^5 + 1)y^2 = x^5 + 1$ in $\mathbb{A}_K^2$. I have two questions. What is $p_g(X)$? Is ...
3
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46 views

$E[n]$ is etale locally $(\mathbb{Z}/n\mathbb{Z})^2$

I don't think we need the entire setup below (from Katz and Mazur's elliptic curve book, pages 74 and 75), but, as a beginner, I am unable to identify the assumptions I need. Let $S$ be the open ...
2
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1answer
32 views

descending smoothness along unramified morphisms

Let $f:X\rightarrow Y$, and $g:Y\rightarrow Z$ be morphisms of schemes locally of finite type. Suppose that $f$ is unramified and surjective, $g$ is flat and $g\circ f$ is étale. Does $g$ is also ...
3
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1answer
45 views

Elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic

I am finding elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic. Elliptic curves over $\mathbb C$ can be identified with $\mathbb C/\land$ for some lattice $\land$. And ...
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1answer
28 views

Laurent expansion at infinity for a weakly modular function with respect to a congruence subgroups

Let $\Gamma\subset \mathrm{SL}_2(\mathbb Z)$ be a congruence subgroup and $h$ the fan width of $\Gamma$ (i.e; the minimum $h>0$ such that $\left(% \begin{array}{cc} 1 & h \\ 0 & 1 \\ ...
0
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1answer
25 views

Recursive writing involving arithmetic progression

I've been trying to figure out this recursion problem but I'm getting stuck trying to find the nth-term sequence for the last recursion. I found one but the second i'm so clueless about. I don't know ...
3
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1answer
47 views

$J$ invariant of elliptic over a number field

Suppose $E$ and $E’$ are elliptic curves over a number field $K$ which are Galois conjugate over $\mathbb Q$. So $\operatorname{End}_C(E)$ and $\operatorname{End}_C(E’)$ are isomorphic. Suppose $\...
2
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0answers
44 views

Asking for some exercises to help me understanding abelian varieties better?

I want to study Mumford's Abelian Varieties in the coming winter break. I tried to study it before, but I didn't find my self really understanding(or memorizing) too much. I guess a better and more ...
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39 views

Weil pairing of curve of genus 2

We know there is Weil pairing for elliptic curve satisfying several nice properties. So do we have Weil pairing for other curves also satisfying the nice property? Especially genus 2 curve?
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34 views

X a genus 2 curve. Element of prime order in Aut(X) has order at most 5.

Let X a genus 2 curve. Aut(X) finite. I want to prove that Element of prime order in Aut(X) has order at most 5. What I know is non-identity endomorphism of X acts nontrivially on the Tate module of ...
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1answer
61 views

2-torsion points in a curve with genus 2

Let X be a genus 2 curve with affine equation y^2 = f(x), where f is a polynomial of degree 6. Write $P_1, ..., P_6$ for the points on X(C) with y=0. Then why every $P_i-P_j$ is a 2-torsion points in ...
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3answers
53 views

Quotienting out polynomial rings by polynomial-generated ideals

I'm studying in Lorenzini's Arithmetic Geometry, and it has been a while since I've taken a rigorous algebra course. I'm trying to understand a certain step in his proof that $\mathbb{Z}[\sqrt{5}]$ ...
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2answers
42 views

Smooth curve defined over K with genus 1 is always isomorphic, over $\overline K$, to an elliptic curve over K

Here, the point is the smooth curve defined over K with genus 1 may not have rational point. But to be an elliptic curve defined over K, the base point must be a rational point. I tried to use ...
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0answers
55 views

Automorphism of m-torsion subgroup of an elliptic curve determines the automorphism of the entire elliptic curve

For $m\ne2$ I want to show that if two automorphisms coincide on $E(m)$, which is the $m$-torsion subgroup of the elliptic curve $E$, then these automorphisms are the same. The statement is very ...
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37 views

Closure of a rational point is irreducible

Let $X$ be a scheme of finite type over a field $k$ which is algebraically closed of characteristic zero. Let $K$ be another field and $\eta \in X(K)$ be a $\mathrm{Spec}(K)$ point of $X$. Is the ...
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49 views

What is the correct generalization of degree of a divisor to the number field case?

When describing smooth algebraic curves over a field $k$, there are (at least) two useful notions of "class group". The first generalizes easily to general schemes: the Picard group can be defined as ...
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54 views

Can I choose the base point of an elliptic curve arbitrarily?

If I define an elliptic curve as a smooth curve with genus one and with base point $\mathbb O$, it seems that I can choose this base point arbitrarily. When I go through the proof that establish ...
3
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72 views

Application of GRR in number theory

In Neukirch Book Algebraic Number Theory page 254, states the Grothendieck-Riemann Roch-Theorem, but missing of applications. Do you know references for applications for this theorem, or may be ...
3
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1answer
61 views

Why is $\mathcal{O}_F \otimes \mathbb{Z}_p$ a Dedekind ring?

In the paper Compactifications de l'espace de modules de Hilbert-Blumenthal (Compositio. '78), M. Rapoport relates some assertions from a letter of Deligne to Serre, dated 1972. At the very beginning, ...