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14 views

Rational points of the kernel of an isogeny of elliptic curves.

Let $E, E'$ be elliptic curves defined over $\mathbb{F}_{q}$ and $$ \phi:E\rightarrow E $$ an isogeny. What can we say about $E(\mathbb{F}_{q^{r}})\cap \ker\phi$? Are there any condition under which ...
1
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80 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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2answers
51 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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1answer
23 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
24 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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47 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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41 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 ...
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72 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 ...
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40 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 ...
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64 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 ...
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1answer
2k views

Learning path to the proof of the Weil Conjectures and étale topology

Assume you are an algebraic geometry advanced student who has mastered Hartshorne's book supplemented on the arithmetic side by the introduction of Lorenzini - "An Invitation to Arithmetic Geometry" ...
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2answers
72 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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19 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
4
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0answers
57 views

Trivialising cover for étale morphisms

Let $f:Y \to X$ be a finite étale morphism of smooth and proper schemes over a field $k$ (not necessarily separable closed). Is there a geometrically connected étale cover $\{U_i\}$ of $X$ which ...
3
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77 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 ...
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42 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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68 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 ...
20
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1answer
255 views

Why are period integrals naïve periods?

Apologies for the long question. I recall the definition of a (naïve) period according to Kontsevitch and Zagier [KS]: A (naïve) period is a complex number whose real and imaginary parts are ...
2
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78 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
77 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 ...
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1answer
41 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 ...
2
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2answers
46 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 ...
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1answer
49 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
60 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 ...
3
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1answer
60 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
64 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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93 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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122 views

On local rings of a normalization

Let $X$ an irreducible singular curve with a singular point $x$. Consider $A$, the normalization of $\mathcal{O}_{X,x}$, and $x_1,\dots,x_r$ the points over $x$ in the normalization of $X$. Why ...
4
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59 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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45 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 ...
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35 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 ...
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 ...
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2answers
3k views

What is the status of the purported proof of the ABC conjecture?

Back in August 2012, Japanese mathematician Shinichi Mochizuki announced a proof of the abc conjecture using Inter-universal Teichmüller Theory. What has been the status of his proof? Has there been ...
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
45 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 ...
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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 ...
2
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0answers
42 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?
2
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1answer
60 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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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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3answers
50 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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54 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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0answers
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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0answers
48 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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60 views

Where does the terminology “fake/false elliptic curve” come from?

In describing, say, the moduli of Shimura curves, people often refer to "fake" or "false elliptic curves" ("les fausses courbes elliptiques"), which are abelian surfaces whose endomorphism ring is an ...
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52 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 ...
31
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1answer
1k views

Has SGA 4$\frac 1 2$ been typeset in TeX?

The title says it all. I've CW'd the question since I'm answering it - this seemed like the best way to get the news out.