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16 views

About characteristic polynomial of Frobenius endomorphism

I was looking for some information related to the values of the characteristic polynomial $\chi(t)$ of the Frobenius of a Jacobian of a hyperelliptic curve $C$ of genus 2 over $\mathbb{F}_q$ and in ...
2
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1answer
42 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
39 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
45 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
43 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
44 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
43 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 ...
1
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1answer
52 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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0answers
74 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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0answers
33 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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0answers
47 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
50 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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0answers
44 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
28 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
40 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
25 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 \\ ...
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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
32 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
34 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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0answers
30 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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0answers
29 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 ...
2
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1answer
51 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
47 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
37 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 ...
1
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0answers
48 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
36 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 ...
1
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0answers
40 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 ...
1
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0answers
49 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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0answers
64 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
55 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, ...
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1answer
46 views

Short Primitive Vectors in a Lattice in $\mathbb{Z}^2$

Given $a,n$ coprime positive integers, let $L = \{(x,y)\in \mathbb{Z}^2, ax=y(n)\}$ be the lattice of all points satisfying $ax=y\pmod{n}$. I want to find an order-of-magnitude bound on the shortest ...
3
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1answer
51 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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2answers
116 views

Not affine, projective, geometrically connected, geometrically reduced, nor geometrically regular…

Is there a field $k$ and a regular integral $k$-variety $X$ that is neither affine, projective, geometrically connected, geometrically reduced, nor geometrically regular?
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1answer
109 views

A Guide to the Proof of Fermat's Last Theorem from the Modularity Theorem

A few years ago, a friend of mine told me that he had taken an advanced undergraduate number theory class (so something that assumed only a knowledge of algebra and mathematical maturity) which ended ...
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0answers
38 views

Arakelov Theory and Arakelov curves

There exists a definition of Arakelov Curve in Arakelov theory? My question is because Neukirch (Algebraic Number Theory, Chapter III) defined Arakelov divisors in the set $X=Spec(\mathcal ...
3
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0answers
37 views

Why do we assume the ring to be torsion free when dealing with formal logarithms in the context of formal group laws?

Let $F$ be a formal group over a ring $R$. Why do we require that $R$ has no additive torsion before we discuss formal logarithms?
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1answer
24 views

connection between absolute irreducibility and smooth+geometrically connected

Let $C_1$ and $C_2$ be two smooth, projective, and geometrically connected curves over a field $K$ of characteristic $0$. Assume there are two non-constant branch mapping of $K$-curves, $\phi_1\colon ...
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0answers
70 views

Fano surfaces all of whose rational points lie on some geometric line

Are there any ? Namely let $X$ be a smooth del Pezzo surface defined over $\mathbb{Q}$ that has rational points and such that the degree of the del Pezzo is small, say $d=3$ or $4$. Is it possible ...
2
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1answer
57 views

Lifting points of étale group scheme.

Consider the following setting: let $(R,\mathfrak{m})$ be a Noetherian complete discrete valuation ring (with maximal ideal $\mathfrak{m}$) and let $K$ be its field of fractions. Now take $L$ be the ...
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0answers
78 views

Some questions about reduction of elliptic curves

Let $E \rightarrow S$ be an elliptic curve (i.e, a smooth proper curve of genus 1). If $S = \text{Spec (K)}$ where $K$ is a local field, the usual way of doing a reduction at a prime $\mathfrak{p} = ...
0
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1answer
41 views

Incommensurable units as ratios

I am having a bit of trouble understanding the concept of an incommensurable unit. From what I have gathered so far, it is simply a magnitude that cannot be expressed as the ratio of two natural ...
0
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1answer
53 views

field of rational functions of a curve

Let $C$ be the algebraic curve defined by the modular polynomial $\phi_N$ of order $N>1$ over the rational numbers, i.e. \begin{equation}C:=\text{specm}(\mathbb{Q}[X,Y]/\phi_N(X,Y)). \end{equation} ...
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0answers
59 views

degrees of L-functions and dimensions of Shimura Varieties

I try to grab a (very) little understanding of what a Shimura variety is, and although I still don't understand the formal definition of that notion, a few vague ideas have come to my mind. Hence ...
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1answer
64 views

Extension of a complete discrete valuation ring

My question came when I was reading the famous Tate's paper on $p$-divisible groups. At the beginning of chapter $(2.4)$ he cites this fact as obvious. If you take a complete discrete valuation ring ...
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0answers
104 views

Understanding the Falting's Theorem

I'm an undergraduate student of mathematics, but soon I'll graduate, and as a personal project I want to understand Falting's Theorem, specifically I want to understand Falting's proof; but yet I have ...
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0answers
28 views

Cartier dual of an exact sequence

Suppose we have an exact sequence of affine finite flat commutative group schemes over an arbitrary ring $R$: \begin{equation} 0\rightarrow H\xrightarrow{i} G\xrightarrow{j} K\rightarrow 0 ...
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0answers
22 views

Rationality of intersection of algebraic groups

Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and ...
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0answers
31 views

Explicit value of $\overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{Z})}(1).\overline{\mathcal{O}}_{\mathbb{P}^1(\mathbb{Z})}(1) $

Let us consider the Fubini-Study metric on the part at infinity of the line bundle $\mathcal{O}_{\mathbb{P}^1(\mathbb{Z})}(1)$ to obtain the Hermitian line bundle ...
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0answers
87 views

Witt vector question

I've started reading various papers and notes on Schemes over the Witt Vectors. In example 8.8 of these: https://www.uni-due.de/~mat903/books/esvibuch.pdf W2 has addition defined as $k \oplus k\cdot ...
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1answer
74 views

Points on an elliptic curve over $\mathbb F_p$

Let $E$ be an elliptic curve over $\mathbb F_p$ (the finite field of $p$ elements) defined by $y^2=x(x-n)(x-m)$ where $p\nmid nm(n-m)$. Let $N$ be the number of $\mathbb F_p$-valued points of $E$. ...