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36
votes
0answers
631 views

Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...
21
votes
0answers
370 views

What is Spec of the Adeles?

Let $K$ be a global field and $A_K$ the ring of adeles. What are the prime ideals of $A_K$? I have been told that a full proof of this is quite subtle, but have been unable to find a reference ...
11
votes
0answers
282 views

Prop. 3.2.15 in Liu: Geometrically reduced algebraic variety

I have a problem with proposition 3.2.15 of Algebraic geometry and arithmetic curves of Qing Liu: Let $X$ be an integral algebraic variety over $k$. If $X$ is geometrically reduced then $K(X)$ is a ...
11
votes
0answers
476 views

Self-Intersection Number $-2$

I am new here, but hopefully you can help me with a concrete problem I have. I try to compute a Self-Intersection Number of a constructed curve in an analytic surface. I know the answer by some ...
8
votes
0answers
41 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 ...
7
votes
0answers
249 views

Do Neron models of hyperbolic curves exist

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$. Does there exist a Neron model $\mathcal X$ for $X$ over $O_K$? By a Neron model, I mean ...
5
votes
0answers
86 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} = ...
5
votes
0answers
55 views

When does a variety have a point over a finite field for sufficiently large primes p?

Let $X$ be an algebraic variety over the rational numbers. Suppose that $X$ has positive dimension. I would like to say that $X(\mathbb{F}_p)$ is non-empty for sufficiently large primes $p$. One idea ...
5
votes
0answers
125 views

Computing a contraction of an exceptional divisor.

For a few days, I have been working on the following problem, from Qing Liu's book: Let $\mathcal{O_K}$ be a discrete valuation ring with uniformizing parameter t and residue characteristic $\neq ...
4
votes
0answers
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 ...
4
votes
0answers
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 ...
4
votes
0answers
67 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 ...
4
votes
0answers
102 views

Contraction of curves on a surface and stalks

Let $$f:X \rightarrow S$$ be a fibered surface over a Dedekind scheme of dimension $1.$ Let $$s_1, \ldots, s_n$$ be closed points of S and $\{E_{ij}\}$ irreducible vertical divisors of $X$ with ...
4
votes
0answers
58 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 ...
4
votes
0answers
150 views

If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$

Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be ...
4
votes
0answers
102 views

Curve - non singular curve and its genus

Help me please with this problem:$ X \subset \mathbb{P}^{2}$ defined as $x^{3}y+y^{3}z+z^{3}x=0$ 1.Prove X - non singular curve and find its genus. 2.Prove X - maximal curve over $F_{8}$ field, and ...
4
votes
0answers
127 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
votes
0answers
92 views

find valuations

consider $f(x)=x^4+5x^3+1$. Let $\alpha$ be a root of this polynomial and consider $K=\mathbb{Q}(\alpha)$. Which are the absolute values on $K$ extending the usual absolute values on $\mathbb{Q}$? Are ...
3
votes
0answers
52 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 ...
3
votes
0answers
80 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 ...
3
votes
0answers
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 ...
3
votes
0answers
70 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
votes
0answers
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 ...
3
votes
0answers
44 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
votes
0answers
42 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?
3
votes
0answers
61 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 ...
3
votes
0answers
94 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 ...
3
votes
0answers
69 views

étale cohomology of valuation rings

Let $S$ be the spectrum of a discrete valuation ring (we can assume complete or henselian if necessary). Is it true that the étale cohomology group $H_{et}^2(S,\mathbb{Z})$ is zero?If not in general ...
3
votes
0answers
55 views

What are the easiest surfaces of general type

The "easiest type" of curves of general type are those of genus two. In this case $\chi(X,\mathcal O_X) = -1$ and $\deg K = 2$, where $K$ is the canonical sheaf. I'm a bit lost when it comes to ...
3
votes
0answers
73 views

Is there a construction known for associating a K3 surface to a curve or cover of curves

Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian. Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to ...
3
votes
0answers
65 views

What applications does the theory of fibered surfaces have

Let $C$ be a smooth projective connected curve over $\mathbf{C}$. Let $X$ be a curve over the function field of $C$. Arakelov and Parshin proved the Mordell conjecture by considering a model for $X$ ...
3
votes
0answers
86 views

Is the degree of a Galois morphism bounded by $84(g-1)$

Let $X\to Y$ be a finite morphism which is Galois in the category of smooth projective connected curves over $\mathbf{C}$. Assume $g=g(X) \geq 2$. Is the degree of $X\to Y$ bounded by $84(g-1)$? I ...
3
votes
0answers
84 views

Does composing the Frobenius with an automorphism give another Frobenius

Let $X_0$ be a variety over $\mathbf{F}_q$. Consider the Frobenius $F_0:X_0\to X_0$. Let $X= X_0\times \bar{\mathbf{F}_q}$ and let $F:X\to X$ be $F_0 \times \textrm{id}$. Let $f:X\to X$ be an ...
3
votes
0answers
130 views

Why is the trace map on an abelian variety continuous

Let $X$ be a (EDIT) variety with a group structure. For $a\in A$, let $t_a$ be the translation on $X$: $t_a(x) = a+x$. Why is the function $f:X\to \mathbf{C}$ given by $$f(a) = \sum_{i} (-1)^i ...
3
votes
0answers
193 views

Minimal resolution of singularities of Fermat curve

Fix a prime number $p$. Let $F$ be the geometrically connected Fermat curve of exponent $p$ over a number field $K$, i.e., $F$ is given by the equation $x^p+y^p = z^p$. Consider the same equation ...
3
votes
0answers
118 views

places valuations and so on

I am looking for complete references on places, valuations and so on. In particular I need to understand the meaning of "a finite place does not divide a prime number $\ell$" and what is the Frobenius ...
2
votes
0answers
65 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 ...
2
votes
0answers
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 ...
2
votes
0answers
43 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 ...
2
votes
0answers
108 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 ...
2
votes
0answers
196 views

What would be the most rigorous book to stydy algebraic geometry and arithmetic curves on my own?

I would like to study algebraic geometry and arithmetic curves on my own but are there suggestions where to start? Namely, I like very rigorous way to do mathematics and I was suggested Liu's book ...
2
votes
0answers
102 views

Deep questions in number theory not accessible by combinatorial results

Number theory and arithmetic geometry were invented to solve many questions about properties of numbers. What are the some of the foundational results or estimates that are accessible to powerful ...
2
votes
0answers
34 views

another representation of the zeta function of a curve over a finite field

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...
2
votes
0answers
54 views

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer

As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= [3;1,2,1,6,1,2,1,6…]$,let $L$ the length of period of simple continued fraction expansion of quadratic algebraic numbers be the number of ...
2
votes
0answers
46 views

The characteristic polynomial of Frobenius of an RM curve

Let $C$ be a genus two curve over $\mathbb{Q}$. We can reduce $C$ modulo a prime $p$ to obtain a curve $\bar{C}$ over $\mathbb{F}_p$. By counting points of $\bar{C}$ over $\mathbb{F}_p$ and ...
2
votes
0answers
45 views

Etale cohomology and restricted direct product

[migrated to mathoverflow: http://mathoverflow.net/questions/161734/etale-cohomology-and-restricted-direct-product] $\newcommand{\h}{\operatorname{H}}$ Let $k$ be a global field, $A$ an abelian ...
2
votes
0answers
63 views

Elliptic curve which attains potential good reduction over an Artin-Schreier extension.

I am looking for an elliptic curve $E$ over the field $\overline{\mathbb{F}_{p}}((t))$, which attains good reduction over an Artin-Schreier extension of $\overline{\mathbb{F}_{p}}((t))$, i.e.: an ...
2
votes
0answers
82 views

Pole of differential

Let $E : y^2 = x^3 + ax + b$ be an elliptic curve over the field $K$, char $K \ne 0$. We know that the differential $\omega = \frac{dx}{y}$ is holomorphic in infinity because we can write it as ...
2
votes
0answers
144 views

Do K3-surfaces have Weierstrass equations

I've been wondering a bit about K3-surfaces and their analogy to elliptic curves. I've just started so this might be a very silly question. Do all K3-surfaces have a Weierstrass equation (up to ...
2
votes
0answers
58 views

What are the branch points of $X(n)\to X(1)$

Let $\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$ be a finite index subgroup. Let $X_\Gamma \to X(1)$ be the corresponding morphism of compact connected Riemann surfaces (obtained by adding the cusps). ...