# Tagged Questions

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91 views

### What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

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}= ...

**2**

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**3**answers

164 views

### How to define the input of computable function or Turing machine over real numbers

Computation or computability over $\mathbb{N}$ can be extended to computation or computability over $\mathbb{R}$ or even computation or computability over $\mathbb{C}$.The following is a formal ...

**4**

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**1**answer

153 views

### Which valuations of a field yield codimension $1$ subschemes of their 'models'

For a field $F$ (for example, a one generated by a finite number of its elements) there is a directed set of its 'models' (in this case those are 'arithmetic' schemes whose fraction field is $F$). It ...

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**0**answers

94 views

### Are there generating functions of rational or integral solutions of Diophantine equation that

As we know,there are generating functions for c.e.languages which are some retricted rational or algebraic or transcendental functions dependent on the class of languages like regular ...

**6**

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183 views

### The Rappoport-Zink spectral sequence vs. the one of the complement of a normal crossing divisor

As far as I understand these matters, for a regular $\mathfrak{X}$ that is proper flat of finite type over $\operatorname{Spec}\mathbb{Z}_p$, the Rappoport-Zink spectral sequence relates the etale ...

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90 views

### Benchmark problems for computing rational points on varieties

Are there standard benchmark problem sets used for empirically evaluating algorithms designed for computing rational points on (various classes of) algebraic varieties?
If so, could you please point ...

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102 views

### algebraicity of Néron-Tate canonical height for Abelian varieties over global function fields

(transcendence of canonical heights)
Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always ...

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615 views

### “Descending cohomology, geometrically” by Mazur:

(Exist texts of that talk or related texts: http://ttv.mit.edu/collections/harris60/videos/13881-problem-session-barry-mazur ?) Article: http://www.math.harvard.edu/~mazur/papers/page37.pdf

**6**

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**1**answer

379 views

### Serre-Tate 1964 Woods Hole notes

I am not sure if this is the right venue to ask this. Apologies in advance.
I would like to clarify the following. When people give as reference:
J.-P. SERRE and J. TATE.-Mimeographed notes from ...

**0**

votes

**1**answer

173 views

### Algebraic varieties in “mixed” affine spaces

Let $K\subset L$ be a field extension and let $K\subset F_1,F_2,...,F_n\subset L$ be proper intermediate fields. Consider the "mixed" affine space $\mathbb{A}_{(F_i)}:=\prod_{i=1}^n F_i$ instead of ...

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198 views

### Where was the arithmetic zeta function of a scheme first defined?

Let $X$ be an arithmetic scheme, that is, a scheme of finite type over the integers. We denote the set of closed points of $X$ by $|X|$. For every $x\in|X|$, write $N(x)$ for the cardinality of the ...

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**0**answers

111 views

### Dualizing sheaf in mixed characteristic for regular schemes.

I've been looking many places, but everything I find seems to either talk about (a) varieties or (b) extremely general situations with dualizing complexes. As I am not in the situation of (a) (i.e. ...

**13**

votes

**1**answer

737 views

### Status of Grothendieck's conjecture on homomorphisms of abelian schemes

In [1] Grothendieck posits the following:
Conjecture. Let $S$ be a reduced connected scheme, locally of finite type over Spec($\mathbf{Z}$) or a field $k$, $A$ and $B$ two abelian schemes over $S$, ...

**2**

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**2**answers

460 views

### Classification of quasi-split unitary groups

Let $U$ be a unitary group defined with respect to an extension $E/F$ of non-archimedean local fields, and assume it is realised with respect to a pair $(V,q)$, where $V$ is an $n$-dimensional vector ...

**12**

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**2**answers

1k views

### Learning path for the proof of the Weil Conjectures

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" ...

**6**

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**2**answers

509 views

### questions on Néron-Tate canonical height

I have three questions regarding height pairings:
In [Serre, Lectures on the Mordell-Weil theorem], p. 85 f., it is stated that the following function is a local height function:
"Let $V/R$ be a ...

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vote

**1**answer

361 views

### Elliptic subfields of a function field

Let $C$ be a curve and $K(C)$ be its function field of genus 2, where $K$ = $\mathbb{C}$.
The number of essential elliptic subfields of $K(C)$ is 0 or 2 or $\infty$.
Edit: I am looking for a proof. ...

**2**

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**0**answers

224 views

### modern reference for Néron's “Quasi-fonctions et Hauteurs sur les Varietes Abeliennes”

Is there a modern reference for Néron's "Quasi-fonctions et Hauteurs sur les Varietes Abeliennes" http://www.jstor.org/pss/1970644 i.e. using Grothendieck's language of schemes and in English?

**2**

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**1**answer

334 views

### good references in moduli stack and stable reduction

Hello,everyone. I'm looking for some good references in moduli stack and stable reduction, so I ask here for some advice.
I knew the famous paper of Deligne-Mumford, but this paper is hard for me ...

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**0**answers

305 views

### Hochschild-Serre for hypercohomology

I need either a proof or a good reference for the following plausible statement:
Let $S$ be a scheme and let $C$ be a bounded complex of abelian sheaves on $S_{\rm{fppf}}$. Let $S^{\prime}\rightarrow ...

**40**

votes

**2**answers

9k views

### What are “perfectoid spaces”?

This talk is about a theory of "perfectoid spaces", which "compares objects in characteristic p with objects in characteristic 0". What are those spaces, where can one read about them?
Edit: A bit ...

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**3**answers

1k views

### Elliptic Curves over Global Function Fields

I am currently thinking about a problem, and I feel that by knowing more about elliptic curves over extensions of $\mathbb{F}_q(T)$, for $q$ a power of $p$ say, might lead to insight. I am also ...

**15**

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**2**answers

828 views

### Why does Tate's conjecture imply semisimplicity of crystalline Frobenius?

I'm trying to find a reference for the following fact:
If Tate's conjecture is true for all smooth projective varieties over $\mathbb{F}_p$, then the Frobenius endomorphism on the crystalline ...

**6**

votes

**1**answer

646 views

### Shot in the dark: Is there an english translation of Deligne-Rapoport “Les schemas de modules…” anywhere?

Extensive googling (and searching here) has yielded nothing, unfortunately.
I knew a language genius once who offered to translate it for me as a favor, but I turned him down because it seemed like ...

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**3**answers

1k views

### Which curves have infinitely many rational points

Question: Assuming finiteness of the Tate-Shafarevich group, is there an algorithm to determine whether a curve $C$ defined over a number field $K$ has infinitely many $K$-rational points?
I ...

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**2**answers

508 views

### Poincaré duality for smooth projective varieties over finite fields

What is exacly the statement of Poincaré duality for smooth projective varieties over finite fields and twisted constant $\mathbf{Z}_\ell$ sheaves? Where can I find a proof?
By twisted constant ...

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**0**answers

201 views

### Is there a reference that treats principal homogeneous spaces for (say) group varieties using schemes?

I was wondering if anyone could recommend a reference that discusses principal homogeneous spaces for general finite type group schemes over a field $k$ entirely in the language of schemes (or even ...

**6**

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**1**answer

2k views

### Where to start reading into p-adic non-abelian Hodge theory?

I'm curious about Faltings' "A p-adic Simpson correspondence ". Do you know more detailed, introductory, expositions, surveys, texts of seminars on that?
Edit: Annette Werner's survey "Vector ...

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**1**answer

544 views

### finite generation of the Mordell-Weil group over finitely generated fields

Does anyone know a reference for the proof of the finite generation of the Mordell-Weil group over finitely generated fields?

**37**

votes

**2**answers

4k views

### What is a good roadmap for learning Shimura curves?

I am interested in learning about Shimura curves. Unlike most of the people who post reference requests however (see this question for example), my problem is not sorting through an abundance of books ...

**6**

votes

**1**answer

806 views

### Elementary questions in arithmetic geometry

In many theories there is a rough divide between elementary problems that can be solved with "one's hands", and "deep results that require powerful tools". For example, I am told that Hodge theory is ...

**16**

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920 views

### Two conjectures by Gabber on Brauer and Picard groups

In a paper I need to make reference to 2 conjectures by Gabber
(see Conjectures 2 and 3, page 1975)
http://www.mfo.de/programme/schedule/2004/32/OWR_2004_37.pdf
1) Let $R$ be a strictly henselian ...

**6**

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**1**answer

1k views

### Reference for the `standard' Tate curve argument.

I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following:
Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant ...