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**6**

votes

**1**answer

169 views

### Question on paper of Stewart and Top about ranks of elliptic curves over Q(t)

I'm reading "On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms" by Stewart and Top, and struggling to understand the argument on pg 962 which shows that the rank of a ...

**0**

votes

**1**answer

161 views

### Sections of proper, flat morphism

Let $f:X \to Y$ be a proper, flat morphism of projective scheme and $Y$ is an irreducible, non-singular surface. Assume further that there exists a Zariski open subset $U$ of $Y$ whose complement is ...

**5**

votes

**2**answers

390 views

### Order of vanishing of an integer polynomial at a point

Let $f(x,y)$ be a polynomial with integer coefficients, and let $\alpha=(\alpha_1,\alpha_2)\in \mathbb{C}^2$ be a complex point. I want to show that $f$ cannot vanish at $\alpha$ to high order unless ...

**3**

votes

**0**answers

113 views

### Is there a difference between the inertia stack and the universal automorphism group

Let $\mathcal M$ be a stack representing some moduli problem. Let $\mathcal X\to \mathcal M$ be the corresponding universal family.
What is the difference between the inertia stack $I\to \mathcal M$ ...

**5**

votes

**1**answer

276 views

### Algebraic spaces which are automatically schemes

Let $S$ be a scheme, and let $f:X\to S$ be a morphism of algebraic spaces.
If $f$ is smooth proper curve of genus at least two, then $X$ is a scheme. (Here I mean that $f$ is a smooth proper morphism ...

**8**

votes

**0**answers

174 views

### Distribution of Mordell–Weil ranks of higher genus curves

By "nice curve", I mean a smooth, projective, geometrically integral curve over $\newcommand{\Q}{\mathbb{Q}}\newcommand{\Jac}{\operatorname{Jac}}\Q$ with at least one $\Q$-rational point. The ...

**7**

votes

**0**answers

173 views

### Almost rational point

Let $X$ be a variety over a number field $K$. Let $S$ be a finite set of places of $K$. Is there a notion of a point $p \in X(\overline{K})$ to be "almost rational" in the following sense?:
$p$ and ...

**3**

votes

**0**answers

114 views

### Classification of compact Shimura curves

Is there a classification that determines all isomorphism classes of compact Shimura curves at least Shimura curves in $A_g$? I did not find this in the literature and appreciate any helpful ...

**1**

vote

**1**answer

153 views

### On functors which are generically representable

Let $F$ be a set-valued (contravariant) functor on the category of schemes. Let $F_{\mathbb Q}$ be the associated functor on the category of schemes over $\mathbb Q$.
Suppose that $F_{\mathbb Q}$ is ...

**0**

votes

**0**answers

79 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 ...

**13**

votes

**2**answers

417 views

### Elliptic curves and supercuspidal representations of conductor $p^2$

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $p \geq 5$ be a prime of additive reduction for $E$.
Let $f$ be the newform associated to $E$, and let $\pi$ be the irreducible admissible ...

**11**

votes

**0**answers

661 views

### Meaningful review of Moriwaki's “Arakelov Geometry”

I have been asked to write a mathscinet review for Atsushi Moriwaki's Arakelov Geometry
book:
http://www.ams.org/bookstore-getitem/item=mmono-244
I could do the review the standard way in a day or ...

**1**

vote

**2**answers

210 views

### Examples of quotients by infinitesimal group schemes

I'm looking for examples of explicit actions of the infinitesimal group schemes $\alpha_{p^n}$ on schemes (maybe as simple as the affine plane) in characteristic $p$ or mixed characteristic, and their ...

**1**

vote

**0**answers

89 views

### Condition for a finite group scheme to be étale [closed]

My question comes from the reading of Tate's paper $p$-divisible groups. In the last few pages there is an argument which gives as trivial the following fact. If we take a $p$-divisible group over a ...

**5**

votes

**1**answer

180 views

### Average height of rational points on a curve

I am seeking a formalism to define the average height of
the rational points on a curve. This is straightforward
if the number of points is finite, but (to me) not straightforward
when the rational ...

**13**

votes

**1**answer

579 views

### What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...

**0**

votes

**1**answer

138 views

### Base change of regular schemes [closed]

Let $R$ be a complete DVR with fraction field $K$, $X$ be a regular scheme flat over $R$. Let $L$ be a finite field extension of $K$ and $Q$ be the integral closure of $R$ in $L$. Denote by $Y:=X ...

**2**

votes

**1**answer

215 views

### 'Stalk' of vanishing cycles at $k$-point

I have a simple question on notation.
Let $S$ be a Henselian trait with closed point $s$ (with finite residue field $k$) and generic point $\eta$. Let $X/S$ be a variety. Then, we have the functor
...

**5**

votes

**0**answers

144 views

### Are all these K3 surfaces supersingular?

Consider all the smooth K3 surfaces given by $X^4+W^2X^2+XW^3 = f(Y,Z,W)$ or $X^4+XW^3 = g(Y,Z,W)$ over $\mathbb F_{2}$ with $f$ or $g$ homogenous of degree 4. There are a lot of choices for $f$ and ...

**9**

votes

**2**answers

756 views

### Understanding Faltings's Theorem

I am soon to become a graduate student and so I started a personal project; I want to understand Faltings's proof of the Mordell conjecture.
I want to get into arithmetic geometry (since I always ...

**8**

votes

**3**answers

358 views

### Deciding a quadratic diophantine equation

Given $a,b\in\Bbb Q_+$, is there an easy way to decide if $$S_{a,b}=\{(x,y)\in\Bbb Z^2:ax^2 + by^2=1\}=\emptyset?$$
I am more interested in seeing if there is a quick way to test for case when ...

**0**

votes

**0**answers

155 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 ...

**4**

votes

**1**answer

167 views

### Subgroups of $Sp_{2g}$ giving rise to Shimura data

Consider the Shimura datum $(GSp_{2g},\mathcal{H}_g)$. Let $G$ be a reductive $\mathbb{Q}$-subgroup of $Sp_{2g}$. I want to know under what condition there exists a point $x\in\mathcal{H}_g$ such that ...

**10**

votes

**2**answers

352 views

### Equivalence of various definitions of arithmetic Chow groups

If I understand correctly, $n$-th arithmetic Chow group of arithmetic variety $X$ is defined as a quotient of the group of pairs of the form $(\sum\limits_in_iZ_i, g)$ where $Z := \sum\limits_in_iZ_i$ ...

**0**

votes

**0**answers

80 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 ...

**1**

vote

**1**answer

164 views

### Dieudonné modules -reference request

I need a reference to start learning about Dieudonn\'e modules, and their application to the arithmetic of abelian varieities. I know that this is a copy of Reference for Dieudonné modules, ...

**6**

votes

**1**answer

271 views

### Representability of morphism of stacks

A morphism of Artin stacks $f:X\to Y$ over $\mathbb Q$ is representable by algebraic spaces if and only if its geometric fibres are algebraic spaces. I would like to know if one can use this to prove ...

**6**

votes

**0**answers

132 views

### Non-embeddable varieties

Suppose that $k$ is a perfect field of characteristic $p>0$, $\mathcal{V}$ is a complete discrete valuation ring with residue field $k$ and quotient field $K$, of characteristic $0$.
Then when ...

**7**

votes

**1**answer

230 views

### Abelian varieties with good reduction everywhere over function fields

There is a famous theorem due to J.-M. Fontaine,
Il n'y a pas de variété abélienne sur Z
(and independently by V.A. Abrashkin) that there are no abelian varieties over Z. I was wondering whether ...

**2**

votes

**0**answers

213 views

### Bruhat Tits buiding to visualize closed points of affine flag varieties?

In his survey "affine springer fibers and affine Deligne-Lusztig varieties", Goertz gives us a tutorial session on how to use Bruhat Tits buildings to visualize subsets of affine Grassmannians or of ...

**5**

votes

**2**answers

377 views

### Obstruction and rational points on curves

Is etale-Brauer the only obstruction to the existence of rational points on projective plane curves over number fields?

**6**

votes

**2**answers

443 views

### How to compute class number of a torus

Let $T$ be an algebraic torus over a number field $K$.
Following notations in Ono's The Arithmetic of Tori,
...

**1**

vote

**0**answers

77 views

### Local duality for abelian varieties

Let $A$ be an abelian variety over a p-adic field $K$. Let $I$ be the inertia group of $K$. There is a Yoneda pairing $$H^n(\hat{\mathbb{Z}},A^I) \times Ext^{2-n}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z}) ...

**2**

votes

**1**answer

159 views

### integral basis for the Lie algebra of the Neron model of an abelian variety

Let $A$ be an abelian variety over a number field $K$. Let $\mathcal{A}$ be the Neron model of $A$ over $O_K$. Let $\Omega_{\mathcal{A}/O_K}$ be the sheaf of invariant differential forms on ...

**18**

votes

**3**answers

587 views

### Variety acquiring rational point over any quadratic extension

Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$?
If ...

**2**

votes

**0**answers

261 views

### When an elliptic curve is a quotient of $\mathbb{G}_a$?

I want to know when an elliptic curve $E \rightarrow S$ is a quotient of $\mathbb{G}_a$.
When $S$ is an analytic space, there is an exact sequence $$0 \rightarrow R^1 \mathbb{Z}^{\vee} \rightarrow ...

**11**

votes

**0**answers

367 views

### Can an abelian variety/Q have no non-trivial points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal ...

**6**

votes

**1**answer

247 views

### Is Scholl construction of modular motives related to Deligne's construction of $\ell$-adic representations?

first of all, I need to declare my extreme ignorance on the topic of modular forms, so , please, does not assume that I know Deligne's construction in details.
In ...

**3**

votes

**1**answer

187 views

### Some clarifications regarding Deligne's paper on $\ell$-adic representations arising from modular forms

I've posted this question few days ago on math.stackexchange because it seems quite superficial. However, since I've got no responses at all, I'm posting it here. If the question is not suitable, ...

**14**

votes

**0**answers

300 views

### Is the absolute Galois group of the rationals Hopfian?

Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?

**61**

votes

**1**answer

7k views

### What is an étale theta function?

Let me start out by urging you to take seriously that whatever I write about the papers surrounding IUTT really are questions. If you would like to use it as a guide to the mathematics in any way, ...

**9**

votes

**0**answers

440 views

### Paths in $\mathrm{Spec} \, \mathbb{Z}$ and Kim's proof of Siegel's theorem for $\mathbb{P}^1 \setminus \{0,1,\infty\}$

This is motivated by a basic number theory question I asked the previous day:
Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$? I noted there that the answer to the ...

**111**

votes

**1**answer

9k views

### What is a Frobenioid?

Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title.
Recently, there has been a flurry of new discussion ...

**3**

votes

**2**answers

309 views

### cohomological obstructions and rational points

Let $X$ a (nice) scheme over $\mathbb{Q}$. Are there cohomolgical obstructions answering the following questions:
1) is $X(\mathbb{Q})$ an empty set ?
2) is $X(\mathbb{Q})$ a finite (non empty) set ...

**3**

votes

**1**answer

221 views

### When does a hyperelliptic Riemann surface admit a map of degree 3

Let $X$ be a hyperelliptic curve of genus $g>1$.
For which $g$ does $X$ admit a map $X\to \mathbb P^1$ of degree $3$?
I think a genus two curve $X$ admits a map of degree $3$.
Proof: Pick $P$ ...

**6**

votes

**2**answers

417 views

### References for general Hasse-Weil zeta function

Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case.
I am ...

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votes

**0**answers

134 views

### Groupoid cardinality of DM stack and point counting on coarse moduli spaces

Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.)
Under which ...

**2**

votes

**1**answer

178 views

### Difference between Frobenii on Tate modules of special and generic fibre

Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$.
If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to ...

**2**

votes

**1**answer

166 views

### rational point of a curve [closed]

Let $X$ be a smooth projective curve over $\mathbb{Q}$. I heard (if I did not misunderstood) that the geometry of the complex points $X(\mathbb{C})$ (flat, hyperbolic case) dicts the shape (group ...

**10**

votes

**1**answer

234 views

### Is there a solvable point on any variety over the field of complex rational functions?

Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$.
Must $V$ have a solvable point?
The variety $V$ is assumed ...