The arithmetic-geometry tag has no wiki summary.

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### Why is Faltings' “almost purity theorem” a purity theorem?

My understanding of purity theorems is that they come in several flavors:
1) Those of the form "this Galois representation is pure, i.e. the eigenvalues of $Frob_p$ are algebraic numbers all of whose ...

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**2**answers

388 views

### Examples of (Phi,Gamma)-modules

What is the (Phi,Gamma)-module of an elliptic curve over Z_p, expressed by a direct construction ?

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### Natural construction of Hodge (Phi,Gamma)-modules

I am looking for a functor from varieties $X/\mathbf{Z}_p$ to $(\varphi,\Gamma)$-modules over the Robba ring over $\mathbf{Q}_p$ (overconvergent ones) that is contructed by differential methods ...

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199 views

### Can one bound the Quadratic Points on Curves?

Let $C$ be a nonsingular projective curve defined over $\mathbb{Q}$, which does not admit a map of degree 1 or 2 to $\mathbb{P}^1$ or to an elliptic curve. It is then a consequence of Corollary 3 of ...

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90 views

### extending truncated Barsotti-Tate group

Let $X$ be a smooth projective curve defined over a finite field of char $p$, let $G[1]$ be a truncated Barsotti-Tate grop of level-1. My question is : can $G[1]$ be extended to a truncated ...

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269 views

### References for period matrix of abelian variety

Hi, everyone.
I am looking for some references for period matrix of abelian variety over arbitrary field, if you know, could you please tell me?
For period matrix of abelian varieties, I means that ...

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136 views

### Hodge filtration over $\mathbb Z_p$

Let $p$ be a prime number.
Let $X\to\operatorname{Spec}\mathbb Z_p$ be smooth and proper. Is it true that
the map $H^i(X,\Omega^{\bullet\geq j}_{X/\mathbb Z_p})\to H^i(X,\Omega^\bullet_{X/\mathbb ...

**4**

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**1**answer

158 views

### pro-$\ell$ etale fundamental group of a semi-abelian variety

Let $A$ be a semi-abelian variety over $K$, $\ell$ a prime number which is not equal to char($K$).
Does the abelianization of geometrically pro-$\ell$ etale fundamental group ...

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863 views

### Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)

A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$.
An arithmetic subgroup is called congruence if it contains a subgroup of type $\Gamma(N)$ for ...

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**1**answer

304 views

### Shafarevich's theorem for elliptic curves defined over function field of algebraic curve over algebraically closed field

Let $K$ be a number field and $S$ a finite set of places of $K$. Then Shafarevich's theorem states that there are only finitely many isomorphism classes of elliptic curves $E$ over $K$ with good ...

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134 views

### Is Hasse-witt map isomorphism?

Fix a level $N \geq 3$ and denote by $\Gamma(N)\subset SL(2,\mathbb{Z})$ the subgroup of matrices which are congruent to the identity modulo $N$. The open modular curve $Y(N)$ corresponding to ...

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**1**answer

229 views

### a question of Galois cohomology

Let $R$ be a complete DVR with algebraically closed residue field $k$ and fractional field $K$ , $PGL(2)$ the automorphic group of projective line over $\overline K$.
My question is:
When ...

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**1**answer

158 views

### morphism between two elliptic curves over a local field

Let $X,Y$ be two elliptic curves over $K:=K(R)$ which have good models over $R$, Char($K$)=$0$, where $R$ complete DVR with algebraically residue field $k$.
If $L$ is a finite extension of $K$, such ...

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**2**answers

479 views

### are moduli stacks deligne-mumford stacks in general

Let M be your favorite moduli stack over the field of complex numbers.
Is it reasonable to expect M to be a Deligne-Mumford stack?
I know this is true for the moduli space of curves of genus g, ...

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**0**answers

188 views

### Is Normalization of log smooth scheme smooth?

Let $f:Y\rightarrow X$ be a finite flat morphism between smooth schemes over $Spec k$, where $k$ is a perfect field. Let $D$ be an irreducible and smooth divisor of $X$, $U=X\setminus D$ the ...

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**1**answer

135 views

### arithmetic group over function fields and its fundamental domain

Let $G$ be a semi-simple algebraic group defined over a global function field $K$.
Let $S$ be a finite set of places of $K$. For a place $v$ of $K$ let $K_v$ be the completion under $v$. We take ...

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90 views

### Bound for the degree of the field of definition for a closed point of a variety

While attempting to prove some existence theorem for matrices over $\mathbb{F}_{2^k}$ I've come across the following problem concerning fields of definition for closed point of, say affine, varieties.
...

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**1**answer

279 views

### Heegner Points on $X_0(N)$ when some primes dividing $N$ are inert in the imaginary quadratic field

If $K = \mathbb{Q}(\sqrt{-D})$ is a imaginary quadratic field with discriminant $-D$, then we get Heegner points on $X_0(N)$ as long as there exists $\mathfrak{n} \subset \mathcal{O}_K$ such that ...

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556 views

### reduction types of elliptic curves

Let $E/K$ be an elliptic curve, where $K$ is a complete local field with residue field $k$ and char$(k) = p$. I'm trying to make sense of Kodaira symbols and Tate's algorithm.
My current ...

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**1**answer

187 views

### When is the determinant of the push-forward of an ample line bundle ample

Let $f:X\to S$ be a "nice" morphism of "nice" schemes. Let $L$ be an ample line bundle on $X$.
When is $\det f_\ast L$ also ample?
A "nice" morphism could be anything from "finite type separated" to ...

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**1**answer

255 views

### Inertia subgroup in the ordinary reduction case when $p=2$

Dear MO,
Let $K/\mathbb{Q}_2$ be a finite extension, and let $E/K$ be an elliptic curve with good ordinary reduction, and such that $\mathbb{Q}_2(j(E))=K$. Let ...

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137 views

### simple proof of relation between H^1 crystalline and Dieudonne module?

Hi,
Let $k$ be a perfect field of characteristic $p > 0$. Let $A/k$ be an abelian variety. Then the first crystalline cohomology group of $A$ with respect to $W(k)$ (= Witt vectors) is canonically ...

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**1**answer

228 views

### Local Norm Mapping for Abelian Varieties

Let $A/K$ be an abelian variety defined over a nonarchimedean local field $K$ of characteristic $0$ and let $L$ be a finite extension of $K$. Consider the norm map $$A(L)\xrightarrow{N_{L/K}}A(K)$$ I ...

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**1**answer

168 views

### Are period domains ever contractible

Which simply-connected period domains are contractible?
Examples. Siegel upper-half space? Poincare upper-half plane? Universal cover of a Shimura variety?
Are these contractible?

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**1**answer

168 views

### scheme of generalizations

Hi,
I have the following problem. Let $\mathcal{O}$ be a valuation ring and $S=Spec(\mathcal{O})$, denote with $s$ the closed point and with $\eta$ the generic one. Let $X\rightarrow S$ be a proper, ...

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202 views

### Isogenous elliptic curves have same conductor

Let $E/K, E'/K$ be elliptic curves defined over a number field $K$. Let $\phi: E \to E'$ be a non-constant isogeny defined over $K$. Why must the conductors be equal?
I know that this is an ...

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**1**answer

206 views

### Tricks to produce examples of hypersurfaces with index greater than $1$

Recall, index of an algebraic scheme $X$ is defined to be the greatest common divisor of the degrees of the space of zero cycles on $X$. I am interested in examples of hypersurfaces in ...

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870 views

### Status of Beilinson conjectures?

(I hesitate to post that question here, but I received on answer on FB:)
Does anyone know how the current status of work on them is? And how the possible generalizations etc. which one thinks ...

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146 views

### Possible reasons why the image of 0 in the modular parametrization would always be O for a family of quadratic twists of elliptic curves

Let $E$ be an elliptic curve and $E_n$ be its quadratic twist by $n$. Let $\phi_n: X_0(N_n) \to E_n$ be the normalized modular parametrization of $E_n$ ($\infty \to O$)
For some particular curves ...

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86 views

### The etale cohomology``ring" structure of torsion sheaves on varieties

For a topological manifold $M$, one can speak of the cohomology ring structure $H^*(M, k)$ where $k$ is a ring. If one replace $M$ by an arithmetic schemes $X$ over a base ring $S$, and replace $k$ by ...

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335 views

### Simple field extension and rational points

Let $F$ be an infinite field and $f$ a homogeneous form on $F$ such that $f$ has no non-trivial zero in $F$. Let $F'$ be a finite extension of $F$ such that $f$ has a non-trivial zero in $F'$. Is it ...

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### computing the order of the image of 0 under the modular parametrization map for an elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve of rank 0, with modular parametrization $\phi: X_0(N) \to E$. Let $\Omega_0$ be the least positive real period. A paper I'm reading (Yoshida, Some variants of ...

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**1**answer

346 views

### Overconvergent/infinitesimal site, base change and six operations

This question is about 6 operations formalism for 'crystalline' cohomology theories - more specifically the infinitesimal cohomology of smooth $\mathbb{C}$-varieties, and the overconvergent cohomology ...

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391 views

### On Determinants of Laplacians on Riemann Surfaces

History of the Formula: In their famous paper "On Determinants of Laplacians on Riemann Surfaces" (1986), D'Hoker and Phong computed the determinant of the Laplacian $\Delta_n^+$ on the space $T^n$ of ...

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### Rational points on a sphere in $\mathbb{R}^d$

Call a point of $\mathbb{R}^d$ rational if all its $d$ coordinates are rational numbers.
Q1.
Is the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$ dense in rational points, i.e. does $S$ include a ...

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226 views

### On the expilicite example of Parshin Construction.

In the Proof of Mordell Conjecture by Gerd Faltings, it is famous that Parshin constructed curve C_P for each Q-rational point P on the given curve C over Q such that genus of C satisfies g(C) > 0.
...

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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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133 views

### cohomological criterion of triviality

Hi,
let $X$ be a proper curve, not necessarily smooth, but reduced and connected and $E$ a vector bundle on $X$ of rank $r$.
For a line bundle $L$ it is true that the conditions $h^0(X,L)>0$ and ...

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votes

**2**answers

547 views

### How to see the geometry and arithmetic of tannakian fundamental groups?

The etale fundamental group is an inverse limit of automorphism groups of finite etale coverings. We can see the geometry of etale fundamental group very well from etale coverings just like ...

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**1**answer

189 views

### S-arithmetic subgroup question

I've been reading a proof concerning S-arithmetic subgroups of algebraic groups and I'm having trouble determining why the following step should be true. First, the setup:
Let $G$ be a connected ...

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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. ...

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**1**answer

174 views

### Which level structures on elliptic curves are twist-invariant?

Let $N \geq 5$ be a prime, and $H$ a subgroup of $GL_2(\mathbb{F}_N)$. As shown in Chapter IV of [DeRap], there is a curve $X_H(N)$, defined over $K_N := \mathbb{Q}(\zeta_N)^{\det H}$, which is a ...

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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 $S$, ...

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749 views

### Do isogenies with rational kernels tend to be surjective?

Dear MO Community,
this is a pretty vague title, so let me tell you the precise observation I have made.
Consider the family of elliptic curves over $\mathbf{Q}$ having a rational $5$-torsion point ...

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**1**answer

192 views

### Results and conjectures on bounds on degrees of isogenies

Dear MO Community,
given an isogeny between two abelian varieties $\varphi: A\rightarrow B$ (everything definied over a number field $K$), we can factor $\varphi$ through a ...

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195 views

### a question about Beauville-Laszlo

Hi,
let $V$ be a complete DVR with uniformizer $\pi$. Let $m$ be a NON zero integer, $a\in V[[u,v]]/(uv-\pi)^{\times}$ and $f=\pi^{m}a$. Consider $F$ as the kernel of the diagram
$$
...

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423 views

### Grothendieck monodromy theorem for l-adic sheaves

Hi,
Suppose that $F$ is a local field, $G_F$ its Galois group, $I$ the inertia subgroup, $k$ its residue field.
Let $X$ be a finite type scheme over $k$. Let $C$ be a constructible $l$-adic sheaf on ...

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346 views

### weight monodromy conjecture for curves?

Hi,
Is there a simple proof of the weight monodromy conjecture in the case of a curve over a mixed characteristic local field?
Thanks!

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### Uniqueness of decomposition of completely reducible representations

Let $X$ be a smooth, separated scheme of finite type over $\mathbb{F}_q$ where $q=p^r$ for some $r>0$. Let $gcd(l,p)=1, \rho:W(X) \to GL_r(\mathbb{Q}_l)$ be a Weil representation which is ...

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**1**answer

274 views

### Minimal semistable model for K3-surfaces.

I wonder if a semistalbe K3 surface over a $p$-adic field has a minimal semistable model. I guess yes but I do not find any reference.
Also, if we have a semistable K3 surface with a log structure, ...