# Questions tagged [arithmetic-geometry]

Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.

2,072
questions

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### Cup product structure on Galois cohomology

Let $G_S$ denote the Galois group of the maximal extension of $\mathbb{Q}$ unramified outside a finite, non-empty set of primes, $S$. Let $p\in S$ and let $V, W$ be a pair of finite dimensional $p$-...

4
votes

1
answer

174
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### Bad prime of torsor and original elliptic curve ; Definition of Tate–Shafarevich group $Ш(E/K)$

Let $E/K$ be an elliptic curve over number field $K$. Let $M_K$ be a set of all places of $K$.
My question is, Does there exist a finite set $S\subset M_K$ such that
$\forall C$: $E/K$-torsor, $\...

5
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0
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258
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### $M\otimes _{\mathbb{Z}_p}\mathbb{Q}_p$ is Banach if $M$ is $p$-adically complete

I was reading the paper On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms of Emerton, in which at page 15 the following fact is claimed (in my notation).
Let $p$ ...

3
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0
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55
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### Practical way of computing bitangent lines of a quartic (using computers)

Are there known practical algorithms or methods to calculate the bitangent lines of a quartic defined by $f(u,v,t)=0$ in terms of the 15 coefficients? Theoretically you can set up $f(u,v,-au-bv)=(k_0u^...

2
votes

1
answer

277
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### Bounding $H^4_{\text{ėt}}$ of a surface

Let $X\longrightarrow X'$ be a smooth proper map of smooth proper schemes defined over $\mathbb{Z}[1/S]$, where $S$ is a finite set of primes. Assume $X'$ is a curve of positive genus, and $X$ is a ...

4
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0
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105
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### How does one compute the group action of the automorphism group on integral cohomology?

Suppose I have a curve $X$ (for concreteness, we can take $X$ to be a smooth, projective curve over a finite field $\mathbb F_q$, and even more concretely consider the family of curves described by ...

3
votes

1
answer

208
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### Action of complex conjugation on etale cohomology

Let $X$ be a genus $g$ smooth projective curve, defined over $\mathbb{Q}$, and let $\overline{X}$ denote the base change of $X$ to $\overline{\mathbb{Q}}$.
It is well known that $H^1_{\text{ét}}(\...

3
votes

1
answer

146
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### Difficulties in the proof of finiteness of n-Selmer group using cohomology

I was reading the proof of finiteness of n-Selmer group $S^n(E/\mathbb{Q})$ from Milne's Elliptic curve book(1st Edition). While reading the proof I had some difficulties in some arguments.
1st ...

18
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1
answer

793
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### What are $L$-functions?

I am coming at this question from the point of view of someone who is working in arithmetic geometry around the Langlands program.
We have $L$-functions associated to many different structures that we ...

1
vote

1
answer

137
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### Zeta function of variety over positive characteristic function field vs. local zeta factor of variety over $\mathbb{F}_p$

Let $X = Y \times_{\mathbb{F}_q} C$, with $Y, C / \mathbb{F}_q$ smooth projective varieties, $C$ a curve. Let $d = \dim_{\mathbb{F}_q} X$. We can consider the local zeta function $Z(X, t) = \prod\...

0
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1
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175
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### Finiteness of Selmer group

I was reading the proof of finiteness of $S^n(E/\mathbb{Q})$ but I am unable to understand from the following lemma how it follows that $S^n(E/L)$ finite.
LEMMA 3.13 For any finte subset $T$ of $\...

4
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0
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184
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### Shouldn't we expect analytic (in the Berkovich sense) étale cohomology of a number field to be the cohomology of the Artin-Verdier site?

Let $K$ be a number field. Consider $X=\mathcal{M}(\mathcal O_K)$ the global Berkovich analytic space associated to $\mathcal O_K$ endowed with the norm $\|\cdot\|=\max\limits_{\sigma:K \...

6
votes

1
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632
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### Understanding the Hodge filtration

Let $X$ be a smooth quasiprojective scheme defined over $\mathbb{C}$, and let $\Omega^{\bullet}_X$ denote its cotangent complex, explicitly, we have:
$\Omega^{\bullet}_X:=\mathcal{O}_X\longrightarrow \...

0
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0
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66
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### Hodge filtration vs Hodge structure on algebraic de Rham cohomology

I have a basic question on the relation between the definitions of the Hodge structure on the algebraic de Rham of a smooth proper scheme defined over a subfield of $\mathbb{C}$ and the Hodge ...

6
votes

1
answer

553
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### What's the relation between analytic stacks and higher complex/non-archimedean analytic stacks?

Recently Clausen and Scholze have developed a theory of analytic stack to unify different analytic geometries. Actually I do not know many details about it. It says an analytic stack is a sheaf $\...

2
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### Action of monodromy on the $p$-adic period domain in Lawrence-Venkatesh

In here, I asked various questions related to Lawrence and Venkatesh's work on the Mordell-Weil conjecture, which failed to receive any answers.
This is my attempt to try and focus the question.
In ...

3
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1
answer

288
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### Bloch–Beilinson conjecture for varieties over function fields of positive characteristic

Is there a version of the Bloch–Beilinson conjecture for smooth projective varieties over global fields of positive characteristic? The conjecture I’m referring to is the “recurring fantasy” on page 1 ...

1
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1
answer

99
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### Grössencharakter or Galois representation associated to a CM elliptic curve in characteristic $p$

When $E$ is an elliptic curve over a number field $L$, with complex multiplication by a maximal order in an imaginary quadratic field $K$, Silverman’s Advanced Topics in the Arithmetic of Elliptic ...

3
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0
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111
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### Monodromy action on the period domain in the Lawrence-Venkatesh paper

Let $Y\longrightarrow X$ be a smooth proper map of smooth quasiprojective schemes over $\mathbb{Z}_p$.
In Lawrence and Venkatesh's paper on Mordell-Weil, the authors consider the $p$-adic period map: $...

7
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0
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136
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### Is the $\ell$-adic cohomology ring of a cubic threefold a complete invariant?

The only interesting $\ell$-adic cohomology of a smooth cubic threefold $X$ is $H^3(X,\mathbb{Z}_{\ell}(2))$, which is isomorphic as a $\mathrm{Gal}_k$-module to $H^1(JX,\mathbb{Z}_{\ell}(1))^{\vee}$ ...

0
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71
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### Bounding the dimension of $H^1(G, V\otimes V^{\vee})$

Let $G_S$ denote the Galois group of the maximal extension of $\mathbb{Q}$ unramified away from a finite set of primes, $S$. Let $V$ be a finite dimensional, $G_S$-representation over $\mathbb{F}_p$ (...

7
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0
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216
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### What justifies the following isomorphism in Cassels' proof of the Cassels–Tate pairing?

In Cassels' paper Arithmetic on curves of genus 1. IV introducing the Cassels–Tate pairing the following lemma is stated.
Lemma 5.1: Let $q$ be a rational prime and $\Gamma$ the Galois group of the ...

0
votes

1
answer

99
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### Kernel of restriction in étale cohomology of curves over number fields

Let $X$ be a smooth projective curve defined over a number field $K$. Let $\overline{K}$ denote the algebraic closure of $K$, and set $\overline{X} := X\otimes \overline{K}$. Denote by $\iota: \...

2
votes

2
answers

224
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### Finding rational points on intersection of quadrics in affine 3-space

Consider the subvariety of Spec $\mathbb{Q}[x,y,z]$ cut out by the equations
\begin{eqnarray*} f_1: a_1x^2 - y^2 - b_1^2 & = & 0 \\
f_2 : a_2x^2 - z^2 - b_2^2 & = & 0
\end{eqnarray*}
...

3
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0
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137
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### Examples of curves $C$ with $\operatorname{Jac}(C) \cong E^3$, $E$ a CM elliptic curve

Let $k$ be a field of your choice— I'm particularly interested in algebraically closed fields. Are there explicit examples of curves over $k$ whose Jacobian is isogenous to the product of three copies ...

1
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0
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109
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### Second group cohomology of a twisted fundamental group

Let $X$ be a smooth hyperbolic projective curve defined over $\mathbb{Z}[1/S]$, where $S$ is a finite set of primes, and let $\pi:=\pi_1^{\text{ét}}(X, \overline{b})$ denote its étale fundamental ...

5
votes

1
answer

371
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### Finiteness of the Brauer group for a one-dimensional scheme that is proper over $\mathrm{Spec}(\mathbb{Z})$

Let $X$ be a scheme with $\dim(X)=1$ that is also proper over $\mathrm{Spec}(\mathbb{Z})$. In Milne's Etale Cohomology, he states that the finiteness of the Brauer group $\mathrm{Br}(X)$ follows from ...

3
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0
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137
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### Motivic $L$-functions came from automorphic representations

Langlands in his 1978 ICM talk made a conjecture that all motivic $L$-functions should arise as automorphic $L$-functions. A part of this conjecture, namely for some Hasse-Weil $\zeta$ functions is a ...

17
votes

1
answer

694
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### Injective ring homomorphism from $\mathbb{Z}_p[[x,y]]$ to $\mathbb{Z}_p[[x]]$

Is there an injective $\mathbb{Z}_p$-ring homomorphism from $\mathbb{Z}_p[[x,y]]$ to $\mathbb{Z}_p[[x]]$?

1
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0
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41
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### Absolute irreducibility implies free action on framed universal deformation ring

Let $\overline{\rho}: G\longrightarrow \text{GL}_n(\mathbb{F}_p)$ be a residual representation of the Galois group of a number field. Let $R_{\overline{\rho}}^{\square,\text{univ}}$ be the universal ...

5
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0
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185
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### Modularity lifting theorem à la Kisin

In its paper "Moduli of finite flat group scheme and modularity", Kisin showed the following theorem:
One of the main tool used is the scheme $\mathscr{GR}_{V_\bf F, \xi}$ defined in ...

3
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0
answers

321
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### The local global principle for differential equations

Are there any good reference to tackle the problem below?
Or, are there any know result?
Problem
Let $f_1...f_n\in \mathbb{Z}[x_1,..,x_n]$ and $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a vector ...

2
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0
answers

153
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### Bounding dimensions of Galois cohomology

Let $G$ be the absolute Galois group of the rationals, and $V$ a finite dimensional $p$-adic representation.
Is there a way to provide a general bound on $H^i(G, V)$ in terms of $\dim_{\mathbb{Q}_p} V$...

3
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0
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165
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### A relative Abel-Jacobi map on cycle classes

I have a question about relativizing a classical cohomological construction that I think should be easy for someone well versed in such manipulations.
Background:
Suppose $X$ is a smooth projective ...

2
votes

1
answer

108
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### Galois action on étale path torsors

TLDR: How is the Galois action on étale path torsors defined?
Let $X$ be a smooth proper scheme, over a field $k$, and let $x,y\in X(k)$ be a pair of points. Let $\pi_1^{\text{ét}}(\overline{X},\...

20
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3
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684
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### Examples when quantum $q$ equals to arithmetic $q$

First, as a disclaimer, I should say that this post is not about any specific propositions, but is more of some philosophical flavor.
In the world of quantum mathematics, the letter $q$ is a standard ...

3
votes

1
answer

218
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### Etale cohomology of relative elliptic curve

Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme.
Let $R^1f_*\mathbb{Q}...

2
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0
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### When is a coherent sheaf on an algebraizable space algebraizable?

Let $\mathcal{X}$ denote an algebraizable rigid analytic space over $\text{Spm}(\mathbb{Q}_p)$,
i.e. there exists a finite type $\mathbb{Q}_p$-scheme $X$ such that $\mathcal{X}$ is its rigid ...

1
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0
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### Multiplicities of Galois representations in the semisimplification of the reduction of a Tate module

Let $C$ be a smooth proper curve, of genus $g$ over a number field $K$. Let $v$ be a prime of good reduction for $C$ above $p>2$, and let $T_pJ$ denote the $p$-adic Tate module of $J$, the Jacobian ...

8
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188
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### Elkies' family of elliptic curves of rank 19

There is a widely cited fact that Elkies had found that infinitely many curves of rank 19 in 2006, in "Z^28 in E(Q), etc. Email to the number theory mailing list
at [email protected]&...

2
votes

1
answer

145
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### Non-torsion points of Tate curves

Let $E$ be a Tate curve over a $p$-adic field $K$. Then there exists $q \in K^*$ with the valuation $v(q)>0$ such that $E(\overline{K})= \overline{K}^*/\left< q \right>$. So it is easy to see ...

2
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0
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98
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### Singularities of curves over DVRs with non-reduced special fibre

Let $R$ be a complete DVR of mixed characteristic with fraction field $K$ of characteristic $0$ and residue field $k$ of characteristic $p>0$. Suppose that $\mathcal{X}$ is a normal $R$-curve such ...

1
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0
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42
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### Frobenius pullback of an integrable connection on a quasi-projective scheme

Let $X_k$ be a smooth quasi-projective scheme over a finite field $k$. Let $X_K$ be a smooth lift of $X_k$ to characteristic $0$, and let $(X_K)^{\text{an}}$ denote the rigid analytic space associated ...

1
vote

1
answer

108
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### Frobenius action on the trivial connection

Let $F$ denote the absolute Frobenius acting on a smooth quasiprojective scheme $X$ over a finite field $k$.
Denote the trivial connection on $\mathcal{O}_X$ by $d$. Denote its pullback by Frobenius ...

3
votes

0
answers

193
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### Is it always true that the complement of an ample divisor is affine?

Consider a proper and integral scheme $X\rightarrow\operatorname{Spec}(A)$ over a Noetherian ring $A$ and $D\in\operatorname{Div}(X)$ an effective ample Cartier divisor on $X$. Is it true that its ...

1
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0
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59
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### $F$-structure implies regular singularities + unipotent local monodromy?

Let $(\mathcal{E},\nabla)$ be a vector bundle with an integrable connection on a smooth quasi-projective $K$ scheme $X$, with $K$ a $p$-adic number field of characteristic $0$. Let $F$ denote a semi-...

3
votes

1
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189
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### Isocrystal with no $F$-structure

$\DeclareMathOperator\Isoc{Isoc}$Let $X_k$ be a quasiprojective $k$ scheme, with $k$ finite, and let $X_K$ be the rigid analytic space lifting it to the fraction field of its Witt ring, which I denote ...

1
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0
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49
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### Frobenius acting by autoequivalence on $\text{Isoc}(X/K)$

Let $X_k$ be a smooth quasiprojective scheme over a finite field $k$. Let $X_K$ be a smooth lift of $X_k$ to the fraction field of the Witt ring of $k$, which I denote by $K$.
In various papers I read ...

4
votes

1
answer

220
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### Equivalence between vector bundles with integrable connections to isocrystals

Let $k$ be a perfect field, $W(k)$ its Witt ring, and $K$ the fraction field of $W(k)$. Let $X_k$ be a smooth proper curve over $k$, and let $X_K$ be the schematic generic fibre of a smooth proper ...

2
votes

0
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124
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### A relative cycle class map

Suppose I have a smooth projective morphism $p: X \to S$ between varieties, and a relative cycle $Z \subset X \to S$ which is assumed to be as nice as can be (rquidimensional with fibers of dimension $...