Questions tagged [arithmetic-geometry]

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

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3 votes
0 answers
137 views

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 ...
1 vote
0 answers
108 views

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 votes
0 answers
141 views

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
97 views

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},\...
1 vote
2 answers
501 views

Finding regions where multi-variate polynomials are positive

Given a constant $k \in \mathbb N$, and a set of $p$ multi-variate polynomials {$P_j:\mathbb N^n\to \mathbb Z$}$\_{j=1...p}$, with $P_j \not\equiv 0$. Is the following true: There exists $n$ sets $...
20 votes
3 answers
549 views

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 ...
4 votes
1 answer
149 views

Meaning of dagger cohomology $H^{1 \dagger}(G^\dagger)$ in "Frobenius and Monodromy Operators" by Coleman and Iovita

Let $G$ be an abelian variety with good reduction over a finite extension $K$ of $\mathbb{Q}_p$. In equation (2.4) on page 179 of my edition of "The Frobenius and monodromy operators for curves ...
3 votes
1 answer
181 views

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}...
3 votes
1 answer
255 views

Global duality theorem for 2-part

$\DeclareMathOperator\coker{coker}\DeclareMathOperator\Sha{Sha}$Let $K$ be a number field. Let $E/K$ be an elliptic curve over $K$. Suppose finiteness of $\Sha(E/K)$. According to Global duality ...
2 votes
0 answers
63 views

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 ...
11 votes
1 answer
372 views

Does every finite affine plane have the doubling property?

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
2 votes
0 answers
152 views

A direct proof that every projectivity between parallel lines is affine

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
1 vote
0 answers
105 views

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 votes
0 answers
173 views

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
136 views

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 votes
0 answers
91 views

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 vote
0 answers
39 views

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
103 views

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
177 views

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 ...
3 votes
1 answer
183 views

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 vote
0 answers
59 views

$F$-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 answer
154 views

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 ...
1 vote
0 answers
49 views

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 ...
2 votes
0 answers
120 views

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 $...
1 vote
0 answers
52 views

Etale local systems and proper base change

I am looking for a reference, or a proof, of the following statement: Let $f:Y\longrightarrow X$ be a smooth proper map of quasiprojective $K$ schemes, and let $\overline{f}:\overline{Y}\...
2 votes
0 answers
218 views

Proof of Remark 6.14(b) of Milne's Arithmetic duality theorems'

Let $E/\mathbb{Q}$ be an elliptic curve.Let $\operatorname{Sha}(E/\Bbb{Q})$ be a Tate-Shafarevich group. Milne's 'Arithmetic Duality Theorems' Remark 6.14(b) describe the following exact sequence. ...
0 votes
0 answers
71 views

Potential typo in "Complete Systems of Two Addition Laws for Elliptic Curves" by Bosma and Lenstra

Here is a link to the article: https://www.sciencedirect.com/science/article/pii/S0022314X85710888?ref=cra_js_challenge&fr=RR-1. Pages 237-238 give polynomial expressions $X_3^{(2)}, Y_3^{(2)}, ...
3 votes
1 answer
187 views

"General position" on $\mathbb{P}^1\times\mathbb{P}^1$

On $\mathbb{P}^2$ we have the notion of general positions: no 3 points on a line, no 6 on a conic, etc. In particular, blowing up points (up to 8) in general positions give ample anti-canonical class, ...
2 votes
0 answers
96 views

Extensions of $F$-isocrystals

Let $X$ be a smooth affine scheme over $k$, a finite field. Let $W(k)$ denote the Witt ring, and $K$ its fraction field. Fix a smooth lift of $X$ to $K$ and denote it by $X_K$. Let $b\in X(k)$ denote ...
1 vote
0 answers
81 views

Extensions in the category $F\text{-Isoc}(X)$

Let $X$ be a smooth affine scheme over a finite field $k$, let $W(k)$ denote its Witt ring, and by $K$ its fraction field. Let $F\text{-Isoc}(X/K)$ denote the category of convergent $F$-isocrystals on ...
1 vote
0 answers
72 views

Ramification of mod $\ell$ representation of elliptic curves [closed]

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $p,\ell$ be two prime numbers. Consider the mod $\ell$ representation $$\rho:Gal(\mathbb{\overline{Q}}/\mathbb{Q})\to Aut(E[\ell])= GL(2,\ell).$$...
2 votes
0 answers
60 views

Fibre functors of the category $F\text{-Isoc}(X)$

Let $X$ be a smooth affine scheme over a finite field $k$. Denote its Witt ring by $W(k)$, and the fraction field of its Witt ring by $K$. Let $F\text{-Isoc}(X)$ denote the category of convergent $F$-...
3 votes
0 answers
215 views

A Brauer group of a double covering of a "well-understood" variety

Let $k$ be a field (it is possible to assume that $k = \mathbb{Q}$ or $= \overline{\mathbb{Q}}$) and $X, Y$ nice varieties over $k$. Let $f \colon Y \to X$ be a finite flat surjective morphism of ...
10 votes
1 answer
336 views

How is Taylor-Wiles patching "horizontal Iwasawa theory"?

I have recently been reading into the proof of modularity of semistable elliptic curves, in particular (what is now known as) the Taylor-Wiles patching argument used to prove the $R=T$ theorem in the ...
3 votes
0 answers
140 views

Congruences between Eisenstein series and cusp forms

Let $k\geq 4$ be an even integer. Let $p>k$ be a prime such that $p\mid B_k$, the $k$th Bernoulli number. Then there is a primitive cusp form $f=\sum_{n\geq1}c(n, f)q^n$ of weight $k$ and level $1$ ...
1 vote
1 answer
95 views

Full Tannakian subcategories and surjection of fundamental groups

Let $(\mathcal{T},w)$ be a neutral Tannakian category over a field $k$, with fundamental group $G$, and $w$ a fibre functor. Let $(\mathcal{S},w|_{\mathcal{S}})$ be a full Tannakian sub-category (i.e. ...
1 vote
0 answers
97 views

Bounding dimension of $H^1(G_{\mathbb{Q}}, (V_pE)^{\otimes n})$

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $p$ be a prime of good reduction, $T_pE$ is its $p$-adic Tate module, $V_pE = T_pE\otimes \mathbb{Q}_p$, and $(V_pE)^{\otimes n}$ its $n$'th tensor ...
3 votes
0 answers
147 views

Computing basis of $\mathrm{Pic}(\bar{X})$ for a Del Pezzo surface

Say we are given a degree 2 del Pezzo $X$ given by $w^2=Q(x,y,z)$ where $Q(x,y,z)$ is degree 4. We can compute the exceptional lines by computing the 28 bitangent lines of $Q$ and look at the ...
2 votes
0 answers
144 views

What is the Galois representation structure of $B_{\text{cris}}^+/(t)$?

In $p$-adic Hodge theory, there is a nice exact sequence for quotients of $B_{\text{dr}}^+$. Denote by $t$ the typical uniformizer of $B_{\text{dr}}^+$ (the cyclotomic character), then there is a $G_{\...
3 votes
0 answers
136 views

Taylor-Wiles systems for higher dimensional deformation rings

Let $R$ be a deformation ring and $M$ be a finitely generated $R$-module. A strategy for proving the theorems $R=T$ is to associate with $(R,M)$ a Taylor-Wiles system denoted $(R_{Q},M_{Q})$. Here I'm ...
2 votes
0 answers
62 views

The Weil height on a generic fiber of family of abelian variety

In the paper Canonical heights on varieties with morphisms by Joseph H. Silverman, in page 184 (which is page 23 in the PDF) Silverman uses Lang's Fundamentals of Diophantine Geometry to show that $$|...
20 votes
3 answers
2k views

what is the maximum number of rational points of a curve of genus 2 over the rationals

Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.) We are ...
2 votes
0 answers
145 views

Connection on relative topological periodic cyclic homology

I have been looking Bhatt-Morrow-Scholze's paper: https://arxiv.org/pdf/1802.03261.pdf and came to a naive question. Let $C$ be a dg-category (with assumptions?) over $\mathbb{F}_p[[z]]$ and view this ...
2 votes
0 answers
196 views

Using the Dold-Thom Theorem to define \'etale cohomology

For reasonable spaces $X$, the Dold-Thom Theorem states that $\pi_i(SP(X)) \cong \tilde{H}_i(X)$ where $SP(X) = \bigsqcup_i \mathrm{Sym}^i(X)$. There is a purely algebro-geometric realization of this ...
2 votes
0 answers
164 views

Real structure(s) of a Shimura curve ("complex conjugation" of abelian surfaces)

For a complete lattice $L \subseteq \mathbb{C}^2$ let $A_L$ denote the complex abelian algebraic surface that is isomorphic (as a complex manifold) to the complex torus ${\mathbb{C}^2}/{L}$ (this ...
5 votes
0 answers
96 views

Equidistribution of Hecke points and Steinitz classes

Let $K$ be a number field and $\mathcal{P}$ be a prime ideal of $K$. Let $k_{\mathcal{P}}$ be the residue field $\mathcal{O}_K/\mathcal{P}$. Consider the following construction used very often in ...
0 votes
1 answer
182 views

Variants of the classical Satake classfication

Let us fix a connected split reductive group $G$ over a local non-archimedean field $K$ with maximal torus $T$, then most notes including that of [Gross] describes as a consequence of the Satake ...
3 votes
1 answer
134 views

Formal étaleness along Henselian thickenings

Assume that $f:X\to Y$ is an étale map between smooth varieties and $(S,I)$ is a Henselian pair. Let $\alpha\in X(S/I)$. Can we say that the lifts of $\alpha$ to $X(S)$ are in bijection with the lifts ...
1 vote
0 answers
241 views

A hard-Lefschetz theorem with torsion coefficients?

Let $X$ be a smooth projective surface over $\overline{\mathbb{F}_{q}}$. Let $\ell$ be a prime distinct from the characteristic. Assume we have a Lefschetz pencil of hyperplane sections on $X$. Let $...
1 vote
0 answers
67 views

Simplicity of abelian varieties and localization

Let $A$ be an abelian variety defined over a number field $K$. Let $v$ be a place of $K$ and denote by $K_v$ the $v$-adic completion of $K$ with respect to $||\cdot||_v$. Assume $A$ is simple, is it ...

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