Questions tagged [arithmetic-geometry]

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

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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 ...
George's user avatar
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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 $\dim_{\mathbb{Q}_p} V$...
kindasorta's user avatar
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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 ...
Asvin's user avatar
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2 votes
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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},\...
kindasorta's user avatar
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20 votes
3 answers
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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 ...
Estwald's user avatar
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1 answer
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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}...
kindasorta's user avatar
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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 ...
kindasorta's user avatar
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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 ...
kindasorta's user avatar
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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]&...
Stepan Nesterov's user avatar
2 votes
1 answer
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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 ...
Desunkid's user avatar
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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 ...
David Hubbard's user avatar
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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 ...
kindasorta's user avatar
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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 ...
kindasorta's user avatar
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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 ...
Kheled-zâram's user avatar
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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-...
kindasorta's user avatar
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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 ...
kindasorta's user avatar
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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 ...
kindasorta's user avatar
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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 ...
kindasorta's user avatar
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2 votes
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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 $...
Asvin's user avatar
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1 vote
0 answers
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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}\...
kindasorta's user avatar
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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. ...
Duality's user avatar
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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)}, ...
Vik78's user avatar
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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, ...
fp1's user avatar
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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 ...
kindasorta's user avatar
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1 vote
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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 $$\rho:Gal(\mathbb{\overline{Q}}/\mathbb{Q})\to Aut(E[\ell])= GL(2,\ell).$$...
ZZP's user avatar
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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 ...
kindasorta's user avatar
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2 votes
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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 $F$-...
kindasorta's user avatar
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3 votes
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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 ...
k.j.'s user avatar
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10 votes
1 answer
338 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 ...
Wojowu's user avatar
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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 $1$ ...
Zakariae.B's user avatar
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. ...
kindasorta's user avatar
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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 ...
kindasorta's user avatar
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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 ...
spiderchips's user avatar
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 ...
Marsault Chabat's user avatar
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_{\...
kindasorta's user avatar
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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 $$|...
Or Shahar's user avatar
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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 ...
Daniel Pomerleano's user avatar
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 ...
Asvin's user avatar
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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 ...
Breakfastisready's user avatar
0 votes
1 answer
183 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 ...
Coherent Sheaf's user avatar
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 ...
DGrimm's user avatar
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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 $...
a17's user avatar
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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 ...
kindasorta's user avatar
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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 ...
ALi1373's user avatar
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2 votes
0 answers
166 views

Is the Weil restriction of an elliptic curve self-dual?

$\DeclareMathOperator\res{res}$Let $K=\mathbb{Q}(\sqrt{-3})$, and let $$p\equiv 1\pmod 3$$ be a prime split in $K$. Assume that $$p=\omega*\overline\omega,\quad\text{where}\quad\omega\equiv 1\pmod 3.$$...
yhb's user avatar
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14 votes
0 answers
1k views

Summary of why Wiles' method does not work for general Fermat curves

It is by now extremely well known that Sir Andrew Wiles proved the Taniyama-Shimura conjecture, and therefore, through the Frey-Hellegouarch curve, that for $n \geq 5$ the only integer solutions to ...
Stanley Yao Xiao's user avatar
3 votes
1 answer
143 views

Examples of curves with fixed genus, number of rational points, and large Jacobian rank

I am asking whether there are known constructions for curves with the following criteria: curves $C$ defined over $\mathbb{Q}$ with genus $g \geq 2$, $|C(\mathbb{Q})| \leq 1$, $C$ is locally soluble, ...
Stanley Yao Xiao's user avatar
1 vote
0 answers
64 views

Tate-Shafarevich group and its twist such that $\text{Sha}(E_D/\Bbb{Q})=0$ or some constant

Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Let $D\in \Bbb{Z}$ be a square free integer and $E_D/\Bbb{Q}$ be its quadratic twist. It is widely known that for all $E/\Bbb{Q}$: elliptic ...
Duality's user avatar
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0 votes
1 answer
285 views

Tate–Shafarevich group and $\sigma \phi(C)=-\phi \sigma(C)$ for all $C \in \operatorname{Sha}(E/L)$

$\DeclareMathOperator\Sha{Sha}\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a quadratic extension of number field $K$. Let $\sigma$ be a generator of $\Gal(L/K)$. Let $E/K$ be an elliptic curve defined ...
Duality's user avatar
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4 votes
1 answer
410 views

Is there an elliptic curve analogue to the 4-term exact sequence defining the unit and class group of a number field?

Let $K$ be a number field. One has the following exact sequence relating the unit group and ideal class group $\text{cl}(K)$: $$1\to \mathcal{O}_K^\times\to K^\times \to J_K\to \text{cl}(K)\to 1$$ ...
Snacc's user avatar
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