Questions tagged [arithmetic-geometry]
Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
2,042
questions
3
votes
0
answers
139
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
142
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},\...
20
votes
3
answers
550
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 ...
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}...
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 ...
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 ...
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
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
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 ...
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 ...
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
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 $$\rho:Gal(\mathbb{\overline{Q}}/\mathbb{Q})\to Aut(E[\ell])= GL(2,\ell).$$...
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 ...
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 ...
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
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_{\...
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
$$|...
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 ...
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 ...
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 ...
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 ...
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 ...
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.$$...
14
votes
0
answers
1k
views
Summary of why Wiles' method does not work for general Fermat curves
It is by now extremely well known that Sir Andrew Wiles proved the Taniyama-Shimura conjecture, and therefore, through the Frey-Hellegouarch curve, that for $n \geq 5$ the only integer solutions to ...
3
votes
1
answer
143
views
Examples of curves with fixed genus, number of rational points, and large Jacobian rank
I am asking whether there are known constructions for curves with the following criteria: curves $C$ defined over $\mathbb{Q}$ with genus $g \geq 2$, $|C(\mathbb{Q})| \leq 1$, $C$ is locally soluble, ...
1
vote
0
answers
64
views
Tate-Shafarevich group and its twist such that $\text{Sha}(E_D/\Bbb{Q})=0$ or some constant
Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$.
Let $D\in \Bbb{Z}$ be a square free integer and $E_D/\Bbb{Q}$ be its quadratic twist.
It is widely known that for all $E/\Bbb{Q}$: elliptic ...
0
votes
1
answer
285
views
Tate–Shafarevich group and $\sigma \phi(C)=-\phi \sigma(C)$ for all $C \in \operatorname{Sha}(E/L)$
$\DeclareMathOperator\Sha{Sha}\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a quadratic extension of number field $K$.
Let $\sigma$ be a generator of $\Gal(L/K)$.
Let $E/K$ be an elliptic curve defined ...
4
votes
1
answer
409
views
Is there an elliptic curve analogue to the 4-term exact sequence defining the unit and class group of a number field?
Let $K$ be a number field. One has the following exact sequence relating the unit group and ideal class group $\text{cl}(K)$:
$$1\to \mathcal{O}_K^\times\to K^\times \to J_K\to \text{cl}(K)\to 1$$
...