Skip to main content
MathOverflow

Questions tagged [arithmetic-geometry]

Ask Question

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

2,220 questions
Newest Active Bountied Unanswered
Filter by
Sorted by
Tagged with
10 votes
0 answers
477 views

It’s a well known fact due to Serre that there is no $\mathbf Q_p$- and $\mathbf R$-valued cohomology theory of “Weil type” for varieties over $\overline{\mathbf F_p}$; but for $\ell\ne p$, a $\mathbf ...
coLaideronnette's user avatar
  • 859
2 votes
0 answers
169 views

Let $A/k$ be an abelian variety over a characteristic $p>0$ perfect field $k$. Usually, we say $A$ admits a lift to $W_2(k)$ if there exists an abelian scheme $\mathcal{A}/W_2(k)$ such that $\...
Razumikhin's user avatar
  • 103
2 votes
0 answers
106 views

Are there references that compute the integral motivic cohomology $\mathrm{H}^{p,q}(BG,\mathbb Z)$, for $G$ finite cyclic and where $BG$ is defined over the rationals. I am particularly interested in $...
kindasorta's user avatar
  • 3,366
0 votes
0 answers
50 views

Suppose $K$ is a number field and we have finitely many elements $\alpha_{1}, \ldots, \alpha_{k} \in K$ and a polynomial $f(x)\in K[x]$ of degree $\geq 2$. Given $m \geq 1$, we define $K_{m}(\alpha_{i}...
Gafar Maulik's user avatar
  • 33
2 votes
0 answers
183 views

Let $K=\mathbb{Q}_3(t)$ be the finite extension of the $3$-adic number field $\mathbb{Q}_3$, where $t=3^{1/13}$. I want to know if the polynomial $$f(x)=(x^9-t^2)^3-3^4x+3^3t^5 \in K[x]$$ is ...
Learner's user avatar
  • 450
2 votes
1 answer
370 views

Let $T_1, T_2, ..., T_g$ be the local co-ordinates of Tangent space of an Abelian variety of dimension $g$ around the identity $O$. Let $P \in A(K)$ and $v$ be a non-archimedean place of number field $...
NumDio's user avatar
  • 317
4 votes
1 answer
199 views

Let $p$ be an odd prime. Suppose an elliptic curve $E/\mathbb{Q}$ has reduction of type $I_n$ at $p$. Let $\mathcal{E}$ be its Néron model, and assume that the component group $\tilde{\mathcal{E}}/\...
elliptic's user avatar
  • 95
2 votes
0 answers
156 views

Let $A$ be an abelian variety over a global function field. Is the Mordell-Weil rank of $A$ known to be at most the analytic rank, i. e. the order at $s = 1$ of the $L$-function? This is stated in ...
Vik78's user avatar
  • 1,135
1 vote
0 answers
64 views

Let $J$ be an abelian variety over a number field $K$, and $x \in J(K)$ be a $K$-ratinal point of $J$. Suppose that for infinitely many non-archimedean valuation $v$ of $K$, (where $J$ has good ...
Hugo's user avatar
  • 453
1 vote
1 answer
137 views

Given a Shimura datum $(G,X)$ of abelian type, for $K_p \subset G(\mathbb Q_p)$ a parahoric subgroup and $K^p \subset G(\mathbb A_f^p)$ an open compact subgroup which is small enough, there is an ...
Suzet's user avatar
  • 801
2 votes
0 answers
78 views

Let $K$ be a number field, $X$ a smooth projective curve over $K$ with genus $g$. Let $J$ be the jacobian of $X$ with $j: X \to J$, let $r$ be the rank of the $\mathbb Z$-module $J(K)$. We may choose ...
Hugo's user avatar
  • 453
6 votes
1 answer
300 views

Let $X$ be an affinoid perfectoid space of characteristic either $0$ or $p$. It is well known that for a $p$-torsion sheaf $\mathcal F$, $H^i(X_{et}, \mathcal F)=0$ for $i>1$ (see Scholze's comment ...
Tyler Perkins's user avatar
  • 113
5 votes
1 answer
377 views

I'm aware of two "geometric" constructions of automorphic forms. The first is in terms of Shimura varieties. A Shimura variety is (roughly speaking) a moduli space of abelian varieties with ...
Doron Grossman-Naples's user avatar
  • 2,255
11 votes
2 answers
749 views

I'm looking for the following result in SGA quoted in Grothendieck's "Groupes de Barsotti-Tate et Cristaux de Dieudonne" via a translation of Peterson: Let $S$ be a scheme over $\mathbb{F}_p$...
Ben Singer's user avatar
  • 313
3 votes
0 answers
407 views

Recall two definitons: Let $\mathrm P_{\mathbb Z} = \mathbb Z[\overline{\mathbb N}] / \mathbb Z[\infty]$ denote the free condensed abelian group on null sequences. A condensed abelian group $M$ is ...
David Kolar's user avatar
  • 71
5 votes
0 answers
225 views

Definition 9.2.2 of Scholze and Weinstein's Berkeley Lectures on $p$-adic Geometry says "Consider the site Perf of perfectoid spaces of characteristic $p$ with the pro-ètale topology. A map $f:\...
Tyler Perkins's user avatar
  • 113
3 votes
0 answers
125 views

Let $X$ be a smooth projective curve over a number field $K$ which has a $K$-rational point, and let $F = K(X)$ be its function field. Consider the map $$ \operatorname{Br}(F) \longrightarrow \prod_v \...
aspear's user avatar
  • 161
0 votes
0 answers
135 views

Thanks to work of many people (Madapusi, Charles, Maulik...) the Tate conjecture is now known for K3 surfaces over any finitely generated field. A smooth quartic surface is K3. Consider instead a ...
Vik78's user avatar
  • 1,135
4 votes
1 answer
194 views

Faltings' isogeny theorem implies that two abelian varieties over a number field $K$ which have the same characteristic polynomial of Frobenius at almost all primes are isogenous. If $K = \mathbb{Q}$ ...
Vik78's user avatar
  • 1,135
2 votes
0 answers
147 views

I'm following Scholze-Weinstein's Berkeley notes (https://www.math.uni-bonn.de/people/scholze/Berkeley.pdf). And there is a theorem by Huber (Thm 2.3.3) in the notes that says the adic spectrum $\...
wxhan's user avatar
  • 21
8 votes
0 answers
245 views

It is a famous open problem in field arithmetic whether $\mathbb{Q}^{\mathrm{solv}}$, the solvable closure of $\mathbb{Q}$, is pseudo algebraically closed (PAC). That is, whether every absolutely ...
aspear's user avatar
  • 161
2 votes
0 answers
246 views

I have a, hopefully not so stupid, feeling that the notion of $(\phi,\Gamma)$-modules is an abstract model for $p$-adic Galois representations. I am wondering what is good for us when we work with ...
User0829's user avatar
  • 1,506
2 votes
0 answers
100 views

Let $X$ be a $p$-divisible group over a perfect field $k$ of characteristic $p>0$. Let $M(X)$ be the covariant Dieudonné module of $X$. Thus, $M(X)$ is a finite free $W(k)$-module equipped with a $\...
Suzet's user avatar
  • 801
3 votes
0 answers
262 views

Illusie's article in volume II of the Grothendieck Festschrift cites a 1988 preprint: "Arithmetic of Fermat varieties I, Fermat motives and p-adic cohomologies", by Suwa-Yui. It does not ...
Vik78's user avatar
  • 1,135
3 votes
1 answer
307 views

In On $l$-independence of Algebraic Monodromy Groups in Compatible Systems of Representations, they stated such an example (Example 6.3): Let $X$ be a connected normal scheme of finite type over $\...
Razumikhin's user avatar
  • 103
3 votes
0 answers
173 views

Surely this is well-known in the arithmetic geometry community, which I (unfortunately) do not regard as a member of, so I will include some basic exposition for the laymen (including myself). We deal ...
Stanley Yao Xiao's user avatar
  • 30.7k
2 votes
0 answers
195 views

Let $X:=\mathrm{LT}_h$ be the Lubin-Tate space of the (unique) formal group law of height $h$ over $k:=\overline{\mathbb{F}}_p$. The adic generic fiber $X_{\eta}=\mathrm{LT}_{h,\eta}$ is then a rigid-...
XYZhou's user avatar
  • 29
0 votes
0 answers
117 views

Application 5.7 in "Monodromy of the $p$-rank strata of the moduli space of curves", by Achter and Pries, seems to imply that for $g \ge 3$, there exists a supersingular smooth projective ...
Vik78's user avatar
  • 1,135
1 vote
0 answers
265 views

Let $A$ be a connected algebraic group defined over a solvably closed field $K$. Are there any results concerning the vanishing of the WC-group $H^1(G_K, A)$?
aspear's user avatar
  • 161
4 votes
1 answer
236 views

If two non-isomorphic smooth cubic surfaces over a field $k$ each contain 27 lines defined over $k$, can they become isomorphic over the algebraic closure?
Vik78's user avatar
  • 1,135
4 votes
0 answers
261 views

Let $k$ be the function field of a $d$-dimensional regular integral finite-type scheme $Y$ over $\mathbb{Z}$. Conjecture 2 in Tate's paper in the Woods Hole proceedings predicts (among other things) ...
Vik78's user avatar
  • 1,135
1 vote
0 answers
246 views

Let $K$ be a $p$-adic number field with ring of integers $\mathcal{O}_K$ and uniformizer $\pi$. Let $F$ and $G$ be two $d$-dimensional formal groups of height $h$ over $\mathcal{O}_K$. Denote: \begin{...
Learner's user avatar
  • 450
4 votes
1 answer
464 views

Let $X$ denote a smooth proper scheme over a finite field $k$ of characteristic $p$. It is my understanding that it is not generally known if the Frobenius action on the rationalized crystalline ...
kindasorta's user avatar
  • 3,366
1 vote
0 answers
67 views

Let $Y\to X$ denote a relative abelian scheme, with $X$ smooth proper over $k$ a finite field of characteristic $p$. Let $\mathcal{V} := R^1_{\mathrm{cr}}f_*\textbf{1}[1/p]$ denote the Gauss-Manin $F$-...
kindasorta's user avatar
  • 3,366
1 vote
0 answers
217 views

I am trying to show that if $\pi: X\rightarrow S, f: S\rightarrow T$ are smooth morphisms over positive characteristic. Then $(\mathbb{H}^i_{dR}(X/S), \nabla_{GM})$ is nilpotent, in the sense of Katz (...
Razumikhin's user avatar
  • 103
7 votes
1 answer
330 views

Excuse me if some parts of the questions are well known, I am still a PhD student and have a lot to learn. This question came to me while doing research on the Coleman's conjecture (as presented in a ...
P. Tchamitchian's user avatar
  • 71
2 votes
1 answer
338 views

Deligne seems to say on page 239 here that the Serre-Tate canonical lift of an ordinary abelian variety over the algebraic closure $\overline{k}$ of a finite field of characteristic $p$ canonically ...
Vik78's user avatar
  • 1,135
7 votes
1 answer
245 views

Let $k$ be a perfect field and $W(k)$ be the ring of Witt vector and define the ring of formal power series $$R:=W(k)[[v_1, \cdots, v_{h-1}]]$$ in $h-1$ variables. Then there is a canonical surjection ...
Learner's user avatar
  • 450
2 votes
0 answers
194 views

By section 2 of https://swc-math.github.io/aws/2024/2024KaremakerNotes.pdf, for $q = p^r$ a prime power, any $g, d \ge 1$, and any $n \ge 3$ prime to $q$, there exists a stratified fine moduli space $...
Bma's user avatar
  • 571
5 votes
1 answer
229 views

Thanks for your help. I'm reading the paper Iwasawa main conjecture for the Carlitz cyclotomic extension and applications (https://arxiv.org/pdf/1412.5957). I find three questions about the motivation ...
Rellw's user avatar
  • 463
5 votes
1 answer
460 views

Let $E/K$ be an elliptic curve over a local field. I understand that the Kodaira type of $E/K$ refers to the isomorphism class of the special fiber of the Néron minimal model of $E/K$ as a scheme over ...
elliptic's user avatar
  • 95
1 vote
0 answers
188 views

Let $A$ be an abelian variety over a finite field $k$ and $\ell$ a prime number. Fix a natural number $n\in \mathbb{N}$ and denote by $k_{n}$ the field $k(A[\ell^{n}](\overline k))$. Let $Q\in A[\ell^{...
aya 's user avatar
  • 327
5 votes
2 answers
508 views

For $X$ a regular (or perhaps just normal) variety over Spec $\mathbb{Z}$, is the etale fundamental group of $X$ topologically generated by Frobenius elements at closed points? This is true when $X$ ...
Bma's user avatar
  • 571
2 votes
0 answers
262 views

When I study Weil-étale cohomology put forward by S. Lichtenbaum in the paper The Weil-étale topology on schemes over finite fields, I find two questions. Let $X$ be a scheme of finite type over a ...
Rellw's user avatar
  • 463
2 votes
0 answers
162 views

Let $F$ be a non-Archimedean local field. For a multiplicative character $\chi$ of $F^\times$, let $a(\chi)$ be the conductor of $\chi$. For a nontrivial additive character $\phi$ of $F$, we denote ...
USER19823's user avatar
  • 53
1 vote
0 answers
98 views

Let $A$ be an abelian variety over a number field $K$ of dimension $g$. Consider $\mathcal{L}$ be a symmetric ample line on $A$ that gives an embedding $ \ i_{\mathcal{L}} : A(\overline{K}) \...
NumDio's user avatar
  • 317
2 votes
0 answers
198 views

Thanks for your help. I know the definition of Weil group and how to get it for $p$-adic local field cases. By reading the answer https://math.stackexchange.com/a/2898449/1007843, I write down my ...
Rellw's user avatar
  • 463
2 votes
0 answers
132 views

Let $K$ be a non-archimedean local field (e.g., $\mathbb{Q}_p$ ), $A / K$ an abelian variety of dimension $g$. Consider $\mathcal{L}$ a line bundle on $A$ and $F \in \Gamma(A, \mathcal{L})$ a global ...
NumDio's user avatar
  • 317
3 votes
0 answers
228 views

Let $A/K$ be an abelian variety where $K = \operatorname{Frac}(R)$ for some discrete valuation ring $R$. Let $\mathcal{A}$ be its Neron model. Assume that $A$ has potential good reduction, so that the ...
user47437's user avatar
  • 393
2 votes
0 answers
158 views

A friend recently nerd-sniped me with a (seemingly elementary) geometry question: Let $\triangle ABC$ (with corresponding side lengths $a$, $b$, and $c$) be an obtuse triangle with $\angle C > 90^\...
abiteofdata's user avatar
  • 43

1
2 3 4 5
…
45