All Questions
974 questions with no upvoted or accepted answers
6
votes
0
answers
295
views
Does a lower bound for models of finite group schemes exist?
Let $R$ be a discrete valuation ring (as beautiful as you like) and set $K:=Frac(R)$. Let $G_K$ be a finite $K$-group scheme, $G_1$ and $G_2$ two affine and flat models of $G_K$ of finite type, i.e. ...
- 73
6
votes
0
answers
1k
views
a naive question about p-adic local monodromy theorem
The question is about whether one can view the p-adic local monodromy theorem as the quasi-unipotence of some monodromy operator.
it is known that the classical local monodromy theorem (i.e. for ...
- 313
6
votes
0
answers
936
views
Can you get Siegel's theorem "for free" from modularity and Mazur's Eisenstein Ideal paper?
There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...
- 3,268
6
votes
0
answers
456
views
On periods of algebraic integers modulo rational primes
I run, somewhat indirectly, into the following problem and I have no hints where to look in the literature in search for answers or clues.
Let $K$ be a number field, which we may assume Galois if it ...
- 797
6
votes
1
answer
266
views
Does exist a "product formula" for arithmetic surfaces?
Let $K$ be a number field, then the Arakelov geometry of $\operatorname{Spec }O_K$ can be interpreted by means of the adelic ring $\mathbb A_K$. In particular, a key ingredient is the product formula ...
- 529
5
votes
0
answers
236
views
Triviality of $\unicode{1064}(T_pE ⊗ T_pE)$ for elliptic curves and Bogomolov's lemma
Consider the case of an elliptic curve $E$ over Q, and let $S$ be a finite set of primes including all places of bad reduction and a place $p$ of good reduction.
Bogomolov's Lemma says that when $p$ ...
- 2,907
5
votes
0
answers
211
views
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 ...
- 617
5
votes
0
answers
110
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 ...
- 885
5
votes
0
answers
261
views
Equations for conic del Pezzo surfaces of degree one
Let $X$ be a del Pezzo surface of degree one over a field $k$ of characteristic not $2$ equipped with a conic bundle $\pi: X \rightarrow \mathbb{P}^1$. By Theorem 5.6 of this paper, $X$ admits a ...
- 241
5
votes
0
answers
348
views
Are there toy models of full poly-isomorphism from Inter-universal Teichmüller theory?
I am learning Inter-universal Teichmüller theory and I am interested in the concept of full poly-isomorphism. According to "INTER-UNIVERSAL TEICHMÜLLER THEORY I: CONSTRUCTION OF HODGE THEATERS&...
5
votes
0
answers
168
views
Generalization of Deuring's theorem
Deuring has the following important theorem on elliptic curves over a finite field: Let $p$ be a prime at least $5$, and $N=p+1-a$ an integer with $|a|\leq 2\sqrt{p}$, then the number of elliptic ...
- 213
5
votes
0
answers
192
views
Image of $\gamma-1$ on etale $(\varphi,\Gamma)$-modules
Let $p\geq 3$ be a prime, $D$ be an etale $(\varphi,\Gamma)$-modules over the classical period ring $A_{\mathbb{Q}_p}=\mathbb{Z}_p[\![T]\!][1/T]^{\widehat{\phantom{xx}}}_p$ and $\gamma$ be a ...
- 549
5
votes
0
answers
387
views
Calculating étale fundamental groups from the usual fundamental group
$\newcommand{Spec}{\operatorname{Spec}}$Let $X$ be a connected affine smooth variety over $\mathbb{Q}$, with a point $x\in X(\Spec(\mathbb{Q})$.
For any algebraically closed field $K$ of ...
- 541
5
votes
0
answers
546
views
Perfect algebraic spaces on a paper of Xinwen Zhu
I have problem reading Xinwen Zhu's paper Affine Grassmannians and the geometric Satake in mixed characteristic about perfect algebraic spaces in Section A.1.
Let $k$ be a perfect field of ...
- 151
5
votes
0
answers
480
views
What does Colmez's conjecture tell us?
There is the well known conjecture of Colmez, which describes the logarithmic derivative of the $L$-function of a character via the periods of CM-abelian varieties. Equivalently it describes the ...
- 4,428
5
votes
0
answers
278
views
Torsion points of an elliptic curve over number fields. Another proof of Silverman AEC theorem
I am studying the following theorem from Silverman's AEC:
I am wondering whether there exists another proof that doesn't make use of formal groups and is still valid for a number field $K$. Could you ...
- 233
5
votes
0
answers
139
views
Liouville property in the Bost theorem on foliations
Let $X$ be a smooth algebraic variety over a number field $K$, and let $\mathcal{F}$ be an involutive coherent subsheaf of the tangent bundle. After a pullback to $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...
- 485
5
votes
0
answers
303
views
2-descent on elliptic curves, and units modulo squares of units
Setup: Let $p$ be a prime, let $f(x) \in \mathbb{Q}_p[x]$ be a separable monic cubic polynomial cutting out the maximal order $\mathcal{O}_{K_f}$ in the etale algebra $K_f := \mathbb{Q}_p[x]/(f(x))$, ...
5
votes
0
answers
328
views
Ramification behavior of field given by adjoining $p$-torsion point on formal group of abelian variety
Setup. Let $p > 2$ be a prime, let $K$ be the completion of the maximal unramified extension of $\mathbb{Q}_p$, and fix an algebraic closure $\overline{K}$ of $K$. Let $A/K$ be an abelian variety ...
- 998
5
votes
0
answers
264
views
Can arithmetic geometry accelerate the search for rational points in high dimensions?
There are several ideas in arithmetic geometry that can help in proving the absence of rational points as well exhibiting rational points on algebraic curves.
I am aware there are some obstructions (e....
5
votes
0
answers
151
views
Reduction theory of higher dimensional algebraic varieties
If $X$ is a nonsigular curve over a number field $K$, one can obtain several arithmetic models of $X$. Namely, we can construct an arithmetic surface $\mathcal X\to\operatorname{spec} O_K$, such that $...
- 1,237
5
votes
0
answers
154
views
Curves of genus 0 over a DVR determined by fibers?
Closely related is this question.
Suppose $R$ is a DVR with fraction field $K$ and residue field $k$ (say finite), and $S = \mathrm{Spec}(R)$.
I am interested in regular, proper, flat schemes $X \to S$...
- 1,338
5
votes
0
answers
184
views
Given a point $P$ on a genus-$1$ curve over $\mathbb{Q}_p$, is there an $R$ such that $2 \mid [P - R]$ and $x(\overline{P}) \neq x(\overline{R})$?
Let $p \in \mathbb{Z}$ be prime, and let $f \in \mathbb{Z}_p[x]$ be a quartic polynomial with nonzero discriminant. Let $C/\mathbb{Q}_p$ be the genus-$1$ curve with affine equation $y^2 = f(x)$. Let $\...
- 53
5
votes
0
answers
148
views
algebraic de Rham cohomology of toric varieties (reference request)
I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled:
...
- 1,142
5
votes
0
answers
212
views
Interpolating between curves in different characteristics
Let $p\neq q$ be two primes. For a given integer $g>0$ choose a smooth proper geometrically connected curve of genus $g$ over $\mathbb{F}_p$ and similarly for $q$. Is there a proper flat morphism $...
user158636
5
votes
0
answers
250
views
A complex analytic version of the eigencurve
I am very much a beginner to the theory of eigencurves so there might be many mistakes in what follows, especially since it is all very speculative.
My understanding of the eigencurve $\mathcal C_{N,...
- 7,746
5
votes
0
answers
344
views
Reduction type of elliptic curves over $p$-adic fields and local Langlands correspondence for $GL_2(F)$
In an introductory note of local Langlands correspondence http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/old_introductory_notes_on_local_langlands.pdf, section $11$ describes a recipe to ...
- 6,622
5
votes
0
answers
163
views
Does the cardinality of coordinate projections of the rational points of affine varieties over finite fields also tend to $\infty$?
We know (basically by Lang-Weil) that for an absolutely irreducible n-dimensional affine variety $V$ the cardinality $\#V(F_{l})$ tends to $\infty$ for $l$ large enough. We could now look at the set ...
- 311
5
votes
0
answers
361
views
Equivalent definitions of the ring $B_{\mathrm{cris}}$
I'm reading Laurie's note about Fargues-Fontaine Curve and I think he uses a different definition of $B_{\mathrm{cris}}$. Usually when $R$ is a perfect ring of characteristic $p$, $A_{\mathrm{cris}}(R)...
- 1,093
5
votes
0
answers
146
views
Finite locally free group scheme killed by its order?
When I saw this paper of René Schoof, there are two questions on the first page and what confuses me is that how to reduce the first question to the second. This is expalined in the first paragraph on ...
user141691
5
votes
0
answers
459
views
A functor on Abelian varieties corresponding to this operation on Weil numbers
Let $A/\mathbb F_q$ be an abelian variety over a finite field with Weil numbers $q^{1/2}\alpha_1,\dots,q^{1/2}\alpha_n$.
Consider the numbers $q^{d/2}\alpha_1,\dots,q^{d/2}\alpha_n$. These are still ...
- 7,746
5
votes
0
answers
315
views
motivations of classifying $p$-divisible groups
Let $k$ be a perfect field of characteristic $p>0$ and $W:=W(k)$ is the witt ring. Let $K$ be a totally ramified extension of $K_0:=W(\frac{1}{p})$ and $\Lambda:=W[[u]]$ is the formal series ring ...
user141691
5
votes
0
answers
209
views
Complex isomorphism class of abelian varieties and $L$-functions
In his famous Mordell paper, Faltings proved that two abelian varietes $A_1, A_2$ defined over a number field $K$ are isogenous if and only if the local $L$-factors of $A_1, A_2$ are equal at every ...
- 27k
5
votes
0
answers
462
views
Over what fields does the Mordell conjecture (Faltings's theorem) hold?
Inspired by this question, over what fields is the Mordel conjecture known to be true?
For instance, is it true over fields of finite type (that is, fields finitely generated over their prime ...
- 7,746
5
votes
0
answers
217
views
Exact differential forms in characteristic $p>0$
Let $k$ be an algebraically closed field of characteristic $p>0$. Suppose $1< e_i <p$ for $i=1,2, \ldots, n$ are integers ($n \ge 2$). What are the conditions on the $e_i$'s so that the ...
- 245
5
votes
0
answers
427
views
Is it true that all smooth group schemes can be deformed?
Consider for instance the map $\mathbb Z/p^2 \to \mathbb Z/p$ and suppose we are given a group smooth scheme $G$ over $\mathbb Z/p$. Is it always possible to lift it to a smooth group scheme $G'$ over ...
- 7,746
5
votes
0
answers
283
views
Smooth morphisms to the moduli stack of elliptic curves
Fix a prime $p$, and let $\overline{M}$ be the Deligne-Mumford compactification of the moduli stack $M$ of elliptic curves (which, concretely, is the open substack of the stack of cubic curves which ...
- 5,780
5
votes
0
answers
197
views
Semisimplicity of the p-adic étale Tate module over $F_p(t)$
Let $k$ be a finitely generated field of positive characteristic p.
Let $A$ be an abelian variety over $k$ and write $T_p(A)$ for the $p$-adic étale Tate module of $A$. Is it known if the natural ...
- 728
5
votes
0
answers
128
views
Do non-constant maps specialize to non-constant maps?
Let $R$ be a dvr with fraction field $K$ and residue field $k$. Let $\mathcal{X}\to \mathcal{Y}$ be a morphism of $R$-schemes such that $\mathcal{X}_K\to \mathcal{Y}_K$ is non-constant.
Is the ...
- 51
5
votes
0
answers
386
views
Is there a relationship between the zeta function of a Laplacian and the Selberg Zeta Function?
Let us define the zeta function of an elliptic differential operator $H$ with eigenvalues $\lambda_n$ like so:
\begin{aligned}
\zeta_H(s)
:=
tr( H^{-s} )
\\
:=
\sum^\infty_{n=1} ...
- 457
5
votes
0
answers
395
views
Geometric Frobenius on $\ell$-adic cohomology
Let $X$ be a smooth projective variety over a finite field $k$, and $F$ the geometric Frobenius on $H^*_{et}(X_{k^{sep}}, \mathbf{Z}_{\ell})$.
Is $F$ only rationally bijective, or integrally ...
user120812
5
votes
0
answers
1k
views
Formal GAGA and étale cohomology
Let $\mathfrak{X}$ be a $p$-adic flat formal scheme over $\mathbf{Z}_p$, whose special fiber has an ample line bundle. Then $\mathfrak{X}$ is algebraizable, that is there exists an algebraic $\mathbf{...
user92332
5
votes
0
answers
398
views
Vector bundles vs algebraic cycles
For integral schemes, the Picard group is isomorphic to the group of Cartier divisors modulo linear equivalence.
What is the correct analog of this isomorphism, for higher codimension Chow groups vs ...
user119470
5
votes
0
answers
243
views
Map associated to linear system onto curve is morphism
In Mumford's first paper on Surfaces in char $p$ [1], part 2 Step (II), he wants to show that, given an indecomposable curve of canonical type $D$ on a smooth projective surface $F$ with $p_g(F)=0, ...
- 61
5
votes
0
answers
140
views
On a family of polynomials
Investigating experimentally a topic (somewhat related to Bernoulli convolutions), I came across families of polynomials and I wonder whether they belong to some well-known family. A closely related ...
- 14.4k
5
votes
0
answers
568
views
Eisenbud-Goto conjecture in Positive Characteristic
Eisenbud-Goto conjecture predicted that the Castelnuovo-Mumford regularity
${\rm reg}(X)$ of a non-degenerate projective variety $X\subset \mathbb{P}^N$
is bounded by the $\deg(X)-{\rm codim}(X,\...
- 1,595
5
votes
0
answers
562
views
Reduction of torsion points on Neron Model
Let $K/\mathbb{Q}_p$ be a finite extension with ring of integers $R$ and residue field $k$. Let $A/K$ be an abelian variety with Neron model $\mathcal{A}/R$. We denote by $\tilde{\mathcal{A}}/k$ the ...
- 421
5
votes
0
answers
445
views
Algebraization of Brauer classes in a paper of Lieblich
I am reading a paper of Lieblich on the unirationality of K3 surfaces and am trying to understand the result of Proposition 4.1:
Proposition 4.1: Let $k$ be an algebraically closed field of ...
- 2,607
5
votes
0
answers
677
views
Basic question on p-adic Hodge theory
I am starting to study the rudiments of p-adic Hodge theory and I have the following basic question. Let $\chi$ be the unramified quadratic character of $G_p = \mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{...
- 111
5
votes
0
answers
328
views
Definition of logarithm for universal vector extension
Let $S$ be a topological $\mathbb{Z}_p$-algebra and $R\to S$ a surjection (where $R$ has $p$ nilpotent) with topologically nilpotent kernel which has a PD structure.
We know that if $G/R$ is a $p$-...
- 843