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Reconstructing a scheme from its quotient stack

Let $X/S$ be a scheme over a base $S$, with a group action by a nice group $G/S$ (typically $G/S$ affine smooth). Can we reconstruct $X$ from its quotient stack $[X/G]$? It seems that we can expect $X$...
RandomMathUser's user avatar
6 votes
0 answers
219 views

Ranks of elliptic curves over cubic fields

We are writing a paper on the ranks of elliptic curves over cubic fields. The curves of different torsion subgroups are created by the formulas in Jeon et al. and by our new parametrizations. D. Jeon,...
Maksym Voznyy's user avatar
6 votes
0 answers
166 views

Fourier transform and Hodge-$*$ operator

Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says $$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$ ...
user avatar
6 votes
0 answers
163 views

Explicit computations of Serre duality for elliptic curves

I have an elliptic curve $E$ defined over a ring $R$, I want to compute the pairing $$ H^1(E,\mathcal{O}_E)\times H^0(E, \Omega_E^1){\rightarrow}R. $$ Clearly we have that $H^0(E, \Omega_E^1)=R \...
marco's user avatar
  • 109
6 votes
0 answers
295 views

The p^n torsion of a supersingular elliptic curve

Let k be an algebraically closed field of characteristic $p$ and $E/k$ a supersingular elliptic curve. It is well known that $E[p]$ is the unique autodual local group $I_{1,1}$ of Lie dimension $1$ ...
RandomMathUser's user avatar
6 votes
0 answers
173 views

Orlik-Solomon algebra and hyperplane complements in positive characteristic

Let $k$ be an algebraically closed field of characteristic $p\geq 0$, $\underline H:=\{H_1,\dots, H_m\}$ a set of hyperplanes in $\mathbb A_k^n$ and $X:=\mathbb A^n-(\bigcup H_i)$. Given a ring $R$ ...
Emiliano Ambrosi's user avatar
6 votes
0 answers
113 views

$S_n$-invariant polynomials on the dual of reflection representation

Let $W=S_n$ (the symmetric group) acting on $V=k^n$ via permutation of the indices, where $k$ is an algebraically closed field. Closely related to this are the reflection (sub-)representation $V_0=\{ (...
Paul Levy's user avatar
  • 1,336
6 votes
0 answers
230 views

Modularity switching for primes $p>7$

In Freitas, Le Hung, and Siksek's 2014 paper proving that elliptic curves over totally real quadratic fields are modular they prove (and use) the following result (Theorems 3 and 4): let $\bar \rho_{E,...
Avi's user avatar
  • 311
6 votes
0 answers
207 views

Concerning the omnipresence of hyper-elliptic curves in the construction of examples

Vague rambling: I hate asking these types of questions, but I feel that I would benefit immensely from hearing some discussion of the use of hyperelliptic curves in constructing certain examples. What ...
AmorFati's user avatar
  • 1,379
6 votes
0 answers
164 views

What are the genus 4 curves with Jacobians that are 4-th powers?

Consider the moduli space of all genus $4$ curves $\overline{\mathscr M_4}$ of dimension $3\times 4 - 3 = 9$. Under the Torelli map, there is a map to $\overline{\mathscr A_4}$ (which has dimension $...
Asvin's user avatar
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6 votes
0 answers
224 views

Existence of elliptic curve with only one rational point over global fields

Let $K$ be a global field, does there always exist an elliptic curve $(E,e)$ over $K$, such that $E(K)=\{e\}$? (Is there an explicit way to find such a curve?)
user avatar
6 votes
0 answers
98 views

Elliptic deformation of the second Chern class

Second Chern class $$c_2 \in H^4(BGL,\mathbb{Q}(2))$$ admits a nice presentation using dilogarithm. The five term relation in this setting becomes a cocycle condition (details can be found here). ...
Daniil Rudenko's user avatar
6 votes
0 answers
387 views

Geometric interpretation of j-invariants of supersingular elliptic curves

In the classical theory of Complex Multiplication, one considers elliptic curves with an endomorphism ring larger than the integers. In this theory, it's possible to determine the j-invariants of all ...
Felipe Lopes's user avatar
6 votes
0 answers
254 views

$\mu=0$ for CM Elliptic curves?

Let $E$ be an elliptic curve defined over $F$ with CM by $\mathcal{O}_K$ where $K$ is an imaginary quadratic field. We may assume the $F$ contains $K$ and also contains the $p$-division points, where $...
debanjana's user avatar
  • 1,283
6 votes
0 answers
342 views

How to decide whether the isogeny between Neron models is etale?

Let there be an isogeny $f:A_1 \rightarrow A_2$ between two abelian varieties over a $p$-adic field $F$ and assume $f$ has degree $p^n$. By the universal property we get a moprhism $f_0: \mathcal{A}_1 ...
Zhiyu's user avatar
  • 6,622
6 votes
0 answers
218 views

Can the base of an elliptically fibered Calabi-Yau threefold be an Enriques surface?

For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$. If ...
doetoe's user avatar
  • 515
6 votes
0 answers
154 views

Descent via an explicit isogeny (genus 2)

This question is related to a previous question posted by me here answered by Prof. M. Stoll. 5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians. Here I ask some technicalities of a ...
Eduardo R. Duarte's user avatar
6 votes
0 answers
370 views

What happens to Neron-Ogg-Shfarevich when characteristic of the residue field equals the prime at which Tate module is considered?

Neron-Ogg-Shafarevich criterion states that an elliptic curve $E$ over a local field $K$ has a good reduction if and only if the Tate module $T_{\ell}(E)$ is unramified for some prime $\ell$ which ...
Johnny T.'s user avatar
  • 3,625
6 votes
0 answers
141 views

Modular forms for non-arithmetic subgroups

Modular forms are usually considered as complex-valued functions on the upper half-plane quite invariant by a discrete subgroup $\Gamma$ of isometries and satisfying smoothness and growth condition. ...
Desiderius Severus's user avatar
6 votes
0 answers
343 views

Are all stabilizer groups of the co-adjoint action smooth?

Let $k$ be a (non-archimedean) local field of positive characteristic $p$ and $\mathfrak{n}$ be any finite-dimensional nilpotent Lie algebra over $k$ with nilpotence length $l<p$. It is well-known ...
m07kl's user avatar
  • 1,702
6 votes
0 answers
467 views

Torsionfree crystalline cohomology implies torsionfree etale cohomology?

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$. Let $W=W(k)$ be the ring of Witt vectors of $k$. Assume that the crystalline cohomology $H^2_{...
Monsie's user avatar
  • 91
6 votes
0 answers
408 views

Kisin module for CM elliptic curve

Let $E$ be a CM elliptic curve with CM by the field $K$ and assume that $p$ is ramified in $K$ so that $\pi^2 = p \in \mathcal{O}_K$. In particular, then $E$ has supersingular reduction at $p$ and by ...
Vincent's user avatar
  • 443
6 votes
0 answers
418 views

Field of definition of a point in $[p]^{-1}E(K)$

Let $E$ be an ordinary elliptic curve defined over a non-perfect field $K$ of characteristic $p$. If $P \in E(K)$ satisfies $P \not\in [p]E(K)$, is it true that its $p^m$-division points of $P$ are ...
Vinicius M.'s user avatar
6 votes
0 answers
273 views

Is there a prime degree endomorphism on supersingular elliptic curves?

Let $E$ be a supersingular elliptic curve which is defined over field $\mathbb{F}_{p^2}$ and $l$ be a prime such that $gcd(l,p)=1$. Is there an endomorphism $\phi\in End(E)$ such that $deg(\phi)=l$? ...
somayeh didari's user avatar
6 votes
0 answers
98 views

Prime divisors of the norm of the first coefficient of an elliptic newform at width-1 cusps.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $f$ be its newform. Suppose $p \geq 5$ is a prime such that $p^2 \mid N$. We assume $f$ is $p$-minimal, which is equivalent to that the ...
Hao Chen's user avatar
  • 221
6 votes
0 answers
673 views

Isogenous elliptic curves have same conductor

Let $E/K, E'/K$ be elliptic curves defined over a number field $K$. Let $\phi: E \to E'$ be a non-constant isogeny defined over $K$. Why must the conductors be equal? I know that this is an ...
stl's user avatar
  • 585
6 votes
0 answers
472 views

elliptic curves over function fields

Let $K$ be a number field, $E$ an elliptic curve over $K$, and $p$ an odd prime. If $v$ is a place of $K$, we know by the Kummer injection that $E(K_v)/pE(K_v) \hookrightarrow H^1 (K_v, E[p])$ for ...
pedro's user avatar
  • 61
6 votes
0 answers
436 views

Do there exist weak Lefschetz-type statements that were proved for varieties over complex numbers, but were not proved in finite characteristic?

As far as I understood the situation (reading section 3.5B of Lazarsfeld's "Positivity in algebraic geometry" and also some of the references), some of weak Lefschetz-type statements known rely on '...
Mikhail Bondarko's user avatar
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 ...
James Weigandt's user avatar
6 votes
0 answers
971 views

Curious propositon in "Les schemas de modules de courbes elliptiques"

Currently I am reading "Les schemas de modules de courbes elliptiques" by Deligne and Rapoport and I got myself seriously confused about the following proposition (in English translation): (II ...
Holger Partsch's user avatar
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 ...
Andrea Mori's user avatar
5 votes
0 answers
126 views

Using Lang–Trotter to get bounds on averages of Fourier coefficients

Let $E$ be an elliptic curve over $\mathbf{Q}$ and let $(a_n)$ be the sequence of Fourier coefficients for the weight two newform attached to $E$. The coefficients $a_p$ are the Frobenius traces given ...
Joseph Harrison's user avatar
5 votes
0 answers
172 views

"Genus theory" for elliptic curve $L$-functions

Let $d$ be a positive integer, so that $-d$ is a fundamental discriminant. This means that $d$ is square-free at odd primes and $\nu_2(d) \in \{0, 2, 3\}$. Further, if $\nu_2(d) > 0$ then $-d 2^{-\...
Stanley Yao Xiao's user avatar
5 votes
1 answer
204 views

Isogenous elliptic curves and canonical modular polynomials

Let $\ell$ and $p$ be two primes. We are looking for a method for checking whether two supersingular elliptic curves over the finite field $F_p$, given through their $j$-invariants, are $\ell$-...
user447243's user avatar
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 ...
Bernie's user avatar
  • 213
5 votes
0 answers
191 views

Primes of supersingular reduction for non-CM elliptic curves

When $E/\mathbb{Q}$ is a non-CM elliptic curve, Serre had shown that there are density 0 primes of supersingular reduction. His proof can be generalized to elliptic curves over arbitrary number fields....
debanjana's user avatar
  • 1,283
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 ...
cartesio's user avatar
  • 233
5 votes
0 answers
209 views

Elliptic curves and localizations at various primes

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime at which $E$ has good reduction. Let $D=D_{E,p}$ be the $p$-torsion in the cokernel of the map $E(\mathbb{Q})\otimes\mathbb{Z}_p\...
Anwesh Ray's user avatar
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 $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...
P. Grabowski's user avatar
5 votes
0 answers
225 views

Making Virasoro uniformization explicit for elliptic curves

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Lie{Lie}\DeclareMathOperator\Der{Der}\newcommand\el{\mathrm{ell}}$Let $\mathcal{M}_{g,1^{\infty}}$ denote the space ...
user avatar
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 $K_f := \mathbb{Q}_p[x]/(f(x))$, ...
Ashvin Swaminathan's user avatar
5 votes
0 answers
275 views

Goldfeld resolution of the quadratic class number problem

Goldfeld proved the following result. Let $E$ be an elliptic curve (with conductor $N$) over $\mathbb{Q}$ whose Hasse-Weil L-function has a zero at $s = 1$ with multiplicity $g$ then for sufficiently ...
Melanka's user avatar
  • 577
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 $\...
IMT's user avatar
  • 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: ...
Somatic Custard's user avatar
5 votes
0 answers
171 views

Touchard / van der Pol's identity for the sum of divisors and an elliptic curve for perfect numbers

In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$, satisifies the following recurrence relation ($n>1$): $$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
Perfect Number's user avatar
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 ...
Zhiyu's user avatar
  • 6,622
5 votes
0 answers
242 views

Counting elliptic curves by discriminant

Enumerating elliptic curves $E/\mathbb{Q}$ sorted by (the absolute value of) their minimal discriminants is a difficult open problem, as is the (likely easier) problem of counting elliptic curves $E/\...
Stanley Yao Xiao's user avatar
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 ...
Huy Dang's user avatar
  • 245
5 votes
0 answers
151 views

Counting elliptic curves over a number field by their Faltings height

In this paper, Hortsch gives an asymptotic formula for the number of elliptic curves over $\mathbb{Q}$, given by their minimal Weierstrass model, of bounded Faltings' height. In general, is it ...
Stanley Yao Xiao's user avatar
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 ...
skd's user avatar
  • 5,780

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