All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
8
votes
0
answers
1k
views
Ramified Geometric Langlands
Is the following a reasonable formulation of (a part of) the geometric Langlands conjecture for $\mathrm{GL}_n$ over a curve $X$?
(*) Let $\mathcal{L}$ be an irreducible rank $n$ $\ell$-adic local ...
- 2,751
2
votes
0
answers
105
views
Possible Context for this "Siegel-like" Modular Form Construction?
The following construction of something very nearly a Siegel modular form of degree 2 arose in my research. I'm outside the worlds of automorphic forms and number theory, so I'm wondering if it ...
- 1,701
3
votes
1
answer
335
views
Comparison of two definitions of the modular sheaf $\omega$
I have seen two equivalent definitions of the modular sheaf $\omega$. Let $S$ be some base scheme. If $p \colon \mathcal{E} \to X$ is the universal generalized elliptic curve over the modular curve $X$...
- 949
23
votes
2
answers
1k
views
Theta functions on an elliptic curve and Serre duality
Given an elliptic curve $E$ (over $\mathbb{C}$) and line bundle $L$, one can identify $H^0(E,L)$ with a particular space of theta functions.
Serre duality gives a perfect pairing between $H^0(E,L)$ ...
- 411
7
votes
0
answers
272
views
Is it a coincidence that $Gal(\mathbb C / \mathbb R) \cong C_2 \cong Aut(E_1)$? (Or: why are $\mathbb C$-algebras with involution so useful?)
The automorphism group $Aut(E_1)$ of the $E_1$ operad is the cyclic group of order 2, $C_2$, and thus $C_2$ acts on any category of algebras (by reversing the multiplication). The seeming coincidence ...
- 64k
1
vote
0
answers
188
views
How small can $u$ be in the Pell equation $u^2-k^3 v^2=\pm 1$?
Let $k$ be positive integer, not a square and let $u_k,v_k$ be non-trivial
solutions to the Pell equation $u_k^2-k^3 v_k^2=\pm 1$.
Q1 How small $u_k$ can be infinitely often as function $k$?
This ...
- 25.4k
3
votes
1
answer
179
views
The local zeta-functions of some cubic plane curves
This question is motivated by the work presented in article 358 of Gauss' Disquisitiones Arithmeticae. For the sake of completeness, let me say something about the background and present the question ...
- 935
4
votes
0
answers
230
views
Extra line bundles from torsors
Another math.stackexchange question (here: $\mathbb{G}_m$-torsors and line bundles) goes over a way to construct a line bundle $L$ from a $\mathbb{G}_m$-torsor $T \to B$, by using a decomposition $\...
- 949
0
votes
0
answers
118
views
Multiplicity of a polynomial in positive characteristic
Let $\mathbb K$ be a field of characteristic $p>0$.
Let $f\in\mathbb K[x_1,\dots,x_n]$ be a multivariate polynomial and let $q\in\mathbb K^n$. Is there a computational method to determine the ...
- 351
3
votes
1
answer
374
views
Computing Mordell-Weil Groups without Rational Torsion
Summary: How does one compute the Mordell-Weil group of an elliptic curve $E / \mathbb{Q}$, in the case where the torsion points are only defined over larger fields?
More detail: I've been reading ...
- 584
1
vote
1
answer
170
views
Maps to additive group scheme
Let $\underline{\mathbb{Q}_p/\mathbb{Z}_p}$ be constant p-divisible group over $\mathbb{F}_p$. And let $\mathbb{G}_a$ be the additive group over $\mathbb{F}_p$. Let me prove
$$
Hom(\underline{\mathbb{...
- 397
3
votes
2
answers
750
views
Witt vectors addition confusing
I raise this confusing because I try to understand the witt vectors for characteristic not equal to p.
Let us assume p=2. The Witt Polynomials is explicitly given by
$$
S_0=X_0+Y_0
$$
$$
S_1=X_1+...
- 397
1
vote
0
answers
256
views
Deformation of $p$-divisible group
To try to understand the deformation of $p$-divisible group more explicit, I am thinking given a connected $p$-divisible group $G_0$ on $\overline{\mathbb{F}_q}$, Choose a deformation $G$ of $G_0$ ...
- 397
6
votes
2
answers
1k
views
About different cohomology theories used to study Shimura varieties
The classical theory of Eichler-Shimura realizes the space of cusp forms of certain weights and levels in the parabolic cohomology of modular curves, which is the image of the cohomology with compact ...
- 159
2
votes
0
answers
107
views
How to obtain the following "trivial" bound on the number of rational points on a hypersurface?
Let $F: \mathbb{R}^{n} \to \mathbb{R}$ be a smooth function.
Suppose $B$ be a closed and bounded box.
I would like to obtain for fixed $q \in \mathbb{N}$
$$
\# \{ \mathbf{a} \in \mathbb{Z}^n : F(\...
- 3,625
4
votes
1
answer
250
views
How to measure how much a rational function/a singularity of variety is complicated?
There are some theorems about various zeta functions which states the rationality of those.
For example, when you consider Igusa's zeta function, roughly the generating series of solutions mod $p^n$ ...
- 485
7
votes
3
answers
553
views
Two queries on triangles, the sides of which have rational lengths
Let us define a "rational triangle" as one in the Euclidean plane, with lengths of all sides rational.
We are aware that a positive integer is called "congruent" only if it is the area of a right ...
5
votes
0
answers
208
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 ...
- 26.9k
3
votes
0
answers
243
views
Interlacing sequences by polynomials?
Given $t=2^\ell$ where $\ell\in\mathbb N_{>0}$ and $M\in\mathbb Z$ and two sets of integers $\{a_1,\dots,a_t\}$ and $\{b_1,\dots,b_t\}$ with $0<a_1\leq \dots\leq a_t<M$ and $0<b_1\leq \...
- 13.9k
4
votes
1
answer
891
views
Isomorphism of the $\ell$-adic Tate module of an elliptic curve with CM
Let $E$ be an elliptic curve over $K$ (totally real number field) with complex multiplication by the field $L$. Let $\psi$ be the Grössencharacter associated to $E$, assume that $\psi$ of type $(-r,0)$...
- 43
0
votes
0
answers
2k
views
Definition of an invariant differential of an elliptic curve
I am somewhat confused by the definition of the invariant differentials in J. Silverman's book The Arithmetic of Elliptic Curves.
Let $E$ be an elliptic curve with Weierstrass equation $F(x,y)=0$. ...
- 2,375
9
votes
2
answers
744
views
Number of solutions mod p and Betti numbers
Suppose $X$ is proper flat scheme over Sepc$\mathbb{Z}$. If $p$ is a large prime, then by Weil conjectures one can recover the Betti numbers of $X$ from the size of $X(\mathbb{F}_{p^n})$ for all $n$. ...
- 123
1
vote
1
answer
271
views
Local heights in Vojta's conjecture
I am a complex geometer trying to parse Vojta's conjecture on rational points, and I have a very basic misunderstanding (I apologize if this is too easy for MO).
Let $X$ be a variety over a number ...
- 188
9
votes
0
answers
475
views
Classification of finite flat group schemes over integers?
One can classify (commutative) finite flat group schemes (with order of $p$-powers) over $\mathbb Z_p$ using semi-linear algebraic datas such as Breuil-Kisin modules. And we can fix the special fiber ...
- 6,622
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
0
votes
0
answers
287
views
On the product in the power series ring
Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$.
Suppose we have two ...
- 563
3
votes
0
answers
77
views
How can I find the integral orthogonal group of a given symmetric positive definite form?
I wonder how one can study the integral orthogonal group of a given (symmetric, positive definite) bilinear form like the one described by the following matrix:
$$M=\begin{bmatrix}
x_1 &...
- 362
3
votes
1
answer
190
views
Igusa zeta functions of univariate polynomials: $\mathbb{Z}_p$ or $\mathbb{Q}_p$ in this statement
Let $f\in\mathbb{Z}_p[X]$ and let $Z_{f,p}(T)\in\mathbb{Z}_{(p)}(T)$ be the $p$-adic Igusa zeta polynomial (i.e. $Z_{f,p}(p^{-s})$ is the $p$-adic Igusa zeta function in the complex variable $s$, with ...
3
votes
0
answers
127
views
Maximal unramified quotient of $E[p]$ for the action of $G_{\mathbb{Q}_p}$
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good and ordinary reduction at an odd prime $p$.
Suppose $E[p]$ denotes the $p$-torsion points of $E$ and $G_{\mathbb{Q}_p} := \text{Gal}(\...
- 303
2
votes
1
answer
622
views
Galois representations associated to the algebraic cycles and transcendental cycles of K3 surfaces
Given a K3 surface $X$, the cup product defines a non-degenerate even unimodular structure on the lattice $H^2(X,\mathbb{Z})$. Inside this lattice we have the Neron-Severi group $\text{NS}(X)$, which ...
- 2,971
0
votes
0
answers
96
views
Elementary constraints for the solutions of $z^{n-2}y(y+z)=x^n$?
Related to FLT and this question.
For natural $n > 4 $ define the curve $C_n : z^{n-2}y(y+z)=x^n$.
$C_n$ has the trivial points with $x=0$ for all $n$.
The answer in the linked question shows ...
- 25.4k
3
votes
1
answer
315
views
How to show that Hodge filtration of CM type is algebraic?
I'm asking for a proof or references of the following claim:
Let $V$ be a rational Hodge structure having CM in the sense that its Mumford-Tate group is abelian. Then there is a filtration $F^{\...
- 467
6
votes
0
answers
434
views
Local Fontaine--Mazur?
Given a finite extension $K$ of $\mathbb{Q}_p$, is there some conjectural statement characterizing which finite-dimensional $p$-adic representations of the absolute Galois group of $K$ are (Tate ...
user138661
9
votes
0
answers
2k
views
Exactly Counting the Number of Lattice Points in an $n$-Dimensional Sphere
Let $S_n(R)$ denote the number of lattice points in an $n$-dimensional "sphere" with radius $R$. For clarification, I am interested in lattice points found both strictly inside the sphere, and on its ...
- 311
2
votes
0
answers
234
views
The abc conjecture modulo variety
It is known that the abc conjecture can't fail with polynomial
identities.
Is the following special case of abc known?
Let $a,b,c,f$ be polynomials with integer coefficients satisfying
$a+b=c+f$. ...
- 25.4k
7
votes
3
answers
401
views
On $\{P(x)+Q(y):\ x,y=0,\ldots,p-1\}$ with $p$ prime
QUESTION: Is my following conjecture (formulated in 2016) true? How to solve it?
Conjecture. For any non-constant polynomials $P(x)，Q(x)\in\mathbb Z[x]$, there is a positive integer $N(P,Q)$ ...
- 15.6k
8
votes
0
answers
405
views
The Frobenius at the infinite prime?
For simplicity, suppose $X$ is a smooth $n$-dimensional variety defined over $\mathbb{Q}$. Then the etale cohomology of $X$, denoted by $H^i_{\text{et}}(X,\mathbb{Q}_\ell)$, gives a representation of ...
- 2,971
4
votes
0
answers
206
views
Higher dimensional generalization of an identity between traces of Hecke operators and number of elliptic curves over finite fields?
In http://www.math.ubc.ca/~behrend/ladic.pdf, the author uses his generalization of Lefschetz trace formula to smooth algebraic stacks to prove an interesting identity (Proposition 6.4.11.):
$\sum_{k}...
- 6,622
2
votes
0
answers
214
views
Cohomology of modular curves: vanishing and decomposition
Let $\pi:E\to Y$ be a universal elliptic curve over an open modular curve $Y$. Take a prime $\ell$ and take $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$ where the dual, $(-)^\vee$, means the internal ...
- 1,428
6
votes
1
answer
652
views
$l$-adic periods?
For an algebraic variety $X$ over $\mathbb{Q}$ the comparison isomorphism between Betti and de Rham cohomologies provides the theory of periods with a motivic context whose reformulation as motivic ...
- 61
4
votes
1
answer
372
views
$p$-adic realisation of Kummer motive and Frobenius matrix
Suppose $M$ is an object in the abelian category of mixed Tate motives over $\mathbb{Q}$, and it is an extension of $\mathbb{Q}(0)$ by $\mathbb{Q}(1)$
\begin{equation}
0 \rightarrow \mathbb{Q}(1) \...
- 2,971
8
votes
1
answer
519
views
Do $p$-adic topological modular forms exist?
Are there $p$-adic topological modular forms? What is the analogue of finite slope and overconvergent?
user139985
24
votes
1
answer
887
views
Universal homotheties for elliptic curves
Let $K$ be a number field and $E_1, \cdots, E_n$ elliptic curves over $K$. Let $\ell$ be a prime. Then there exists an element $\sigma \in \text{Gal}(\overline{K}/K)$ such that $\sigma$ acts on $T_\...
- 23k
9
votes
0
answers
291
views
Searching for hypergeometric motives that split
Motivation: It seems that the splitting of a hypergeometric motive is closely related to some highly non-trivial hypergeometric identities discovered by Ramanujan, Guillera et al. The splitting of ...
- 3,337
3
votes
2
answers
907
views
Finding coefficient of multivariate polynomial
$f(x_1,x_2,\ldots x_n)$ is polynomial with integer coefficients.
$f$ is rather large to be computed explicitly, but an algorithm can
compute it efficiently at integers and complex number and "...
- 25.4k
2
votes
0
answers
93
views
The prime spectrume of integral-valued polynomial ring
Let $ D $ be an integral domain with quotiont field $K $ and let $Int (D) $be the set of all integral-valued polynomials on $D $, that is, $ Int (D):=\{f \in K[x]\mid f (D) \subseteq D\} $. The ...
- 21
7
votes
0
answers
222
views
Space of algebraic closures of $\mathbb{Q}$
The ambiguity inherent in defining the absolute Galois group $G_\mathbb{Q}$ - that it is determined only up to inner automorphisms - arises from the fact that one has to choose an algebraic closure of ...
- 71
7
votes
1
answer
479
views
Rigid versus log-rigid cohomology for semistable varieties
If $K$ is a p-adic field, with maximal unramified subfield $K_0$, and $X$ is a proper semi-stable $O_K$-scheme, then there's a canonical way to make the special fibre $X_k$ into a log-scheme; and ...
- 37k
2
votes
1
answer
184
views
Centralizers of Cartan subgroups
Let $E$ be an elliptic curve with CM by an order $\mathcal O$ in an imaginary quadratic field $K$. Choose a basis for $E[N]$ to get an isomorphism $\operatorname{Aut}(E[N])\cong \operatorname{GL}_2(\...
- 2,375
12
votes
2
answers
2k
views
how do we prove that a sum of two periods is still a period?
Kontsevich and Zagier define periods as the values of absolutely convergent integrals $\int_\sigma f$ where $f$ is a rational function with rational coefficients and $\sigma$ is a semi-algebraic ...
- 129