All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
2
votes
2
answers
817
views
7
votes
0
answers
616
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periods of modular forms
Let $f$ be a cusp modular form of weight $k$ (assume it has rational coefficients to get things simpler). To $f$ one associates some periods $\Omega^{\pm}(f)$ by looking at the subspace of the ...
- 71
0
votes
1
answer
322
views
bilinear pairing on elliptic curve
I'm sorry if this question is not good for mathoverflow.
In this article
http://www.staff.uni-oldenburg.de/florian.hess/publications/pairing-lattice.pdf
Florian Hess defined $a_{s,h}$ and $a_{s,h}^{...
- 2,576
1
vote
1
answer
537
views
Functional equations of zeta functions over global fields
The functional equations for Dedekind zeta functions (zeta functions attached to rings of integers in algebraic number fields) come from functional equations of theta functions like $\sum_{n \in \...
- 7,062
3
votes
1
answer
136
views
q-torsion points of nonsingular elliptic curve
Suppose we have non-supersingular elliptic curve $E$ over $GF(q)$.
How to find minimal k that $|E(GF(q^k)[q]| = q$?
- 2,576
11
votes
1
answer
659
views
Unramified Galois representations not from smooth and proper stacks
Are there any irreducible representations of $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$ over $\mathbb Q_l$ that are unramified away from $l$ and crystalline at $l$ but are not known to arise ...
- 149k
3
votes
0
answers
306
views
Abelian Varieties over C(t) and Galois representations
I am interested if (part of) Tate's theorem can be extended to Abelian Varieties over
$\mathbb{C}(t)$. Specifically, let $A$ be an Abelian Variety of dimension $g$ over
$\mathbb{C}(t)$, and consider ...
- 2,824
13
votes
1
answer
1k
views
Etale homology via étale cosheaves
Can one develop a theory of étale homology via étale cosheaves? The hope is that this would, for example, return the Tate module (and not its dual) for an elliptic curve, and it would return group ...
- 15.4k
3
votes
2
answers
752
views
hyperelliptic curves over finite fields
What is information about the existence of rational points on hyperelliptic curves over finite fields available?
- 2,576
3
votes
0
answers
465
views
Weierstrass Models and Canonical Models
In the book "Algebraic Geometry and Arithmetic Curves," Liu mentions in passing that the minimal Weierstrass model of an elliptic curve plays a similar role to the canonical model of curve of (...
- 51
0
votes
0
answers
244
views
Uniqueness of decomposition of completely reducible representations
Let $X$ be a smooth, separated scheme of finite type over $\mathbb{F}_q$ where $q=p^r$ for some $r>0$. Let $gcd(l,p)=1, \rho:W(X) \to GL_r(\mathbb{Q}_l)$ be a Weil representation which is semi-...
- 1,040
1
vote
1
answer
405
views
calculate function from its divizor
There is elliptic curve $C (y^2 = x^3 + Ax + B)$ over $GF(q)$.
There is algebraic function f on C.
We have div(f).
How calculate f as rational function ( $f = (f_1(x) + yf_2(x)) / (g_1(x) + yg_2(...
- 11
7
votes
1
answer
955
views
Vanishing cycles of a locally constant sheaf for a smooth morphism in the $l = p$-case
$\DeclareMathOperator{\Spec}{Spec}$
My question is concerned with vanishing cycles of a locally constant sheaf for a smooth morphism in the case $l = p$. In the case $l \neq p$ this is a statement in ...
- 146
10
votes
2
answers
961
views
What is a(n algebro-geometric) family of modular forms?
We know that a family of elliptic curves is a morphism of schemes $f:X \to Y$ such that the fiber of every point of $Y$ is an elliptic curve (and we usually require the morphism to be smooth, proper, ...
- 15.4k
9
votes
2
answers
328
views
Is the following the right definition of $L$-functions (on the Galois side)?
This question may be too elementary for this forum, but I have asked it on math stackexchange and didn't get an answer. I have now deleted it so there wouldn't be duplicates... Here is the question as ...
- 91
9
votes
2
answers
1k
views
Are all anabelian Galois actions faithful?
Let $C/\mathbb Q$ be a smooth projective curve of genus $g\geq 2$ or a smooth affine curve of genus $g \geq 1$. The exact sequence
$1 \to \pi_1^{et}(C \otimes_\mathbb Q \bar{\mathbb Q}) \to \pi_1^{et}...
- 149k
2
votes
1
answer
468
views
Absorbing ramification and factoring finite flat maps
In his Algebraic surfaces book, Beauville gives a result allowing one to "absorb ramification" for certain maps (see below). There are also something similar one can do with number fields. I would ...
- 3,555
8
votes
1
answer
654
views
Diferent abelian varieties over local field with the same p-adic representation?
Let $K$ be a local field with residue field of char $p$, denote $G$ its Galois group. Is it possible that we have two Abelian varieties $A_1$ and $A_2$, defined over $K$, such that they are not ...
- 709
5
votes
0
answers
2k
views
What do I need to understand Wiles' proof of Fermat's Last Theorem? [closed]
I'm interested in the proof of FLT, but I'm new to algebra, only knowing a little thing about abstract algebra, group theory.
Can anyone recommend a learning path to me? Particularly, can anyone give ...
- 351
6
votes
3
answers
1k
views
Are there Heronian triangles that can be decomposed into three smaller ones?
Is there anything known about the existence of Heronian triangles ABC (i.e. with rational side lengths and rational area) that can be decomposed into three Heronian triangles ABD, BCD, CAD? ...
- 13.4k
27
votes
2
answers
2k
views
Etale site is useful - examples of using the small fppf site?
Edit: After the answers and comments, I'm hoping for a little bit of elaboration (in the comment to the answer below.) Also, question 2 was discussed here:
Points in sites (etale, fppf, ... )
There, ...
- 3,555
6
votes
4
answers
1k
views
Hodge numbers of reduction mod $p$
Let $X$ be a projective variety defined over a number field $K$, and $p \in \textrm{Spec }\mathcal{O}_K$ a maximal ideal, so that reduction mod $p$ makes sense, and the resulting scheme (mod $p$) $\...
- 3,555
9
votes
1
answer
528
views
jacobians with non-abelian complex multiplication
Hi friends,
I am looking for examples of curves over a number field such that their jacobians are CM abelian varieties by a field whose Galois group is non-abelian. Does anybody know how to produce ...
- 93
32
votes
1
answer
4k
views
How should a number theorist learn a modest amount of algebraic geometry?
A little bit vague, but I hope useful for the entire community.
I am, by training, an analytic number theorist. I have managed to learn some algebraic geometry, by reading parts of Silverman's ...
7
votes
2
answers
595
views
Representation theory of Discrete Subgroups of Lie groups
My question is the following. Which representations of $Sp(2g, \mathbb Z)$ are extendable to representations of $Sp(2g, \mathbb C)$ or $Sp(2g, \mathbb R)$. Is there a general theory and a good ...
- 327
15
votes
0
answers
591
views
For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?
An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...
- 605
2
votes
0
answers
473
views
$\sigma$-conjugate iff $p$-adically close
First some notations. Let $p$ be a prime, $k$ a perfect field of characteristic $p$, $W=W(k)$ the ring of Witt vectors over $k$, $\sigma : W \rightarrow W$ the Frobenius, $R$ a commutative $\mathbb{Z}...
- 271
12
votes
1
answer
1k
views
On the derived category of constructible étale sheaves
The derived category $D^{\flat}_{c}(X,R)$ of constructible sheaves of $R$-modules on $X_{et}$ is defined as the full subcategory of $D^b(X,R)$ whose cohomology sheaves are all constructible.
Clearly, ...
- 15.4k
58
votes
3
answers
4k
views
What is the geometry of an undecidable diophantine equation?
As an arithmetic algebraic geometer of the highest moral fiber, I am trained to look at Diophantine equations in terms of the geometry of the corresponding scheme. For instance, if the Diophantine ...
- 149k
3
votes
0
answers
198
views
Jacobians for arithmetic curves
We know that the Jacobian of an algebraic curve play an important role in the study of curves. My question is: Is there a "Jacobian" for an arithmetic curve such as $Spec Z$, which parameterizes some ...
- 247
6
votes
1
answer
2k
views
Scheme defined over $\mathbb{Z}$
I'd like to check a definition:
If $X$ is a scheme, what does it mean to say that $X$ is "defined over $\textrm{Spec }\mathbb{Z}$"? Is this a precise statement? Certainly this statement ...
- 3,555
14
votes
3
answers
753
views
The boundedness of the rank of twists of a fixed curve
It is conjectured that there are do not exist elliptic curves over $\mathbb Q$ of arbitrarily high rank. I was wondering wether someone made a similar conjecture if one restricts to a fixed $j$-...
- 2,224
2
votes
0
answers
128
views
supersingular curve detector
Suppose I give you a prime $p$ and ask for a non-CM supersingular elliptic curve over $\mathbb{F}_p.$ Can this be done in polynomial time (so, polynomial in $\log p$)?
- 96.4k
5
votes
3
answers
1k
views
Orders of Number Fields
Let $K$ be a number field over $\mathbb{Q}$ of degree $n$, and $\mathcal{O} \subset \mathcal{O}_K$ an order.
$\textbf{Questions:}$
$\newcommand{\Spec}{\textrm{Spec }}$
$\newcommand{\cO}{\mathcal{O}}$
...
- 3,555
7
votes
0
answers
881
views
Rigid Uniformization vs Grothendieck's Local Monodromy Theory
I've noticed that some interesting results about abelian varieties can each be proven using one of two ways: the theory of rigid uniformization of abelian varieties or Grothendieck's local monodromy ...
- 15.4k
16
votes
0
answers
546
views
What can be the dimension of a pointless smooth proper Z-scheme?
What is the smallest dimension $d$ such that there is a smooth proper morphism $X \to \operatorname{Spec} \mathbb Z$ of relative dimension $d$, with $X$ nonempty, without a section?
Of course, there ...
- 149k
2
votes
0
answers
632
views
Kernel of an \'etale isogeny of prime degree $\ell$ between elliptic curves
I recently try to read Vatsal's paper ``Multiplicative subgroups of $J_0(N)$ and applications to elliptic curves.'' He seemed to use the following fact freely:
Let $E$ be a semistable elliptic ...
- 21
6
votes
3
answers
2k
views
Is there any theorem like implicit function theorem in $\mathbb{Q}$ ?
My qeustion is that,
is there any theorem like implicit function theorem in $\mathbb{Q}$ ?
More precisely, let $p(\bar{x},\bar{y})$ be in $\mathbb{Z}[\bar{x},\bar{y}]$ such that in $\mathbb{Q}$, for ...
- 69
6
votes
4
answers
1k
views
Brauer group elements of order $2$
Let $K$ be a field and let $Q$ be a quaternion algebra over $K$. Then it is well-known that the class $[Q]$ of $Q$ in $Br(K)$ has order $2$. One can show this by constructing an explicit isomorphism $...
- 21.3k
1
vote
1
answer
815
views
Bielliptic curves of genus 2
Given a hyperelliptic equation y^2 = f(x) for a curve of genus 2, is there a method to decide whether the curve is bielliptic?
- 23
6
votes
2
answers
401
views
A weird function related to the denominators of rational squares
Between any consecutive integers $a$ and $a+1$ there are infinitely many rational squares of the form $t^2 / s^2$. I have been working to understand the following question: How small can $t$ and $s$ ...
- 525
3
votes
1
answer
612
views
Why is there no "regular etale fundamental group"?
Let $K$ be a number field, and let $X_K$ be a $K$-variety.
The etale fundamental group of $X_K$, as defined in SGA1, classifies the automorphisms of finite etale covers of $X_K$. Some of these etale ...
- 6,348
13
votes
3
answers
2k
views
Estimates for Bezout coefficients
The answer to my question is probably well-known, but I was unable to find a reference.
The Bezout's identity states that for any positive non-zero integers $a_1, \ldots , a_n$ there exist integers $...
- 2,648
8
votes
0
answers
938
views
How do you get algebraic models for modular/shimura curves?
I've got a few questions related to a paper by Lei Yang - "Exotic Arithmetic Structure on the First Hurwitz Triplet" http://arxiv.org/pdf/1209.1783v3.pdf
We know that there are exactly three Hurwitz ...
- 10.7k
10
votes
1
answer
749
views
Is the Hasse principle a birational invariant?
Is the Hasse principle a birational invariant?
It is probably a very trivial question, but I am a beginner in arithmetics.
- 3,779
3
votes
0
answers
877
views
Higher direct image of the structure sheaf and the Hodge bundle.
Hi all
I have a question from a statement made in van der Geer's paper "Cycles on the
Moduli Space of Abelian Varieties".
The statement is as follows. In the paper we are looking at the Hodge ...
- 63
296
votes
8
answers
143k
views
Philosophy behind Mochizuki's work on the ABC conjecture
Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...
0
votes
1
answer
273
views
local galois representation with higher coefficient
Suppose K is a local field , G is its galois group, V a fine dimensional Vector space over F, which is a sub field of K, and totally ramified over $Q_p$. Consdider the linear action of G on V (V is ...
- 709
2
votes
0
answers
74
views
Weak admissibility in algebraic families
Let $M$ be an algebraic family of isocrystals over a base scheme $R/\mathbb{Q}_p$ (not a rigid analytic space).
The question is: is the set of weakly admissible points (i.e., the points $r\in R$ ...
- 1,841
3
votes
1
answer
853
views
Tate's thesis for varieties over finite fields
Tate showed that the functional equation for zeta functions of number fields can be proven with fourier-analytic methods on the adele ring. Can the same be done for zeta functions of varieties over ...
- 804