All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
13
votes
2
answers
1k
views
Galois cohomologies of an elliptic curve
I asked this question on Mathematics Stack Exchange but did not get any answer and I was suggested to post the question here.
I am studying basic theory of elliptic curves. I encountered Galois ...
user46846
12
votes
1
answer
1k
views
What's wrong with my understanding of the scheme $\text{Isom}(E_\lambda, E_{\lambda'})$?
Let $\mathcal{M}_{1,1}$ be the moduli stack of elliptic curves (over the complex numbers). Define
$$\begin{eqnarray*} X &:=& \Bbb{A}^1_{\lambda} - \{0,1\}\\
X' &=& \Bbb{A}^1_{\lambda'} ...
- 2,607
7
votes
1
answer
985
views
Why are integer points on elliptic curves interesting and useful?
I read some papers which dealed with integer points on elliptic curves. One of these papers are
http://projecteuclid.org/euclid.rmjm/1214947612.
My question is: Why are integer points on elliptic ...
- 321
3
votes
1
answer
336
views
If $F(x,y)$ is a polynomial which is not a square, then how often is the specialization $F(x,a)$ a square?
Suppose $F(x,y)$ is a polynomial in two variables over a field $K$, and $F(x,y)$ is not a square. When is $F(x,a)$ a square for $a\in K$? I would guess that Hilbert's Irreducibility Theorem might help ...
- 671
16
votes
1
answer
2k
views
Relation between Weil Conjecture and Langlands Program
Recently I read Gelbart's An Elementary Introduction To The Langlands Program, which explained the origin of the program, and this question came to me. For an elliptic curve over finite field, the ...
- 363
8
votes
2
answers
2k
views
Explicit $2$-descent on elliptic curves
Let $k$ be a field of characteristic $0$ and let
$$E: y^2 = f(x)$$
be an elliptic curve over $k$, with $\mathrm{deg}(f) = 3$. Kummer theory yields a map
$$\varphi:\mathrm{H}^1(k, E[2]) \to \mathrm{H}^...
- 21.3k
10
votes
1
answer
414
views
Integral points on elliptic curves of the form $y^2=x^3+px$
As the title says. Can we determine all the integral points on elliptic curves of the form
$$y^2=x^3+px$$
for a prime $p$? If yes, can someone explain me how? A good reference would also be ...
- 321
5
votes
1
answer
332
views
are the coarse moduli schemes of finite etale covers of $\mathcal{M}_{1,1}$ smooth?
Let$\newcommand{\mM}{\mathcal{M}}$ $\mM_{1,1}$ be the moduli stack of elliptic curves. Let $R$ be a Dedekind domain, say $\mathbb{Z}[1/N]$ for simplicity, and suppose we have a finite etale cover:
$$\...
- 10.7k
1
vote
0
answers
81
views
The genus of specialization of rational surface (compared to genus of parametrization)
Let $f(u,v),g(u,v),h(u,v)$ be polynomials with rational
coefficients or rational functions.
Assume (this happens in general) they parametrize
rational surface $F(x,y,z)=0$.
For rational $a$, assume ...
- 25.4k
3
votes
1
answer
665
views
Reducibility of resultants
This is closely related to this question. Suppose I have the resultant $\mathcal{R}$ of two (or more polynomials) over $\mathbb{Q},$ and suppose $\mathcal{R}$ is not irreducible. What is the ...
- 96.4k
12
votes
1
answer
572
views
Non-algebraic Hecke characters
Algebraic Hecke characters are ubiquitous in modern number theory. They are in 1-1 correspondence with one dimensional complex Galois representations, and in some precise sense they are the building ...
- 17.6k
8
votes
1
answer
610
views
what are the finite etale covers of $\mathbb{Z}_p((x))$?
Let $R$ be the ring of integers of some $p$-adic field $K$ (finite over $\mathbb{Q}_p$) with uniformizer $\pi$ and residue field $k$. I'd like to understand the finite etale extensions of $R((x)) := R[...
- 10.7k
34
votes
3
answers
3k
views
irreducibility of discriminant
This must be well-known to everyone but me, but here goes: take a general (monic) polynomial $p(x) = x^d + a_{d-1} x^{d-1} + \dotsc + a_0.$ The discriminant is a polynomial $D(a_0, \dotsc, a_{d-1}).$ ...
- 96.4k
16
votes
3
answers
1k
views
First formulation of the Dedekind and Hasse-Weil conjectures
I'm looking for the original statement of two important conjectures in number theory concerning L-functions. I'm particularly interested in pinning down the year in which they were first formulated:
...
- 17.6k
6
votes
2
answers
1k
views
Motivation for Hirzebruch-Jung Modified Euclidean Algorithm
Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows:
Let $e_i \in \mathbb{N} >2$, and $ r_k \in \mathbb{N}$...
3
votes
0
answers
175
views
polynomial relations between modular functions
$\newcommand{\Qbar}{\overline{\mathbb{Q}}}$
We define a modular function to be a meromorphic modular form of weight 0 for some subgroup (not necessarily congruence) $\Gamma\le\text{SL}_2(\mathbb{Z})$ ...
- 10.7k
7
votes
1
answer
398
views
Higher-dimensional Artin L-functions
I begin by clarifying that the "higher-dimensional" in my question refers to analogues of Artin L-functions over higher dimensional base schemes than $\mathrm{Spec}(\mathbb{Z})$.
Now for the set-up. ...
- 21.3k
3
votes
1
answer
182
views
Does a modular function primitive for $\Gamma$ generate the function field of $\mathcal{H}/\Gamma$?
Let $f$ be a modular function (that is, a meromorphic modular form of weight 0) holomorphic on $\mathcal{H}$ which is invariant under $\Gamma\le SL_2(\mathbb{Z})$ (not necessarily congruence!), and ...
- 10.7k
7
votes
2
answers
686
views
confounding riddle about fine moduli schemes and twists of elliptic curves
I've encountered a strange situation while thinking about modular curves... Consider the modular curve $Y(3)$ parametrizing elliptic curves with a symplectic basis for their 3-torsion. This curve has ...
- 10.7k
6
votes
3
answers
449
views
Smooth complete intersections and sharpness of the Chevalley-Warning theorem
Let $X$ be a complete intersection in $\mathbb{P}^n$ of multidegree $(d_1,\ldots,d_r)$. If we're working over a finite field $\mathbb{F}_q$, the Ax-Chevalley-Warning theorem says that if $X$ is in the ...
- 153
9
votes
1
answer
446
views
Is Frac $\mathbb{Z}((x))$ Hilbertian?
Note that Frac $\mathbb{Z}((x)) \ne\mathbb{Q}((x))$.
As a result of Some questions about the ring Z((x)), we know that it is a Dedekind domain with uncountably many primes, each of which is of the ...
- 10.7k
7
votes
0
answers
944
views
Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres
The Wikipedia article on Exotic Sphere displays this sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classses as
$$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; 2,\; ...
- 10.5k
8
votes
0
answers
403
views
rings of modular functions on the upper half plane
Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index.
Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index.
Let $M_0(\Gamma_i)$ denote the ring of modular ...
- 10.7k
7
votes
3
answers
908
views
Do there exist elliptic curves over schemes which have all primes as residue characteristics?
It's well known that there are no elliptic curves over Spec $\mathbb{Z}$, but it's unclear (to me at least) if the proof generalizes.
My question is: If $S$ is a connected scheme such that has every ...
- 10.7k
6
votes
0
answers
489
views
A problem on universally locally acyclic
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ and $S$ be two smooth varieties over $k$ and $\mathcal F$ a constructible \'etale sheaf of $\mathbb F_\ell$-modules on $X$ (...
- 135
2
votes
1
answer
487
views
On quasi-algebraically closed fields
By Lang's theorem, a complete valued field which is the fraction field of a discrete valuation ring with an algebraically closed residue field is quasi-algebraically closed (or $C_1$).
How much is ...
- 1,981
15
votes
0
answers
427
views
Counting abelian varieties over finite fields in a given isogeny class
Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the ...
- 148k
12
votes
2
answers
489
views
Twists of cubic threefolds
Let k be a field of characteristic $0$.
Let $X$ be a variety over k which is isomorphic to a smooth cubic threefold over $\bar{k}$. Then is $X$ isomorphic to a smooth cubic threefold over $k$?
...
- 21.3k
6
votes
1
answer
541
views
Why does $GL_n(F)\backslash GL_n(\mathbb{A}_F)/\prod_xGL_n(O_x)$ classify vector bundles over $X$?
Suppose $X$ is an algebraic curve, $F$ its function field, $\mathbb{A}_F$ its adele, $O_x$ the ring of integers at local field of $x$. Why does $GL_n(F)\backslash GL_n(\mathbb{A}_F)/\prod_xGL_n(O_x)$ ...
user39380
3
votes
1
answer
164
views
Mestre-type algorithm for higher-genus curves?
Is there an analogous algorithm for genus $g>2$ curves that, given a complete set of invariants, outputs a curve with those invariants?
(I'm interested in particular in $g=3$.)
Any references ...
- 61
31
votes
2
answers
15k
views
A road to inter-universal Teichmuller theory
What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study inter-universal Teichmuller (IUT) theory? I know that it heavily relies on anabelian ...
- 1,099
1
vote
0
answers
154
views
Rank of the Jacobian of a family of hyperelliptic curves of genus 2
Assume tha $C$ be the hyperelliptic curve $y^2 = (x-a_1)\cdots (x-a_5)$ of genus $g=2$ and $a_i \in \mathbb{Z}$ and we know that the integers $a_i$ has the form $a_i= d_1^2 - d_i^2$ for some positive ...
- 11
3
votes
0
answers
489
views
Does the Bombieri-Lang conjecture imply severe restrictions on rational points on twists of hyperelliptic curves?
According to Silverman, the Bombieri-Lang conjecture implies
that the rational points of surface on general type lie on
finite set of curves, except for a finite set of points.
Let $f$ be univariate ...
- 25.4k
2
votes
0
answers
246
views
Morphism of Shimura varieties and differential equations
Is there a way of constructing a morphism between Shimura varieties using differential equations? Maybe, this looks like a completely ridiculous question, so I think that I should explain the context ...
- 21
3
votes
0
answers
558
views
Rank of the Jacobian of twists of hyperelliptic curves
Suppose that a hyperelliptic curve $C$ of genus $g \geq 4$ is given by the equation
$$\displaystyle C: y^2 = a_0 x^{2g+2} + a_1 x^{2g+1} + \cdots + a_{2g+2} = f(x).$$
The Jacobian variety $J(C)$ of ...
- 26.9k
9
votes
1
answer
864
views
Definition of p-adic modular forms
I have been reading Hida's book "p-Adic automorphism forms on Shimura varieties" and I don't understand a point.
He first describes p-adic modular forms of tame level N as functions on the Igusa ...
- 845
8
votes
1
answer
776
views
Hyperelliptic curves with fixed genus and many rational points
It is a famous theorem of Faltings, previously a conjecture by Mordell, that any algebraic curve of genus at least $2$ defined over the rational numbers have at most finitely many rational points. A ...
- 26.9k
3
votes
2
answers
480
views
Is it normal surface of general type to have infinitely many positive rank elliptic curves?
Cross-posted from MSE.
I am not good at algebraic geometry and almost surely am
misunderstanding something.
Got an alleged argument against Bombieri-Lang conjecture and
would like to know what the ...
- 25.4k
2
votes
1
answer
408
views
Will a tilting sheaf over Z which is a generator over Q be a generator modulo every prime?
Assume you have a smooth quasi-projective scheme $X$ (you can actually assume $X$ is projective over an affine scheme of finite type) defined over $\mathbb Z$ (or if you prefer, a discrete valuation ...
- 44.7k
9
votes
0
answers
649
views
Motivic fundamental group of the moduli space of curves?
Suppose I have a smooth projective family of varieties of varieties over $\mathcal M_g$ - i.e. a universal functor, commuting with deformations, from curves to smooth projective varieties. Can I ...
- 148k
2
votes
0
answers
245
views
Help for reference of moduli stack of fake elliptic curves
I see everywhere the following:
Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$.
...
- 21
6
votes
2
answers
417
views
How simple does a $\mathbb{Q}$-simple group remain after base change to $\mathbb{Q}_{\ell}$?
Of course the general answer to the question in the title is: not very simple.
I could not think of a better title, so let me explain my question in more detail.
I have a number field $E/\mathbb{Q}$, ...
- 5,504
2
votes
0
answers
416
views
In how many ways can one extend the zero section of the affine line with a double origin
Let $X$ be the affine line with a double origin over $\mathrm{Spec}\,\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over $\mathrm{Spec}\,\mathbb Q$.
Let $0$ be ...
- 21
8
votes
3
answers
1k
views
Ranks of elliptic curves depend only on the field?
Let $K/\mathbb{Q}$ be an algebraic extension, and let $E_1,E_2/\mathbb{Q}$ be elliptic curves. Is it possible that the Mordell-Weil rank of $E_1(K)$ is finite while that of $E_2(K)$ is infinite?
- 11.3k
2
votes
0
answers
182
views
An elliptic curve trivial over any extension unramified outside 7 and infinity?
Is there an elliptic curve $E/\mathbb{Q}$ such that $E(K)$ is trivial for every finite extension $K/\mathbb{Q}$ with discriminant a power of $7$ ?
- 11.3k
10
votes
0
answers
541
views
Is the compositum of all quadratic extensions of the rationals an ample field?
Let $K$ be the compositum of all quadratic extensions of $\mathbb{Q}$, that is $$K = \mathbb{Q}(\sqrt{d} \ : \ d \in \mathbb{Q}).$$
Is there a (geometrically irreducible) smooth variety $V/\mathbb{...
- 11.3k
6
votes
1
answer
1k
views
Pure motives and compatible systems of $\ell$-adic representations
I am trying to understand the statement of the conjectures of Deligne on special values of certain $L$-functions, from his article titled, "Valuers de Fonctions L et periodes d'integrales" which ...
- 1,563
13
votes
2
answers
633
views
Automorphisms of finite order in $Out(\widehat{F_2})$
Let $\widehat{F_2}$ be the pro-$\ell$ completion of the free group of rank 2, where $\ell$ is some prime.
Every outer automorphism of $F_2$ induces an outer automorphism of $\widehat{F_2}$, hence an ...
- 10.7k
7
votes
1
answer
409
views
What is the normal closure of $GL_2(\mathbb{Z})$ inside $GL_2(\mathbb{Z}_\ell)$?
This weird problem popped up in my research:
Let $\ell$ be a prime. Is there a description of the smallest normal subgroup of $GL_2(\mathbb{Z}_\ell)$ containing $GL_2(\mathbb{Z})$?
Is there a ...
- 10.7k
8
votes
2
answers
804
views
Field of definition of dominant morphisms
Let $k$ be an algebraically closed field and $k_0$ a sub-field. Let $X,Y$ be two projective varieties defined over $k_0$. Suppose that that there exists a dominant morphism $f$ between $X_k=X\otimes k$...
- 382