All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
3
votes
2
answers
530
views
isogeny and congruence subgroup
Let $G_1$ and $G_1$ be two semisimple algebraic groups defined over $\mathbb{Q}$, suppose we have a surjective homomorphism $f: G_1\to G_2$, with finite kernel contained in the center of $G_1$.
By ...
- 699
2
votes
1
answer
208
views
Integral values of rational map
This question is related to this post on Math.MO.
A theorem of B.Segre tells us that if there is one rational point on a non-singular cubic surface $X$ over $\mathbb{Q}$, then the surface is ...
- 3,337
3
votes
2
answers
921
views
Gross's paper on Heegner points
I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points:
Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered ...
- 433
4
votes
0
answers
331
views
Counting Special Rational Points on Cubic Surfaces
A paper of Heath-Brown gives an heuristic argument for the density of rational points on two cubic surfaces: $x^3+y^3+z^3=kw^3,k=2,3$, say, the number of rational points of height less than $N$ on ...
- 3,337
6
votes
1
answer
539
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Fundamental group of the moduli stack of ordinary generalized elliptic curves
Let $M$ be the moduli stack of ordinary but possibly nodal elliptic curves over the field $\overline{\mathbf{F}_p}$. Then $M$ has a $\mathbb{Z}_p^{\times}$-torsor over it, given by the moduli scheme ...
- 25.6k
1
vote
1
answer
348
views
Torsion in relative Picard group
Let $f:X\rightarrow Y$ be a projective map of schemes, $Y$ is finite type over $\mathbb Z[1/N]$ for some big $N$ and moreover $R^1 f_* \mathcal O_X=0$. As far as I understand all this implies that $\...
- 790
15
votes
1
answer
1k
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Derived categories of arithmetic schemes?
Let $X$ be a smooth scheme of finite type over $\mathbb Q$ or $\mathbb Z$. It is natural to consider $D(X)$, the derived category of coherent sheaves on $X$. Beyond the definition, have there been any ...
- 5,186
9
votes
3
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2k
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How many integer points does my favorite ellipse go through?
The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is
$$
x^2+y^2 - \frac{2(d-1)}{d}xy-x-...
- 174
3
votes
0
answers
267
views
How do I compute the Azumaya locus?
I have an algebra $A$ that is finite over its centre $Z(A)$ and I want to compute the Azumaya locus of $Z(A)$ (or equivalently, its ramification locus).
Looking in McConnell-Robson Noncommutative ...
- 1,841
4
votes
1
answer
431
views
Automorphicity of L-Factors of Zeta Functions
Associated to a variety over a number field $K$, one has a family of "Hasse-Weil" L-functions, which can be combined (as an alternating product) to give the Hasse--Weil zeta function of the variety, ...
- 643
3
votes
0
answers
593
views
"Extended" Weil Cohomology Theories
According to Wikipedia, a Weil cohomology theory is a functor from the category of smooth projective varieties over a field $k$, to graded algebras over a field $K$ of characteristic zero, together ...
- 1,838
3
votes
1
answer
352
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Question about the definition of the genus 0 curves in Gross' paper "Heights and the Special values of L-series"
Let $N \in \mathbb{Z}$ be a prime number, and let $B = \left( \dfrac{a, b}{\mathbb{Q}} \right)$ be the unique quaternion algebra over $\mathbb{Q}$ ramified at $N$ and at $\infty$. Then, in section 3 ...
16
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0
answers
878
views
L-Functions of Varieties, Zeta Functions of Their Models
Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...
- 643
1
vote
0
answers
275
views
Integral points on affine rational curves over $\mathbb{Q}$
Given a rational curve $C:(f_1(t),f_2(t))$, where $f_i(t),i=1,2$ are rational functions with rational coefficients.
Question: Is there any criterion(proved or conjectural) for the existence of ...
- 3,337
6
votes
0
answers
719
views
Deligne-Rapoport stack and reduction mod p of X0(p)
I'm trying to better understand some results contained in Deligne and Rapoport's paper on the moduli spaces of elliptic curves.
For convenience, I'll briefly summarize the parts of the paper that I'm ...
- 597
10
votes
0
answers
466
views
Galois action on $E_n$-operads
Let $E_n$ be the little $n$-cubes operad which acts on $n$-fold loop spaces (up to group completion, an $E_n$-action is precisely the data needed to perform an $n$-fold delooping). I am looking for ...
- 25.6k
5
votes
1
answer
723
views
Cubic forms and Hasse Principle
It's well-known that quadratic forms over the rational numbers $\mathbf{Q}$ satisfy the Hasse-Minkowski theorem. This is to say that they are isotropic over $\mathbf{Q}$ if and only if they are ...
- 3,625
2
votes
1
answer
252
views
Height on a semiabelian variety
Let $A$ be a semiabelian variety over $\bar{\mathbb{Q}}$ and $B$ a semiabelian subvariety of $A$. Let $\pi:A\to A/B$ be the canonical morphism. Let $h:A(\bar{\mathbb{Q}})\to\mathbb{R}$ and $h':(A/B)(\...
- 21
2
votes
0
answers
326
views
PAC field : Algebraically closed field :: ? : Henselian local ring
I'm wondering if the following exists in the world as a definition. I'll use the word "pseudo-Henselian." I'll restrict to DVRs for simplicity.
I'd want to call a DVR $(R,\mathfrak{m})$ pseudo-...
- 1,499
7
votes
1
answer
300
views
Generators for exact representations of 3-manifold groups
Does anyone have a list of matrix generators for a bunch of hyperbolic 3-manifold groups? I am testing an algorithm and am looking for a collection of test cases. I am looking for exact values of ...
- 96.4k
2
votes
0
answers
85
views
different and discriminant for finite invariants
Let $k$ be an algebraically closed field.
Let $B$ a $k$-algebra of finite type, normal and Cohen-macaulay. Let $G$ a finite group acting on $B$. We assume that the order of $G$ is prime to the ...
- 3,472
9
votes
0
answers
315
views
congruences of level 1 and level p modular forms
I've been carrying out some experiments on the computer and I noticed the following congruence phenomenon: fixing a prime $p$, it seems that any modular form over $SL_2(\mathbb{Z})$ and of weight $k \...
- 597
4
votes
1
answer
480
views
relation between Faltings height and periods
Let $E$ be an elliptic curve defined by an equation $y^2=4x^3+ax+b$ where $a$ and $b$ are algebraic numbers. What is the relation between the Faltings height $h_F(E)$ and the periods
$$
\int_{\gamma} ...
3
votes
1
answer
424
views
Are Isom-schemes geometrically connected
This question is about properties of Isom-schemes that are well-known over algebraically closed fields.
Let $K$ be a field of characteristic zero, let $C$ be a smooth projective geometrically ...
- 123
17
votes
1
answer
875
views
How fast can we numerically calculate Kloosterman sums?
Define the usual Kloosterman sum by $$S(m,n;c) = \sum_{\substack{x \pmod{c} \\ (x,c) = 1}} e\Big(\frac{mx + n\overline{x}}{c}\Big),$$
where $x \overline{x} \equiv 1 \pmod{c}$, and $e(x) = e^{2 \pi i x}...
- 4,671
1
vote
0
answers
111
views
topological invariance of direct image in the \'etale topology
Let $R$ be a complete local ring (even of dimension one if it helps) and write it as limit of artinian rings $R_n$. Let $X\rightarrow S=Spec(R)$ be proper, finite type even smooth outside the maximal ...
- 11
1
vote
1
answer
510
views
Rational points of non-rational curves
An algebraic curve (in this question) is the zero set $C = f^{-1}(X\ Y)$ of any polynomial $f\in\mathbb R[X\ Y]$; we say then that $f$ represents $C$. ...
- 7,500
2
votes
1
answer
304
views
Purely additive reduction of Jacobian of Hyperelliptic curve
For general, let X be an abelian variety of dimension g.
We say that X has 'purely additive reduction' at prime p if the dimension of the unipotent radical of the special fiber of the Neron Model of ...
- 41
11
votes
1
answer
786
views
A frustrating cohomology class on the moduli of abelian surfaces
Here's a very frustrating question that I have been stuck on for some time. I believe that my question could fit in a general framework of what happens when you restrict $L^2$-cohomology classes on a ...
- 40.2k
2
votes
0
answers
180
views
Jacobian of hyperelliptic curve over local field
For elliptic curve $E$ defined over $K_v$ we know that $E(K_v)$ = $Z_p^{[K_v:Q_p]} + T$(direct sum) where v is prime of K above p and T is finite abelian group(By prop 6.3 in Silverman's book). In ...
- 41
9
votes
1
answer
350
views
Degenerations of modular curves
Has anyone come across anything along the following lines?
Let $X_1(p)$ be the compactification of the quotient of upper half plane by $\Gamma_1(p)$ for some unspecified large prime. Let $X_1(p) \to ...
- 5,186
4
votes
1
answer
398
views
Properties of divisors when moving from char 0 to char p.
Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...
- 141
1
vote
2
answers
472
views
Equations of elliptic curves
First part of question I have asked on mathoverflow already: https://math.stackexchange.com/questions/467088/explict-form-of-the-equation-of-elliptic-curve
1) Let $E(\mathbb{F}_{q^2})$ is elliptic ...
- 2,576
1
vote
0
answers
155
views
Isogenies in multidimensional formal groups
Let $K/\mathbb{Q}_p$ be a local field, $A$ the ring of integers of K, $\pi$ a uniformizer element for $A$, $F$ an n-dimensional formal group with coefficients in $A$ and $f$ an endomorphism of $F$. ...
- 11
7
votes
3
answers
550
views
How few terms may appear in a polynomial with given (cyclotomic) roots and nonnegative coefficients?
Given $W \subset \mathbb C$, let $S_W$ be the set of polynomials in $\mathbb R[x]$ that vanish on $W$ and have only nonnegative coefficients.
Warm-up question: It's clear that if $W$ contains a ...
- 1,513
2
votes
1
answer
188
views
Division fields of isogenous abelian varieties over a p-adic field
Let $p$ be a rational prime number, and let $K$ be a finite extension of $\mathbb Q_p$. Let $A$ be an abelian variety over $K$. For any rational prime number $\ell$, let $K(A[\ell])$ be the field of ...
- 23
7
votes
3
answers
572
views
Siegel's theorem with real coefficients
Let $h(x,y)$ be a polynomial with real coefficients. Suppose there are infinitely many integer solutions to $|h(x,y)|<1$. What can I say about $h$?
When $h$ itself has integer coefficients, a ...
- 156k
5
votes
1
answer
715
views
Weil pairing, fixed field of a $p$-adic Galois representation
Let $A$ be an abelian variety over a $p$-adic field $K$. If $K(A_{p^\infty})$ is the field extension of $K$ obtained by adjoining the coordinates of all $p$-power division points of $A$. By the Weil ...
- 379
7
votes
2
answers
639
views
Is there an algebraically normal function from $\mathbb{Z}^{n}$ to $\{ 0 , 1\}$?
Definition: Let $h$ be a polynomial in $n$ variables, then :
$\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$
Let $\omega : \mathbb{Z}^{n} \to \{ 0 , 1\}$...
3
votes
0
answers
184
views
Ampleness on the P^1 bundle over Siegel threefold
I am looking at the Shimura variety for $\mathrm{GSp}_4(\mathbb Q)$, with hyperspecial level structure at $p$. Let $X$ denote the special fiber over $\mathbb F_p$. For simplicity, let us pretend ...
- 61
5
votes
1
answer
522
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Modular Functions with Rational Fourier Expansions
I have been reading the paper of Cox, McKay and Stevenhagen "Principal Moduli and Class Fields", http://arxiv.org/pdf/math/0311202v1.pdf, and I have a question regarding the nature of the function ...
- 135
2
votes
0
answers
312
views
Parshin construction
In the famous proof of the Mordell conjecture by Gerd Faltings, the so-called Parshin construction is known.
For example, let $E/\mathbb{Q}$ be an smooth elliptic curve, and let us pick up a $\...
- 781
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votes
2
answers
2k
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How did Weil prove the Weil conjectures for curves?
I understand that Weil proved the Weil conjectures for curves. I have seen his proof of the third and trickiest part, the "Riemann Hypothesis for curves," but I am curious about how he showed ...
- 71
4
votes
1
answer
600
views
Is it expected that every natural number is the rank of some elliptic curve over the rationals?
It is a well-known problem on the theory of elliptic curves that the rank of an elliptic curve (the number of generators of the free part of the Mordell group of the elliptic curve) cannot be ...
- 26.9k
1
vote
6
answers
1k
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List of structure theorems for vector valued Siegel modular forms (esp. of genus 2)
What are Siegel modular forms?
We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts.
The symplectic group $...
3
votes
0
answers
486
views
Arithmetic of Fano varieties of lines
Let $k$ be a number field and let $X \subset \mathbb{P}^n$ be a non-singular hypersurface of degree $n-1$. Let $F(X)$ denote the Fano variety of lines of $X$. Then it is known that for general $X$ the ...
- 21.3k
7
votes
1
answer
2k
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Riemann hypothesis and Kakeya needle problem
The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are ...
- 2,106
5
votes
1
answer
570
views
Rigidity, moduli space, and moduli field
In his comment to the question Algebraic numbers and the complex projective line minus three points JSE says that for algebro-geometric objects defined over the complex numbers "in practice, 'the ...
- 11.1k
3
votes
1
answer
231
views
Relative density of images of diophantine polynomials
My current research project involves sampling a sequence along non-constant polynomials with non-negative integer coefficients defined over the natural numbers, and I'd like to be able to say when two ...
- 155
3
votes
1
answer
1k
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Surjectivity of frobenius
I have a question about Faltings' paper "Crystalline cohomology and p-adic Galois representations". Suppose $R$ is a smooth $\mathbb{Z}_p$-algebra, of relative dimension $1$, such that there is an ...
- 699