All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
13
votes
0
answers
595
views
What's about N. M. Katz' "over-world" of exp. sums?
Having just read in N. M. Katz' beautiful old survey on exponential sums a d differential eq.s, I wonder what became out of his question (on p. 297 - 300) on a "general conceptual framework in which ...
- 10.8k
3
votes
0
answers
474
views
Comparison between singular and etale cohomology in Batyrev's paper on Birational Calabi-Yau varieties
My question refers to the paper http://arxiv.org/pdf/alg-geom/9710020.pdf where Batyrev proves that birational Calabi-Yau algebraic varieties have equal Betti numbers by counting points over finite ...
- 806
2
votes
0
answers
358
views
Counting points on an algebraic set over a finite field
Let $q=p^n$, for $p$ a prime. Let $C$ be an Artin–Schreier curve defined by $y^p-y=f(x)$ where $f(x) \in \mathbb{F}_{q}[x]$.
Let $C_g$ be $y^p-y=f(x)$ where $x \in g(\mathbb{F}_{p^{n}})$ for some $...
- 21
3
votes
1
answer
306
views
many rational points on an algebraic curve
Given polynomial $f(x,y)$ with integer coefficients, may be reducible, but without linear factors. For positive integer $n$ denote by $a_n$ the number of points $(x,y)\in \frac1n \mathbb{Z}^2$ on a ...
- 109k
7
votes
1
answer
1k
views
An interesting double coset in the theory of automorphic forms
Does anyone have some idea to describe the double coset $P(F)\backslash G(F)/H(F)$ , say using Weyl group elements ? Here $G=GL_n\times GL_{n-1}$ is defined over a number field $F$ , $H=GL_{n-1}$ ...
- 809
14
votes
1
answer
1k
views
Is every finite group a quotient of the Grothendieck-Teichmuller group?
The Grothendieck-Teichmuller conjecture asserts that the absolute Galois group $Gal(\mathbb{Q})$ is isomorphic to the Grothendieck-Teichmuller group. I was wondering, would this conjecture imply the ...
- 5,929
5
votes
4
answers
518
views
What is the obstruction for a local set of models of a curve to come from a global model?
If $X_{\mathbb{Q}}$ is a curve over $\mathbb{Q}$, we get a curve $X_{\mathbb{Q}_p}$ over $\mathbb{Q}_p$ for every prime $p$.
My question is about the reverse process. Say we are given curves $X_{\...
- 6,348
1
vote
1
answer
263
views
Is the other extreme of Hilbert Irreducibility true?
Let $K$ be a number field (or perhaps more generally a Hilbertian field). Let $X_K\rightarrow \mathbb{P}^1_K$ be a regular (i.e. without extension of scalars) $G$-Galois branched cover. Hilbert's ...
- 5,929
6
votes
4
answers
1k
views
Nonstandard Methods ( or Model Theory ) and Arithmetic Geometry
I hear that the nonstandard methods are applied to many problems in various fields of mathematics such as functional analysis, topology, probability theory and so on.
So, I have become interested in ...
- 945
7
votes
2
answers
870
views
Polarizations of K3 surfaces over finite fields
Suppose that $X$ is a (projective) K3 surface over a field $k$. A polarization of $X$ is an element $\lambda\in Pic_X(k)$ that is represented over an algebraic closure $\overline{k}$ by an ample line ...
- 3,032
2
votes
0
answers
464
views
a small question about abelian variety
Given an abelian variety over $k$, say $A$, and suppose $O_L\hookrightarrow End(A)$, let $P$ be a fractional ideal of $L$, I'm a little confused why $A\otimes_{O_L} P$ is an abelian variety. In my ...
- 21
4
votes
1
answer
664
views
Ideal class groups and extension of number fields
[I already posted this question on stackexchange a while ago, but did not get any response: http://math.stackexchange.com/questions/93437/ideal-class-groups-and-extension-of-number-fields]
Let $(X, \...
- 1,961
7
votes
0
answers
909
views
Deformation of ordinary p-divisible groups via Grothendieck-Messing
I am hoping that someone can point out the error in the "proof" of the following "theorem":
Theorem: Let $k$ be a perfect field of characteristic $p>2$ and let $G$ be an ordinary $p$-divisible ...
- 1,609
3
votes
0
answers
308
views
Invertible Hasse-Witt for non-ordinary curves
Assume that $S$ is a smooth curve of over a field $k$ of characteristic $p>0$ and $f\colon X\to S$ is a relative curve over $S$ (i.e., the fibers are curves). It is well-known that when all the ...
- 395
6
votes
1
answer
554
views
Class field theory using only ideles of norm 1
I am a total non-expert, so the answer to this question may be obvious, but here goes.
In Chevalley's formulation of CFT we get Artin maps $J_k \rightarrow Gal(L/k)$, where $J_k$ is the group of all ...
- 806
8
votes
1
answer
506
views
Is the extension of the Abel-Jacobi map to the smooth locus of the minimal regular model of a curve an immersion?
Let $S$ be the spectrum of a discrete valuation ring with generic point $\eta$. Let $C/\eta$ be a smooth connected curve with an $\eta$-valued point, and let $\mathcal{C}/S$ be the smooth locus of the ...
- 646
5
votes
0
answers
376
views
Does Grothendieck-Teichmuller tell us something about Galois actions, or just about Gal(Q)?
Taking the approach in: http://www.msri.org/realvideo/ln/msri/1999/vonneumann/schneps/1/
I view the Grothendieck-Teichmuller conjecture as saying that $Gal(\mathbb{Q})$ is isomorphic to a well ...
- 5,929
4
votes
1
answer
626
views
Elliptic curve multiplication on the generic point
Given an elliptic curve $E$ there is the multiplication by $n$ map $[n]: E \rightarrow E$.
If $K(E)$ is the fraction field then this map makes $K(E)$ a degree $n^2$ extension of itself.
My question ...
- 43
7
votes
1
answer
617
views
l-adic Turrittin
What would be an l-adic analogue of the Turrittin-Levelt decomposition theorem?
Turrittin-Levelt is a structure theorem of meromorphic connections on a complex curve in the formal neighbourhood of a ...
6
votes
0
answers
295
views
Does a lower bound for models of finite group schemes exist?
Let $R$ be a discrete valuation ring (as beautiful as you like) and set $K:=Frac(R)$. Let $G_K$ be a finite $K$-group scheme, $G_1$ and $G_2$ two affine and flat models of $G_K$ of finite type, i.e. ...
- 73
2
votes
2
answers
666
views
Is the integrality of the zeta function easy?
I'm trying to get the gist of the proof of the Weil conjectures. Let $X$ be a variety over $\mathbb{F}_{p^n}$. A priori $Z(X,t)\in \mathbb{Q}[[t]]$. Due to the Grothendieck-Lefschetz fixed point ...
- 5,929
7
votes
1
answer
1k
views
References for bad reduction of Jacobians of modular curves?
Hi,
Where can I learn about the reduction of the Jacobians of modular curves
such as X_0(N) and X_1(N) at primes p dividing N?
Thanks!
- 2,842
1
vote
0
answers
429
views
Witt rings and prime number generator?
Let $p$ be a fixed prime number. We define the ring of Witt vectors $W(R)$ for any commutative ring $R$ as follows:
For every ring morphism $R \rightarrow R'$ the induced morphism $W(R) \rightarrow ...
- 397
12
votes
1
answer
518
views
Are there nonobvious cases where equations have finitely many algebraic integer solutions?
Let $X$ be a scheme of finite type over $\mathbb{Z}$. Let $R$ be the ring of algebraic integers. My intuition is that $X(R)$ is practically always infinite.
More specifically, suppose that $X$ is ...
- 156k
7
votes
3
answers
1k
views
Simple basis for Barnes-Wall lattices in dimension `$2^n$`
I'm searching for a "simple" description of the basis of the Barnes-Wall lattices
in (real) dimension $2^n$, if possible in a basis of minimal vectors, so that I can
do some calculations.
Can ...
26
votes
2
answers
4k
views
A route towards understanding Shimura varieties?
I'm in the embarrassing situation that I want to ask a question that
was already asked, but (for complicated reasons) never answered. I'd
like to try with a blank slate.
Shimura varieties show ...
- 445
6
votes
0
answers
462
views
Semistable reduction theorem over higher dimensional schemes
Let $k$ be a field, $S/k$ a smooth variety with function field $K$ and $U$ a nonempty open subscheme of $S$. For every finite separable extension $E/K$ we denote by $S^E$ (resp. $U^E$) the ...
- 1,673
3
votes
0
answers
204
views
Computing Elliptic Curves of Conductor Divisible by a Large Prime Factor
A little while ago, I came across a paper (or slides from a talk or something) that seemed to suggest that the modular symbol method for computing elliptic curves over $\mathbf{Q}$ of prescribed ...
- 309
12
votes
2
answers
430
views
Records for low-height points on elliptic curves over number fields
Elkies maintains a list of nontorsion points of low height on elliptic curves over Q; does anyone know of anything similar for curves over number fields?
Everest and Ward give examples of points of ...
- 873
15
votes
1
answer
769
views
Crystalline realization of mixed Tate motives
Deligne and Goncharov, in their article of 2005, mention that the crystalline realization functor has yet to be worked out. What's the current state of the literature on this? And how big of an issue ...
- 313
9
votes
3
answers
863
views
Is there a connected non-affine scheme $S$ such that it is the union of rings of integers of number fields?
I was woolgathering about the notion of a scheme, and it occurred to me that I know of no non-affine scheme $S$ that is the union of $Spec(O_K)$'s of some number field $K$ (I allow $K$ to vary - so ...
- 6,348
7
votes
1
answer
1k
views
Gauss Theorem and Weil Conjectures for elliptic curves
It is known (by Gauss) that for a prime $p \equiv 1 \pmod 3$ there is a "unique" writing of $4p=A^2+27B^2$ where $A=1+p-M_p$ and $M_p$ is the number of solutions of $X^3+Y^3+Z^3=0$ in the projective ...
- 95
2
votes
3
answers
1k
views
Finiteness of étale Cohomology Groups
Mr. Milne, in "Étale Cohomology", gives the following proposition (p.224, Corollary VI.2.8):
Proposition: Let $F$ a constructible sheaf on $X_{et}$, the small étale site of $X$, $X$ proper over a ...
- 23
14
votes
3
answers
3k
views
Does isomorphic generic fibre imply isomorphic special fibre for smooth morphisms?
Let $X$ and $Y$ be regular integral Noetherian schemes. Assume that $X$ and $Y$ are smooth and proper over a base scheme $S=Spec R$, where $R$ is a discrete valuation ring.
If $X$ and $Y$ have ...
- 21.4k
0
votes
1
answer
249
views
-1
votes
1
answer
802
views
Genus of algebraic curves with unknown degree
I am not sure if this is a valid question but posting any way:
Say I am over $\mathbb{F}_{p}$ for a prime $p$.
I have a curve of form $x^{2} = f(y)$ where $f(y)$ has an unknown form (and hence ...
10
votes
1
answer
2k
views
How does the conjectural Langlands group fit into the Tannakian point of view?
I've read that one way to formulate the Langlands program is the following:
Let $\mathcal{L}_ {\mathbb{Q}}$ be the conjectural Langlands group. Then the category of semi-simple (continuous) ...
18
votes
1
answer
1k
views
Torsion points of abelian varieties in the perfect closure of a function field
The following is a problem, which was recently brought to my attention by H. Esnault and A. Langer.
Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ...
- 3,076
12
votes
3
answers
1k
views
Sequences of Squares with all square differences
Background
The following question was first asked by Alex Rice, who was thinking about small subsets $A\subset [1,\ldots , N]$ with lots of square differences. Certainly for any set $A$ the maximum ...
- 3,625
7
votes
2
answers
2k
views
What is the relationship between the finiteness of the Tate-Shafarevich group and the Tate conjectures?
(I asked this on math-stackexchange, but it seems more appropriate to this forum, so I took it off from there and am posting it here)
After the great answer I got for my previous question about the ...
- 71
14
votes
2
answers
2k
views
How would a motivic proof of the Riemann hypothesis over finite fields go?
It is well known that Grothendieck had a different idea than Deligne about how one should go about proving the Riemann hypothesis for finite fields. However, since Grothendieck's desired proof never ...
- 6,348
3
votes
1
answer
777
views
Is the "L-function of the complex cohomology" of a motive equal to the L-function of its l-adic realization?
Let's say I have a motive in $\mathcal{M}_{num}(K)$ ($K$ a number field). For each prime $l$ there is a realization of this motive in terms of etale cohomology with coefficients in $\mathbb{Q}_l$. ...
- 6,348
3
votes
0
answers
365
views
Is there a notion of a zeta function of a morphism?
The Hasse-Weil zeta function is defined only for arithmetic schemes. By an arithmetic scheme I will mean a scheme $X$ together with a morphism of finite type $X\rightarrow S$, where $S$ is an affine ...
- 6,348
2
votes
2
answers
606
views
Axioms for zeta functions
The Selberg class is an axiomatization of arithmetically significant zeta functions (a.k.a. L-functions) by a few analytic properties (functional equation etc.) However there do exist other zeta ...
- 333
12
votes
9
answers
6k
views
Proofs of Mordell-Weil theorem
I would like to ask if there exist pedagogical expositions of the Mordell-Weil theorem (wikipedia). What parts of number theory (algebraic geometry) one should better learn first before starting to ...
- 14.3k
24
votes
1
answer
3k
views
What is the precise relationship between Langlands and Tannakian formalism?
As anyone who's been reading the forums closely can see, I've been averaging a question a day about Tannakian formalism for the past few days. It's quite an interesting concept!
In any case, I wish ...
- 6,348
1
vote
1
answer
291
views
is a closed subscheme of the projective line closed under the action of Gal(Qbar/Q)
Let $S$ be a non-empty closed subscheme of $P^1_K$, where $K$ is a number field. Assume that the cardinality of $S$ is finite.
Is $S$ closed under the action of the absolute Galois group of the ...
- 19
6
votes
2
answers
532
views
Expressing Galois actions on fundamental groups explicitly
Let $X$ be some variety over $\mathbb{Q}$, and let $\pi_1(X\times_{\mathbb{Q}}\mathbb{C},x)$ denote its (topological) fundamental group. As is well known $Gal(\mathbb{Q})$ acts on this fundamental ...
- 5,929
4
votes
2
answers
1k
views
on the Zeroes of Hasse -weil L-function
my question is that
already we know that the Birch and Swinnerton Dyer conjecture ,formally conjectures that the Hasse-weil L-function should have a zero at $s=1$ when curves have infinitely many ...
Trust God
6
votes
1
answer
3k
views
Intuition behind the Tamagawa numbers
i have read many books concerning the definition of tamagawa numbers ,but none of the books explained an intuition behind the concept ,
i mean what could be the intuitive definition of tamagawa number ...
Trust God