User tzanko matev - MathOverflow

Answer by Tzanko Matev for Motivic generalisation of Neron-Ogg-Shaferevich criterion

2012-12-03T12:53:34Z

Unfortunately I don't know much about motives in genereal, but this might be relevant to your question. One result of my thesis, that I am currently writing, is to prove Neron-Ogg-Shafarevich for 1-motives. The proof is not particularly difficult and it ultimately reduces to the corresponding results for the components of the 1-motive. I will describe below what good reduction means in this particular case.

A 1-motive $M = [u\colon Y\to G]$ over a scheme $S$ consists of a group scheme $Y$, which is locally etale isomorphic to $\mathbb{Z}^r$, a group scheme $G$ which is an extension of an abelian scheme $A$ by a torus $T$ and a homomorphism $u\colon Y\to G$. If $S$ is the spectrum of a field $K$, this means that $Y$ is a free finitely-generated $\mathbb{Z}$-module with a continuous action of the absolute Galois group $\Gamma_K$ and that $u$ is a $\Gamma_K$-equivariant homomorphism $u\colon Y\to G(\bar K)$.

If $R$ is a complete discrete valuation ring with a fraction field $K$ we say that a 1-motive $M$ over $K$ has good reduction if there exists a 1-motive $\widetilde{M}$ over $R$ whose generic fiber is isomorphic to $M$. This is equivalent to the following:

$G$ has good reduction $\widetilde{G}$ over $R$, which is equivalent to saying that both $A$ and $T$ have good reduction;
The action of $\Gamma_K$ on $Y$ is unramified;
The image of $u(Y)$ is contained in the set of those points in $G(K')$ which can be reduced, where $K'/K$ is some finite field extension. Equivalently, $u(Y)$ is contained in the maximal compact subgroup of $G(K')$;

With this definition, the criterion of Neron-Ogg-Shafarevich is as follows: Let $l,p$ be primes, with $l\neq p$. A 1-motive $M/\mathbb{Q}$ has good reduction mod p if and only if the Tate module $T_l(M)$ is unramified at $p$. For general number fields replace $p$ by a prime ideal.

If you want to learn more about reduction of 1-motives you can look at M. Raynaud's paper 1-Motifs et Monodromie Géométrique.

Is the n-torsion of an extension of an abelian variety by a torus, finite and flat?

2012-11-17T19:32:31Z

I am looking for reference or hints how to prove the following result.

Let $G$ be a commutative $S$-group scheme which is the extension of an abelian scheme $A$ by a torus $T$. Then the n-torsion $G[n]$ is a finite flat $S$-group scheme.

Specifically, I have difficulties in showing that $G[n]$ is finite. For a general semi-abelian scheme we know that it is quasi-finite and flat, but not necessarily finite (see e.g. the book Neron Models, Lemma 7.3/2).

Thanks in advance,

Reference request for Cartier Duality of algebraic tori

2012-05-30T11:49:52Z

Hi,

I need a reference for the following result:

Let $S$ be a scheme and let $X$ be an algebraic torus over $S$. Then the functor $F_X :S'\mapsto Hom_{S'}(X\times S',\mathbb{G}_M\times S')$ is representable by an $S$- group scheme $Y$ which is locally etale isomorphic to $\mathbb{Z}^n$. Furthermore, the similarly defined functor $F_Y$ is represented by $X$.

This result can probably be found somewhere in SGA7. However I have very basic understanding of French and the style in which SGA is written makes it practically impossible for me to find the exact reference. Are there any other references (in English or in simpler French) which one could use? If not, do you know exactly which statements in SGA7 imply the Cartier duality theorem?

Thanks in advance, your help will be very much appreciated.

Detecting linear dependence on multiplicative groups

2012-01-20T17:59:25Z

Let G = $\mathbb{G}_m^2/\mathbb{Q}$ and let $\Gamma \subseteq G(\mathbb{Q})$ be a free abelian group of rank 2. Assume that the set of primes $p$ for which $\Gamma \mod p$ is cyclic has positive density. Does this imply that $\Gamma$ is contained in a proper algebraic subgroup of G?

Torsion points on commutative $Z_p$-group schemes

2011-07-08T12:54:27Z

Hi,

Let G be a smooth commutative $\mathbb{Z}_p$-group scheme of finite type and let $G_0$ be the $\mathbb{Q}_p$-fiber. We have an embedding $G(\mathbb{Z}_p)\subseteq G_0(\mathbb{Q}_p)$. My question is does every torsion point in $G_0(\mathbb{Q}_p)$ come from a torsion point in $G(\mathbb{Z}_p)$? I am mostly interested in the prime-to-p part of the torsion group. I think that the answer to this quesiton is yes, but I can't figure out how to prove it. Any ideas or references would be greatly appreciated.

Thanks in advance!

Estimating the size of reduction of rational points on $\mathbb{G}_m^2$

2011-05-30T15:19:32Z

Hi,

Let $\Gamma$ be a free subgroup of rank 2 in $\mathbb{G}_m^2(\mathbb{Q})$. For all but finitely many primes p we can reduce $\Gamma$ modulo p. Let $S$ be the of primes for which $\Gamma$ does not reduce modulo p, and for any $p$ not in $S$, let $\gamma_p$ be the size of $\Gamma \mod p$. My question is what is known about the function

$f(x)= \sum_{p\not\in S,\ p\leq x}\frac{\log p }{\gamma_p}$

In particular what is the asymptotic behavior of $f$? Is the corresponding infinite series convergent whenever $\Gamma$ is not contained in an algebraic subgroup of $\mathbb{G}_m^2$? Do you know of any references that might be relevant to those questions?

Thanks in advance,

How big is the Fourier transform of the log of a polynomial over the p-adic numbers

2010-10-20T14:34:00Z

Let $f(z_1,\dots,z_n)$ be a polynomial with $p$-adic coefficients, and let $g(z):=log\lvert f(z) \rvert$. If $\xi$ is a complex character of $\mathbb{Z}_p^n$ there exists a number $v=v(\xi)$ such that $\xi$ is trivial on $p^v\mathbb{Z}_p^n$, but not on $p^{v-1}\mathbb{Z}_p^n$. The question is:

Find an upper bound of $\lvert\hat{g}(\xi)\rvert$ in terms of $v(\xi)$.

If $f$ is a polynomial in one variable, then I get

$\lvert\hat{g}(\xi)\rvert \ll p^{-v(\xi)}$

The solution is quite simple: One solves the problem for f(z)=z, in which case $\hat g(\xi)$ can be explicitly computed. The general case then follows easily, because we can factor f into linear factors times a non-vanishing function. This solution, however, doesn't work in higher dimensions.

My guess is that the same bound holds for higher dimensions, but I still haven't managed to show that. Has this problem already been solved? Do you know of any books or papers that might be helpful? Any help would be appreciated. Thanks.

Counting points on lattices

2010-07-19T14:53:50Z

I expect that the following is a standard problem from analytic number theory, but I don't know where exactly to look for an answer.

Let f: ℤ^r→ H be a surjective homomorphism into a finite group. Let

$S(N)= \frac{1}{N^r}\#\{(x_1,\dots,x_r)\in \ker f\colon 0\leq x_i <N\}$ .

One expects that S(N) is roughly |H|^-1. My question is:

What is the best known estimate of the error term S(N)-|H|^-1 in terms of N and the structure of H? I am especially interested in the case, when H=(ℤ/pⁿℤ)^d, for some d<r.

To give some idea of what kind of results I am looking for, I will give the estimate, that I managed to find myself. If "e" is the exponent of the group H, and "h" is its size, then by estimating character sums one gets

$S(N) - h^{-1} \ll h^{-1}(\log \min\{h,N\})^r \max\{eN^{-1},e^rN^{-r}\}$ ,

where the implicit constant depends only on r. I think, that this can be improved at least when N is small with respect to e.

Comment by Tzanko Matev

2012-11-18T08:50:57Z

@Qing Liu: I am sorry if the question was not stated well. I know that for a general semi-abelian scheme what I ask is false. I am only interested in the case when the scheme is an extension of an abelian scheme by a torus.

Comment by Tzanko Matev

2012-11-18T08:43:16Z

@nosr Thanks a lot! I will accept the answer when I manage to verify it.

Comment by Tzanko Matev

2012-11-17T20:56:52Z

I have seen a generalization of this statement in several papers on 1-motives, however no proof or reference is given there. For example: Deligne [Hodge III, 10.1.10] or M. Raynaud, [1-Motifs et Monodromie Géométrique, 3.1]. This is why I think that it should be true.

Comment by Tzanko Matev

2012-11-17T20:48:33Z

The statement should be true for an arbitrary scheme but I would be happy with an answer for $S= Spec R$ when $R$ is the ring of integers of a finite $\mathbb{Q}_p$-extension.

Comment by Tzanko Matev

2012-05-31T08:54:25Z

That is exactly what I was looking for. Thanks a lot!

Comment by Tzanko Matev

2011-07-19T15:02:56Z

Thank you all for the comments. They were very helpful to clear up my confusions.

Comment by Tzanko Matev

2011-07-10T09:43:20Z

@Jason Starr: Thanks for the remark. I do require that $G$ is of finite type over $Z_p$. I fixed the question.

Comment by Tzanko Matev

2011-05-31T07:31:26Z

Yes, that is what I meant. I corrected the question to make the statement clearer. Thank you for for the remark.

Comment by Tzanko Matev

2011-03-28T16:28:28Z

There is a conjecture due to Bjorn Poonen (www-math.mit.edu/~poonen/papers/leopoldt.pdf) which implies that if $J$ is simple, then $r'=r$. So, if you believe it you will have to pick $J$ to be a product of two elliptic curves, with ranks 2 and 0 respectively.For any Jacobian of this type $r'$ will be 1.