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Abstracts for Number theory lunch seminar

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The Oort Conjecture on Lifting Covers of Curves

Posted in
Speaker: 
Florian Pop
Affiliation: 
UPenn
Date: 
Wed, 2012-05-30 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

I plane to give a sketch of a proof of Oort's conjecture on lifting covers of curves from
positive characteristic to characteristic zero. The main ingredients in the proof is a very
recent special but critical case of the conjecture proved by Obus--Wewers and a deformation
argument in characteristic p.
 

Arithmeticity for periods of automorphic forms

Posted in
Speaker: 
A. Raghuram
Affiliation: 
IISER, Maharashtra, India
Date: 
Wed, 2012-06-06 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

An automorphic form f on a group G is said to to be distinguished with respect to a subgroup H if the integral of f along H is nonzero. Such period integrals are related to (non)vanishing of interesting L-values and also to Langlands functoriality.

New estimates in Vinogradov's mean value theorem

Posted in
Speaker: 
Kevin Ford
Affiliation: 
University of Illinois at Urbana-Champaign
Date: 
Mon, 2012-06-11 11:15 - 12:15
Parent event: 
Number theory lunch seminar

We discuss new estimates for the Vinogradov mean value, which are based on the Wooley's new method of "efficient congruencing". The new bounds are best possible in a certain range of the parameters, and improve upon existing bounds in another ranges. Applications are given to Waring's problem and other Diophantine problems. This is joint work with Trevor Wooley.

Noncommutative Iwasawa theory for Hida families

Posted in
Speaker: 
Ramdorai Sujatha
Affiliation: 
Tata Institute of Fundamental Research , Mumbai
Date: 
Fri, 2012-06-15 15:30 - 16:30
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

We shall study the variation of Iwasawa theoretic invariants for the dual Selmer groups of representations in a Hida family. This is joint work with Sudhanshu Shekhar.

Artin's Primitive Root Conjecture and the Euclidean Algorithm

Posted in
Speaker: 
Kathleen Petersen
Affiliation: 
Florida St. U./MPI
Date: 
Wed, 2012-06-27 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Artin's famous primitive root conjecture states that if n is an integer
other than -1 or a square, then there are infinitely many primes p such
that n is a primitive root modulo p. I will discuss a number field version
of this conjecture and its connection to the following Euclidean algorithm
problem. Let O be the ring of integers of a number field K. It is
well-known that if O is a Euclidean domain, then O is a unique
factorization domain. With the exception of the imaginary quadratic number

Arithmetic PDEs

Posted in
Speaker: 
Alexander Buium
Affiliation: 
U. of New Mexico/MPI
Date: 
Wed, 2012-07-04 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

We develop an arithmetic analogue of linear partial differential equations in 2 independent  "space-time" variables.
The spatial derivative is a Fermat quotient operator while the time derivative is a usual derivation. This allows
one to "flow" points in algebraic groups with coordinates in rings with arithmetic flavor. In particular we show
that elliptic curves  possess certain canonical ``arithmetic flows" which are analogous to the convection, heat,
and wave equations. Canonical convection and heat (but no wave) equations also exist on  modular curves;

Computing coefficients of modular forms

Posted in
Speaker: 
Peter Bruin
Affiliation: 
z.Z. MPI
Date: 
Wed, 2012-07-11 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

We consider normalised Hecke eigenforms of level n and weight k.  In
recent years, Edixhoven, Couveignes et al. (for n = 1) and the speaker
(generalisation to n > 1) developed an algorithm that, given such an f
and a positive integer m in factored form, computes the m-th coefficient
of the q-expansion of f, and whose running time is polynomial in n, k
and log m under the generalised Riemann hypothesis.  I will explain this
algorithm and the fundamental idea behind it, namely the computation of

Good Reduction of Three-Point Covers

Posted in
Speaker: 
Andrew Obus
Affiliation: 
MPI
Date: 
Tue, 2012-07-17 11:15 - 12:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

We study Galois covers of the projective line branched at three


points with Galois group G.  When such a cover is defined over a p-adic


field, it is known to have potentially good reduction to characteristic p


if p does not divide the order of G.  We give a sufficient criterion for


good reduction, even when p does divide the order of G, so long as the


p-Sylow subgroup of G is cyclic and the absolute ramification index of a


field of definition of the cover is small enough.  This extends work of

On integral well-rounded lattices

Posted in
Speaker: 
Lenny Fukshansky
Affiliation: 
Claremont McKenna College/MPI
Date: 
Wed, 2012-07-25 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

A lattice in R^n is called well-rounded, abbreviated WR, if it contains n linearly independent shortest vectors. Such lattices have many symmetries and play a central role in discrete optimization, extremal lattice theory, and related number theoretic problems. WR lattices corresponding to integral quadratic forms are of specific interest in the arithmetic context. In this talk, I will discuss some results on distribution properties of integral WR lattices, mostly concentrating on the planar case.

Modularity of M24-Twisted Siegel Product Expansions

Posted in
Speaker: 
Martin Raum
Affiliation: 
MPI
Date: 
Wed, 2012-08-29 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

One can attach Siegel product expansions of degree $2$ to elliptic genera of symmetric powers of K3 surfaces.

 

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