On the Gross-Rubin-Stark conjecture

Speaker: 

Cristian Popescu

Institution: 

UCSD

Time: 

Tuesday, April 25, 2017 - 2:00pm to 3:00pm

Location: 

RH 340P

A special case of the GRS Conjecture predicts a surprising link between values of derivatives of p-adic and global L-functions. Recently, Dasgupta-Kakde-Ventullo have used Hida families of modular forms to make progress towards the proof of a rational form of this special case. In  this lecture I will report on an independent approach and progress  towards the integral GRS conjecture, building upon my  joint work with Greither in equivariant Iwasawa theory.

Slopes of modular forms and the ghost conjecture

Speaker: 

Liang Xiao

Institution: 

University of Connecticut

Time: 

Tuesday, April 18, 2017 - 2:00pm to 3:00pm

Location: 

RH 340P

The topic of this talk will be understanding the p-adic slopes of modular forms. Recently, Bergdall and Pollack, based on computer calculations, raised a very interesting conjecture on the slopes of overconvergent modular forms, which predicts that the Newton polygons of the characteristic power series of U_p are the same as the Newton polygons of another explicit characteristic power series, which they call ghost series. This conjecture would imply many well-known conjectures regarding slopes of modular forms, like Gouvea's conjecture, Gouvea-Mazur conjecture, and etc. The goal of our joint project is to prove this conjecture under some mild hypothesis, and to explore some further application.  I will report on the progress so far.

Slopes of L-functions of Z_p-covers of the projective line

Speaker: 

Michiel Kosters

Institution: 

UCI

Time: 

Tuesday, February 21, 2017 - 2:00pm to 3:00pm

Location: 

RH 340P

Let P: ... -> C_2 -> C_1 -> P^1 be a Z_p-cover of the projective line over a finite field of characteristic p which ramifies at exactly one rational point. In this talk, we study the p-adic Newton slopes of L-functions associated to characters of the Galois group of P. It turns out that for covers P such that the genus of C_n is a quadratic polynomial in p^n for n large, the Newton slopes are uniformly distributed in the interval [0,1]. Furthermore, for a large class of such covers P, these slopes behave in an even more regular way. This is joint work with Hui June Zhu.

Period and index of higher genus curves

Speaker: 

Shahed Sharif

Institution: 

Cal State University San Marcos

Time: 

Tuesday, May 2, 2017 - 2:00pm to 3:00pm

Location: 

RH 340P

The period and index of a curve C are two quantities which describe the failure of C to have rational points. The mismatch between the two is of interest for its impact on the Shafarevich-Tate group of the Jacobian of C. The period-index problem is to determine what values of period and index are possible for a given genus g. We will give a complete answer when g=1, and an almost complete answer when g ≥ 2.

A Family of p-Dimensional Lattices

Speaker: 

Carmelo Interlando

Institution: 

San Diego State University

Time: 

Tuesday, May 9, 2017 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

In this talk a lattice will mean a discrete subgroup Λ of n-dimensional Euclidean space; the sphere packing associated to Λ is the arrangement of congruent spheres of radius equal to one half the minimum distance of Λ and centered at the lattice points.  The main parameter under consideration will be the packing density of the arrangement of spheres.  With this in mind, a family of p-dimensional lattices will be constructed from submodules M of the ring of integers of a cyclic number filed of degree p, where p is an odd unramified prime in L/Q.  The density of the associated sphere packing will be determined in terms of the nonzero minimum of the trace form in and the discriminant of L.

Adelic points of elliptic curves

Speaker: 

Peter Stevenhagen

Institution: 

Universiteit Leiden

Time: 

Tuesday, January 17, 2017 - 2:00pm to 3:00pm

Location: 

RH 340P

We show how the Galois representation of an elliptic curve over a number field can be used to determine the structure of the (topological) group of adelic points  of that elliptic curve.

As a consequence, we find that for "almost all" elliptic curves over a number field K,  the adelic point group is a universal topological group depending only on the degree  of K. Still, we can construct infinitely many pairwise non-isomorphic elliptic curves  over K that have an adelic point group not isomorphic to this universal group.

This generalizes work of my student Athanasios Angelakis (PhD Leiden, 2015).

Higher moments of arithmetic functions in short intervals: a geometric perspective

Speaker: 

Vlad Matei

Institution: 

University of Wisconsin

Time: 

Tuesday, November 29, 2016 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

In joint work with Daniel Hast, we recast the paper of Jon Keating and Zeev Rudnick "The variance of the number of prime polynomials in short intervals and in residue classes" by studying the geometry of these short intervals through an associated highly singular variety. We manage to recover their results for a a general class of arithmetic functions up to a constant and also obtain information about the higher moments. Recently work of Brad Rodgers in "Arithmetic functions in short intervals and the symmetric group" gives new insight into the geometry of our variety.

Torsion subgroups of elliptic curves in elementary abelian 2-extensions

Speaker: 

Ozlem Ejder

Institution: 

USC

Time: 

Tuesday, November 22, 2016 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

Let E be an elliptic curve defined over Q. The torsion subgroup of E over the compositum of all quadratic extensions of Q was studied by Michael Laska, Martin Lorenz, and Yasutsugu Fujita. Laska and Lorenz described a list of 31 possible groups and Fujita proved that the list of 20 different groups is complete.

In this talk, we will generalize the results of Laska, Lorenz and Fujita to the elliptic curves defined over a quadratic cyclotomic field i.e. Q(i) and Q(\sqrt{-3}).

Dynamically distinguishing polynomials

Speaker: 

Derek Garton

Institution: 

Portland State University

Time: 

Tuesday, February 28, 2017 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

Given two polynomials with integer coefficients, for how many primes p do the polynomials induce nonisomorphic dynamical systems mod p? This question will lead us to the study of the statistics of wreath products, the Galois theory of dynatomic polynomials, and other topics. This work is joint with Andrew Bridy.

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