Unit root L-functions coming from families of exponential sums

Speaker: 

Douglass Haessig

Institution: 

University of Rochester

Time: 

Tuesday, March 10, 2015 - 2:00pm to 3:00pm

Host: 

Location: 

RH340P

Motivated from his p-adic study of the variation of the zeta function as
the variety moves through a family, Dwork conjectured that a new type of
L-function, the so-called unit root L-function, was always p-adic
meromorphic. In the late 1990s, Wan proved this using the theory of
sigma-modules, demonstrating that unit root L-functions have structure.
Little more is known.

This talk is concerned with unit root L-functions coming from families of
exponential sums. In this case, we demonstrate that Wan's theory may be
used to extend Dwork's theory -- including p-adic cohomology -- to these
L-functions. To illustrate the technique, the unit root L-function of the
Kloosterman family is studied in depth.

Arcs in the Projective Plane

Speaker: 

Nathan Kaplan

Institution: 

Yale University

Time: 

Tuesday, May 12, 2015 - 2:00pm to 3:00pm

Location: 

RH 340P

An arc in the projective plane over a finite field Fq is a collection of points, no three of which lie on a line.  Segre’s theorem tells us that the largest size of an arc is q+1 when q is odd and q+2 when q in even.  Moreover, it classifies these maximal arcs when q is odd, stating that every such arc is the set of rational points of a smooth conic.  

We will give an overview of problems about arcs in the plane and in higher dimensional projective spaces.  Our goal will be to use algebraic techniques to try to understand these extremal combinatorial configurations.  We will also see connections to special families of error-correcting codes and to modular forms.

 

Galois groups of Mori polynomials, semistable curves and monodromy

Speaker: 

Yuri G. Zarhin

Institution: 

Pennsylvania State University

Time: 

Tuesday, May 5, 2015 - 2:00pm

Host: 

Location: 

RH 340P

We study the monodromy of a certain class of semistable hyperelliptic curves over the rationals that was introduced by Shigefumi Mori forty years ago (before his Minimal Model Program). Using ideas of Chris Hall, we prove that the corresponding $\ell$-adic monodromy groups are (almost) ``as large as possible". We also discuss an explicit construction of two-dimensional families of hyperelliptic curves over an arbitrary global field with big monodromy.

On the strong multiplicity one for the Selberg class

Speaker: 

Haseo Ki

Institution: 

Yonsei University, Korea

Time: 

Tuesday, February 3, 2015 - 2:00pm to 3:00pm

Host: 

Location: 

RH340P

The strong multiplicity one in automorphic representation theory says that if two
automorphic cuspidal irreducible representations on $\text{GL}_n$ have isomorphic
local components for all but a finite number of places, then they are isomorphic. As
the analog of this, the strong multiplicity one for the Selberg class conjectures
that for functions $F$ and $G$ with $F(s) = \sum_{n=1}^\infty a_F(n)n^{-s}$ and
$G(s) = \sum_{n=1}^\infty a_G(n)n^{-s}$ in this class, if $a_F(p)=a_G(p)$ for all
but finitely many primes $p$, then $F=G$. In this article, we prove this
conjecture.

Eigencurve over the boundary of the weight space

Speaker: 

Liang Xiao

Institution: 

University of Connecticut

Time: 

Wednesday, January 7, 2015 - 1:00pm

Host: 

Location: 

RH340N

Eigencurve was introduced by Coleman and Mazur to parametrize
modular forms varying p-adically. It is a rigid analytic curve such that
each point corresponds to an overconvegent eigenform. In this talk, we
discuss a result on the geometry of the eigencurve: over the boundary
annuli of the weight space, the eigencurve breaks up into infinite disjoint
union of connected components and the weight map is finite and flat on each
component. This was first observed by Buzzard and Kilford by an explicit
computation in the case of p=2 and tame level 1. We will explain a
generalization to the definite quaternion case with no restriction on p and
the tame level. This is a joint work with Ruochuan Liu and Daqing Wan,
based on an idea of Robert Coleman.

The $\mu$-ordinary Hasse invariant and applications

Speaker: 

Marc-Hubert Nicole

Institution: 

Institut mathématique de Luminy / UCLA

Time: 

Tuesday, October 28, 2014 - 2:00pm to 3:00pm

Location: 

RH340P

Let $p>2$ be a prime number. The classical Hasse invariant is a modular form modulo p that vanishes on the supersingular points of a modular curve. Its non-zero locus is called the ordinary locus. While the Hasse invariant generalizes easily to moduli spaces of abelian varieties with additional structures, it happens often that the generalized ordinary locus is empty, and therefore the Hasse invariant is then tautologically trivial. We present an elementary and natural generalization of the Hasse invariant, which is always non-trivial, and which enjoys essentially all the same properties as the classical Hasse invariant. In particular, the usual applications generalize nicely, and we shall highlight the state-of-the-art in our talk.

Euler systems and the Birch--Swinnerton-Dyer conjecture

Speaker: 

Sarah Zerbes

Institution: 

University College London and MSRI

Time: 

Saturday, October 18, 2014 - 4:00pm to 5:00pm

Location: 

NS2 1201

The Birch--Swinnerton-Dyer conjecture is now a theorem, under some mild hypotheses, for elliptic curves over Q with analytic rank ≤ 1. One of the main ingredients in the proof is Kolyvagin's theory of Euler systems: compatible families of cohomology classes which can be seen as an "arithmetic avatar'' of an L-function. The existence of Euler systems in other settings would have similarly strong arithmetical applications, but only a small number of examples are known.

In this talk, I'll introduce Euler systems and their uses, and I'll describe the construction of a new Euler system, which is attached to the Rankin--Selberg convolution of two modular forms; this is joint work with Antonio Lei and David Loeffler. I'll also explain recent work with Loeffler and Guido Kings where we prove an explicit reciprocity law for this Euler system, and use this to prove cases of the BSD conjecture and the finiteness of Tate--Shafarevich groups.

On a problem related to the ABC conjecture

Speaker: 

Daniel Kane

Institution: 

UCSD

Time: 

Saturday, October 18, 2014 - 2:30pm to 3:30pm

Location: 

NS2 1201

The ABC Conjecture, roughly stated says that the equation A+B+C=0 has no solutions for relatively prime, highly divisible integers A, B, and C. If the divisibility criteria are relaxed, then solutions exist and a conjecture of Mazur predicts the density of such solutions. We discuss techniques for proving this conjecture for certain ranges of parameters.

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