February 17

Number Theory Seminar

2:30 - 3:20 pm.,

PSA 108

Paulo Lima-Filho,

Texas A & M University

Classifying arithmetic toric varieties

Abstract. Given an arbitrary field k and a conbinatorial fan F, we study the classification of the various “toric K-forms” of the toric variety X_K,F where K is a separable algebraic closure of k containing F. This classification generalizes the work of Delaunay on “real toric varieties” and has a particularly simple description in the case of complete non-singular toric surtaces. We show how to use the Cox construction to perform explicit calculations and make a few applications. This is joint work with Javier Elizondo, Frank Sottile and Zach Teitler.

February 24

Number Theory Seminar

2:30 - 3:20 pm.,

PSA 108

John Jones,

SoMSS

Discriminant bounds and extensions unramified away from 2

Abstract. TBA.

September 16

Number Theory Seminar

2:00 - 3:20 pm.,

PSA 118

Andrew Bremner,

SoMSS

Rational points on y² = xⁿ + k

Abstract. Nonsingular rational curves of genus greater than 1 have only finitely many rational points. The set of points on a few specific curves has been explicitly determined (eg xⁿ+yⁿ=1), but in general this in itself is a very difficult problem. There seem to be few examples in the literature of lower bounds for the number of points on curves of high genus. Here, we give some explicit parameterizations of points on the title curve and related curves (joint work with Maciej Ulas).

September 30

Number Theory Seminar

2:00 - 3:20 pm.,

PSA 118

Chase Franks,

SoMSS

A homological approach to determining Iwasawa modules up to isomorphism

Abstract. Click here to view.

October 1

Colloquium

4:30 - 5:30 pm.,

PSA 118

Michael Spieß,

Universität Bielefeld

Special zeros of p-adic L-functions

Abstract. Zeta functions (like the Riemann Zeta function), or more generally L-functions of algebraic varieties, are of great importance to number theory. For example their values at integral points often reveal deep relations between different types of arithmetic invariants. In this talk we introduce p-adic L-functions (in particular of elliptic curves). These are functions of a p-adic variable. They are of similar significance to number theory as classical L-functions. In the last part of my talk I shall explain the phenomena of trivial zeros of p-adic L-functions of elliptic curves.

October 7

Number Theory Seminar

2:00 - 3:20 pm.,

PSA 118

Michael Spieß,

Universität Bielefeld

p-adic periods and p-adic L-functions of Hilbert modular forms

Abstract. We will construct p-adic periods (i.e., L-invariants) and p-adic L-functions of Hilbert modular forms. The first non-trivial Taylor coefficient of the latter at s=1 can be expressed in terms of L-invariants (provided that the complex L-function does not vanish at s=1). This is the so-called weak exceptional zero conjecture for Hilbert modular forms.

October 21

Number Theory Seminar

2:00 - 3:20 pm.,

PSA 118

N. Saradha,

Tata Institute of Fundamental Research, Mumbai

Perfect powers in arithmetic progression

Abstract. Click here to view.

October 28

Number Theory Seminar

2:00 - 3:20 pm.,

PSA 118

Chad Awtrey,

SoMSS

Computing Galois groups of dodecic local fields

Abstract. Given a prime number p and an irreducible degree-12 polynomial f(X) in Z_p[X], we discuss an original algorithm that identifies the Galois group of f from among the 301 transitive subgroups of S₁₂.

November 4

Number Theory Seminar

2:00 - 3:20 pm.,

PSA 118

Ahmed Matar,

SoMSS

Selmer groups and class field towers

Abstract. We will discuss a control theorem for the p-primary Selmer group of an Abelian variety with respect to an infinite p-Hilbert class field tower of a number field.

November 17

Colloquium

4:30 - 5:30 pm.,

PSA 104

Romyar Sharifi,

University of Arizona

Galois groups and ramification

Abstract. In number theory, one is most concerned with the arithmetic of finite extensions of the rational numbers, known as number fields.  The arithmetic of such fields is reflected in the Galois groups of its field extensions of finite degree, these groups acting as symmetries of roots of polynomials with coefficients in the ground field.  One is therefore interested in knowing which isomorphism classes of Galois groups can occur for such extensions.  Much of the finer structure of the arithmetic of a number field can be found in objects such as its class group, the quotient group of nonzero ideals modulo principal ideals of its ring of integers.  In understanding this, the decomposition of prime ideals of this integer ring in the ring of integers of extension fields comes into play.  For instance, a prime is said to ramify if some prime of the larger field containing it occurs to power at least two in the decomposition, this being the arithmetic analogue of branching in covering spaces in topology.  One might then ask which Galois groups can occur as field extensions that are ramified at most at these primes.  We will discuss a sampling of what is known in answer to this remarkably deep question, as well as other questions to which this leads.

November 18

Number Theory Seminar

2:00 - 3:20 pm.,

PSA 118

Romyar Sharifi,

University of Arizona

Reciprocity maps and their applications

Abstract. We will discuss the construction of so-called S-reciprocity maps attached to a number field and a set of primes. These are a direct analogue of the reciprocity maps occurring in local class field theory. We will consider their Iwasawa-theoretic analogues and use in the study of Iwasawa modules over certain noncommutative extensions, and we will discuss connections with the Selmer groups of certain abstract reducible representations. We may use this to hint at a conjecture relating the values of these reciprocity maps for cyclotomic fields on cyclotomic p-units and p-adic L-values of newforms congruent to Eisenstein series at primes over an odd prime p.

December 2

Number Theory Seminar

2:00 - 3:20 pm.,

PSA 118

Deepa Krishnamurthi,

SoMSS

Elliptic curves

Abstract. In this talk, we will see some historical background of elliptic curves, group structure on them, some results related to their group structure and some applications of elliptic curves in various fields.