January 22

Colloquium (DLS)

4:20 – 5:20 pm., PSA 309

Karl Rubin,

University of California, Irvine

Ranks of elliptic curves

Abstract. In this lecture, we will discuss elliptic curves and some of the fundamental questions about them. The rank of an elliptic curve is a measure of the number of solutions of the equation which defines the curve.  In recent years, there has been spectacular progress in the theory of elliptic curves, but the rank remains very mysterious. Even basic questions such as how to compute the rank, or whether the rank can be arbitrarily large, are not settled. We will survey what is known, as well as what is conjectured but not known about ranks of elliptic curves.

January 23

Number Theory Seminar

11:00 – 11:50 am., PSA 311

Karl Rubin,

University of California, Irvine

Refined class number formulas and Kolyvagin systems

Abstract. In this talk, we will discuss refined class number formulas conjectured by Gross and by Darmon. We will prove (a slight variant of) Darmon’s conjecture, using the theory of Kolyvagin systems. This is joint work with Barry Mazur.

February 6

Number Theory Seminar

11:00 – 11:50 am., PSA 311

Ahmed Matar

Euler characteristics and elliptic curves

Abstract. This talk will be a survey of some recent results concerning Euler characteristics of Selmer groups of elliptic curves over p-adic Lie extensions of number fields.

February 23

Number Theory Seminar

2:30 – 3:20 pm., PSA 109

Philip Kutzko,

University of Iowa

Smooth representations: an introduction to types

Abstract. The representation theory and character theory of finite groups form an important chapter in modern algebra. On the other hand the representation theory of topological groups – real Lie groups for example – is usually approached using analytic methods. (This is especially the case with character theory which, in this context, is subsumed into harmonic analysis.) Interestingly, there is a class of topological groups, those which are locally compact and totally disconnected, that comes up naturally in number theory. In the case that these groups are compact they are in fact profinite; that is, they are inverse limits of finite groups.  Because of this, it is possible to employ algebraic methods in studying the representation theory of these groups. In this talk, we will give an introduction to these ideas, provide some examples, and give a survey of recent results.

March 4

Number Theory Seminar

2:30 – 3:20 pm., PSA 109

François Charles,

Ecole Normale Superiéure

Zeroes of normal functions and étale cohomology

Abstract. Normal functions arise in the study of algebraic cycles on complex projective varieties, generalizing the usual Abel-Jacobi map for divisors on curves. They are holomorphic functions defined through Hodge theory, and as such, their algebraic behaviour is not well-understood yet.  In this talk, we use an analog of those defined through continuous étale cohomology to extract arithmetic information on the zero locus of normal functions. We will recall some of the key definitions to make the talk accessible to non-algebraic geometers.

March 20

Number Theory Seminar

11:00 – 11:50 am., PSA 307

Rachel Wallington

Number fields with solvable Galois groups and small Galois root discriminants 

Abstract. In a paper by Jones and Roberts, they pose the problem of identifying Galois number fields with root discriminant less than or equal to Omega where Omega is the Serre-Odlyzko bound. In this paper, we will identify some of these fields which have solvable Galois groups and discuss the methods used to find these fields.

March 20

Number Theory Seminar

3:30 – 4:20 pm., PSA 304

David Roberts,

University of Minnesota, Morris

Arboreal dessins d'enfants

Abstract. Dessins d’enfants (children’s drawings) are bipartite planar graphs.  They are in bijection with Möbius equivalence classes of rational functions in C(x) with critical values within {0,1,∞}. They form an elementary pathway into a deep area of arithmetic geometry.  The dessins corresponding to polynomials are exactly the trees.  In my talk I will explain how there is a natural larger class of dessins for which the same techniques go through.  These dessins correspond to appropriately weighted trees, and accordingly I call them arboreal. One aspect of the arboreal theory is that two partitions of a positive integer n with a total of v parts determine a number field (or exceptionally, a product of several number fields) of degree ≤ v–2. This number field can be wildly ramified only for primes p ≤ v–2.  It can be tamely ramified only for these primes and the primes v–2 < p ≤ n.  The main focus of the talk is to completely describe tame ramification at the latter primes by a combinatoric procedure that avoids calculating a defining polynomial.

April 22

Number Theory Seminar

11:00 – 11:50 am., PSA 304

Alejandra Alvarado

Arithmetic progressions on curves

Abstract. Click here to view.

April 22

Number Theory Seminar

2:30 – 3:20 pm., PSA 109

Gary Roth

Projective curves and their intersections

Abstract. We will discuss the projective plane, and projective curves, from two separate perspectives: homogeneous triples and the affine plane together with points at infinity. We will then explore the ways in which projective curves intersect based on the concept of intersection index. The talk will conclude with a proof of Pascal’s Theorem using Bezout’s Theorem and other ideas developed during the talk.

April 29

Number Theory Seminar

2:30 – 3:20 pm., PSA 109

Gary Roth

The group of rational points on cubics

Abstract. We will discuss a method for finding rational points on conics, and then use that method as a steppingstone towards developing a method for finding rational points on cubics. We will then develop explicit formulas for the group law of rational points on non-singular cubics that have been put into Weierstrass normal form.

September 24

Number Theory Seminar

2:30 – 3:20 pm., PSA 109

Ahmed Matar

Selmer groups and the Mazur control theorem, part I

Abstract. Let K be a number field and E/K an elliptic curve. Suppose L/K is a Z_p-extension. In this talk we discuss the following question: Is the rank of E(K_n) as K_n runs over finite subextensions of L/K bounded? Or ever better: Is E(L) finitely generated? Via the Mazur control theorem, we will see that the answer to this question is affirmative if the elliptic curve has certain properties. This will lead us to a discussion of the control theorem and several of its consequences. Some of the ideas used in its proof will be explained.

October 1

Number Theory Seminar

2:30 – 3:20 pm., PSA 109

Ahmed Matar

Selmer groups and the Mazur control theorem, part II

Abstract. Let K be a number field and E/K an elliptic curve. Suppose L/K is a Z_p-extension. In this talk we discuss the following question: Is the rank of E(K_n) as K_n runs over finite subextensions of L/K bounded? Or ever better: Is E(L) finitely generated? Via the Mazur control theorem, we will see that the answer to this question is affirmative if the elliptic curve has certain properties. This will lead us to a discussion of the control theorem and several of its consequences. Some of the ideas used in its proof will be explained.

November 19

Number Theory Seminar

2:30 – 3:20 pm., PSA 109

Alejandra Alvarado

y-Arithmetic progressions on elliptic curves

Abstract. Let E be an elliptic curve over the rationals with coordinates (x,y). Previous work on x-arithmetic progressions has shown that there exist infinite families of elliptic curves with length 8 x-arithmetic progressions. Examples have been found with x-arithmetic progressions of length 14. But not much work has been done on the length of y-arithmetic progressions on elliptic curves In this talk we will discuss arithmetic progressions in the y- coordinates of certain types of elliptic curves.

November 21

Number Theory Seminar

2:30 – 3:20 pm., PSA 109

Solomon Friedberg,

Boston College

Euler products and twisted Euler products

Abstract. Euler products with functional equation are at the heart of the Langlands program and are fundamental to modern number theory.  We introduce a new variation on the standard notion of Euler product, namely a twisted Euler product, and explain how twisted Euler products with functional equation arise naturally.  In particular, we exhibit examples of these objects, which are infinite sums of Gauss sums.  The series are given by attaching number-theoretic quantities to combinatorial objects that arise in representation theory.

December 3

Number Theory Seminar

2:30 – 3:20 pm., PSA 109

Chase Franks

Greenberg’s conjecture for Abelian number fields

Abstract. Click here to view.