Number Theory Seminar at Johns Hopkins University

Abstract: According to the Harish-Chandra philosophy, cuspidal representations are the basic building blocks in the representation theory of finite reductive groups. Similarly for supercuspidal representations of $p$-adic groups. I will prove a folklore theorem that says that all such groups have such representations, but I will go farther. Self-dual representations play a special role in the study of parabolic induction, so it is of interest to know whether self-dual (super)cuspidal representations exist. With a few exceptions involving some small fields, I will show precisely when a finite reductive group has irreducible cuspidal representations that are self-dual, of Deligne-Lusztig type, or both. Then I will look at implications for the existence of irreducible, self-dual supercuspidal representations of p-adic groups. This is joint work with Manish Mishra.

Abstract: The Tate conjecture and its applications play important role in arithmetic geometry and number theory. This conjecture is known for curves, and its divisor version is known for abelian varieties, K3 surfaces and, more generally, for general genus one surfaces. In this talk, we will look at this conjecture from the Galois representation aspect. In particular, for certain elliptic surfaces over P^1 over Q which have ``small'' self-dual transcendental parts, we can make use of the known cases of Fontaine-Mazur conjecture to prove their Tate conjecture. This work provides examples to support the the Tate conjecture for surface with genus greater than one. This is a joint work with Xiyuan Wang.

Abstract: For rings, there is a natural bijection between (two-sided) ideals and congruences, but for semirings (where we do not require additive inverses), this correspondence breaks. Many basic notions concerning ideals in rings thus split into two different notions for semirings, one for ideals and the other for congruences. Currently there is interest in extensions of scheme theory to commutative semirings, motivated especially by ideas concerning the "field of one element" in arithmetic geometry. To do so, we must decide if we would rather talk about prime ideals or about prime congruences; if the latter then we must decide what we mean by a prime congruence. Of the several definitions of a prime congruence put forward, the most promising is due to Joó and Mincheva.

In this talk, we will see that the congruences in a semiring R are themselves special ideals in a different semiring R_{fd}, the semiring of formal differences for R. Then Joó and Mincheva's prime congruences are precisely those congruences that are prime ideals in R_{fd}. We will also examine Joó and Mincheva's natural arithmetic of congruences in comparison with classical arithmetic of ideals.

The natural numbers N is already a rich example for this story. We will end with an alternative multiplication law for congruences on N that better extends the arithmetic of ideals in Z than Joó and Mincheva's. In particular, this alternative multiplication law distributes over sum of congruences and admits unique factorization of congruences into prime congruences.

Abstract: Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work with Chandrashekhar Khare and Jack Thorne.