G-BRL Number Theory Seminar at CNU

Abstract:  Montgomery in 1973 introduced the Pair Correlation Conjecture (PCC) for zeros of the Riemann zeta-function. He also conjectured that asymptotically 100\% of the zeros are simple. His reasoning to support these two conjectures used the Riemann Hypothesis (RH). Building on Montgomery's approach, Gallagher and Mueller proved in 1978 that PCC under RH implies that 100\% of the zeros are simple. Actually, the method of Gallagher and Mueller does not depend on RH, and thus Montgomery's second simplicity conjecture follows unconditionally from his PCC conjecture. We clarify this result by explicitly not assuming RH and considering PCC as a conjecture only concerning the vertical distribution of zeros. We then show that, for the first time, PCC can also be used to obtain information on the horizontal distribution of zeros. Using Gallagher and Mueller's method and a new idea concerning \lq\lq horizontal multiplicity", we use PCC to prove that asymptotically 100\% of the zeros are not only simple but also on the critical line. We also formulate an appropriate form of the Alternative Hypothesis (AH), which determines a different PCC, and, using the same method as above, prove that asymptotically, 100\% of the zeros are both simple and on the critical line. As in our previous paper, we do not assume RH.

Abstract:  We compute the Jordan constants of the multiplicative subgroups of the endomorphism algebras of simple abelian surfaces over fields of positive characteristic, with the aid of a similar computation on the Jordan constants of some arithmetic objects. As a recent update, we also briefly record a similar result on the case of simple abelian fourfolds over finite fields

Abstract:  Let p(n) be the partition function. Erdős conjectured that for each integer m, there exists a positive integer n such that p(n) = 0 (mod m). Later, Newman conjectured that for each integer m and r, there are infinitely many positive integers n such that p(n) = r (mod m). In this talk, I will talk about known results on Newman's conjecture on the partition function and our recent progress on the conjecture. This is a joint work with Youngmin Lee.

Abstract:  The local Langlands program relates smooth representations of p-adic groups over complex number to representations of the p-adic Galois group (or Weil-Deligne group) over complex number. The mod p Langlands program does the same but for complex number replaced by a field of characteristic p. It is far less understood than the complex case. In particular, the classification of irreducible mod p representations of p-adic groups remains mysterious. Among these, the most difficult class is supercuspidal representations, also called supersingular in the mod p setting. In this talk, I will briefly overview the mod p representation theory and explain how supersingular representations are "very strange", as their name suggests. This is based on a joint work with Zachary Feng, Ray Li, Vaughan McDonald, and Nischay Reddy.

Abstract:  Counting algebraic objects often leads to generating functions with surprisingly deep structure. In this talk, I will introduce several zeta functions that enumerate subobjects such as sublattices, subforms of quadratic forms, and (if time permits) submodules of quiver of Lie-type structures. These functions originate in number theory but naturally connect with algebraic and combinatorial enumeration problems. I will present recent developments in this area and explain how zeta functions and number theory helps us understand these diverse counting problems.

Abstract:  Friedman--Washington proved that the limiting distribution of the cokernel of Haar-random p-adic matrix is given by the Cohen--Lenstra distribution. Later, Wood vastly generalized this result to epsilon-balanced random p-adic matrix, which is known as the universality theorem. In this talk, we present a sharp threshold for this universality phenomenon. This is joint work with Jiwan Jung and Jungin Lee.