Here we will discuss speakers, topics, and time slots for upcoming seminar talks. The webpage will be appropriately updated afterwards. Please join us if you are interested in coming to future talks and/or would like to give a talk.

       The existence of a Landau-Siegel zero leads to the Deuring-Heilbronn phenomenon, here appearing in the 1-level density in a family of quadratic twists of a fixed genus character L-function. We obtain explicit lower order terms describing the vertical distribution of the zeros, and realize the influence of the Landau-Siegel zero as a resonance phenomenon. preprint.

       We will given an overview of some recent results on the μ-invariants in the title. These results are based on a strategy found by Professor Haruzo Hida using linear independence of modular forms mod p.

       We review the well known microscopic correspondence between random zeros of the Riemann zeta-function and the eigenvalues of a random unitary or Hermitian matrix, and discuss evidence that this correspondence extends to larger 'mesoscopic' collections of zeros or eigenvalues. In addition, we discuss interesting phenomena that appears in the statistics of even larger 'macroscopic' collections of zeros. The terms microscopic, mesoscopic, and macroscopic will be defined in the talk. This talk is based in part on results in the papers link1, link2. The slides used for this talk can be found here.

       We will give an introduction to the basic objects involved in the MTT conjecture; namely, we will discuss p-adic L-functions L_p(E, s) and important invariants of elliptic curves E over the rationals. Under certain assumptions on the prime p and curve E, the p-adic L-function satisfies L_p(E, 1) = 0. We will state the MTT conjecture, which is a formula for the value of the derivative at s = 1.

       Let q > 3 be a prime congruent to 3 modulo 4. Suppose also that the real quadratic number field generated by the square root of q has class number 1. Then the continued fraction expansion of the square root of q allows one to read off the class number of the imaginary quadratic number field generated by the square root of -q. This was noticed by Friedrich Hirzebruch as a restatement of a Kronecker limit formula on narrow ideal classes by Curt Meyer. Similarly, for an odd prime p, we can use the same continued fraction expansion to read off the corresponding p-adic lambda invariant of the imaginary quadratic number field. This is a consequence of a very general Kronecker limit formula on narrow ray classes by Shuji Yamamoto.

       In this talk, we will show that every classical self-dual Hecke eigensystem can be deformed into a p-adic family of classical Hecke eigensystems. This is a stronger version of the inverse of Ash-Pollack-Steven's conjecture. To do this, we constructed an eigenvariety of Gl_n parameterizing all self-dual Hecke eigensystems of Gl_n.

       The local Shimura correspondence relates representations of PGL₂ to those of the metaplectic cover of SL₂, where both groups are p-adic and the representations are on C-vector spaces. Many pieces of the usual construction break down, due to non-semisimplicity, when the representations are instead taken over a field of positive characteristic. The mod p case is especially problematic. In the specific case of the pair ((SL₂)^~, PGL₂), I will discuss an alternate approach, which involves a comparison of certain Iwahori Hecke algebras of both groups, as well as progress towards a classification of the mod p representations of the metaplectic group.

       According to the Hilbert-Pólya conjecture, the imaginary parts of the Riemann zeros (and, more generally, the zeros of L functions) are eigenvalues of a self-adjoint operator on some Hilbert space. I will review the evidence for this, and discuss recent attempts by Berry and Keating and by Connes to construct the Riemann operator.