G-BRL Number Theory Seminar at CNU

• Abstract:  There is a great deal of evidence, both theoretical and experimental, that the distribution of zeros of zeta and L-functions can be modeled using statistics of eigenvalues of random matrices from classical compact groups. In particular, we expect that there are arbitrarily large and small normalized gaps between the ordinates of (high) zeros zeta and L-functions. Previous results are known for zeta and L-functions of degrees 1 and 2. We discuss some new results for higher degrees, including Dedekind zeta-functions associated to Galois extensions of the rationals and principal automorphic L-functions.

• Abstract:  Let $M$ be a compact Riemannian manifold. It was proved by Weyl that number of Laplacian eigenvalues less than $T$, is asymptotic to $C(M)T^{dim(M )/2}$, where $C(M)$ is the product of the volume of $M$, volume of the unit ball and $(2π)^{dim(M)}$. Let $Γ$ be an Arithmetic subgroup of $SL_2(Z)$, $H^2$ be an upper-half plane. When $M = Γ\H^2$, Weyl’s asymptotic holds true for the discrete spectrum of Laplacian. It was proved by Selberg, who used his celebrated trace formula.

Let $G$ be a semisimple algebraic group of Adjoint and split type over $Q$. Let $G(R)$ be the set of $R$-points of $G$. For simplicity of this exposition let us assume that $Γ ⊂G(R)$ be an torsion free arithmetic subgroup. Let $K_{\infty}$ be the maximal compact subgroup. Let $L^2(Γ\G(R))$ be space of square integrable $Γ$ invariant functions on $G(R)$. Let $L^2_{cusp}(Γ\G(R))$ be the cuspidal subspace. Let $M = Γ\G(R)/K_{\infty}$ be a locally symmetric space. Suppose $d= dim(Γ\G/K_{\infty})$. Then it was proved by Lindenstrauss and Venkatesh, that number of spherical, i.e. bi-$K_{\infty}$ invariant cuspidal Laplacian eigenfunctions, whose eigenvalues are less than $T$ is asymptotic to $C(M)T^{dim(M )/2}$, where $C(M)$ is the same constant as above.

We are going to prove a similar Weyl’s asymptotic estimates for $K_{\infty}$-finite cusp forms for the above space. We are going to closely follow the pre-trace formula and the test functions used in Lindenstrauss and Venkatesh.

• Abstract:  We will discuss the Langlands functoriality principle and explore its relation with the L-functions.

• Abstract:  In this talk, we will discuss the complete classification of faithful rectangular representations of a complex semisimple Lie algebra. We will also discuss its application to $\lambda$-independence on the algebraic monodromy groups of compatible systems of $\lambda$-adic Galois representations of number fields. This is a joint work with Chun-Yin Hui.

• Abstract:  In the 1950s Selberg derived his well celebrated trace formula to prove the existence and infinitude of Maass cusp forms. Explicit examples of such forms have been rare, only occurring from examples that Maass originally constructed and those arising from Galois representations. From this, we mainly rely on numerical approximations to study specific cusp forms. In this talk I will give an overview and history of numerical computations of Maass cusp forms and furthermore, I will discuss personal progress in this area in using the trace formula for numerical computations.

• Abstract:  We obtain the n-th centered moments of one level densities of a large orthogonal family of L-functions associated with holomorphic Hecke newforms of level q, averaged over $q \sim Q$.  We verify the Katz-Sarnak conjecture for these statistics, in the range where the sum of the supports of the Fourier transforms of test functions lies in (-4, 4). In so doing, we need to understand certain phantom oversized terms, which allow us to extract the right off-diagonal contributions.  We further need to resolve the combinatorial problem that arises when matching our main terms with random matrix predictions. This is a joint work with V. Chandee and X. Li. Preprint is available at https://arxiv.org/abs/2510.07647

• Abstract:  We discuss how certain families of Heegner points still know the finiteness of p-primary part of Tate-Shafarevich groups under mild hypotheses. This observation grew out from the discussion with Minhyong Kim.

• Abstract:  TBA