Assuming the Riemann Hypothesis, Montgomery established results concerning pair correlation of zeros of the Riemann zeta function. Rudnick and Sarnak extended these results to automorphic $L$-functions and all level correlations. We show that automorphic $L$-functions exhibit additional geometric structures related to the correlation of their zeros. This is joint work with Cruz Castillo and Alexandru Zaharescu.
The problem of primes with restricted digits has been pondered and studied since the mid twentieth century, and has seen much progress in recent years thanks to the work of mathematicians such as James Maynard. But what if the ring of integers is replaced with $\mathbb{F}_q[T]$? In this talk we will look at function field analogues of questions about primes with restricted digits. We will consider both classical approaches, such as the circle method, and more modern ideas. I will present some very recent results about asymptotic behaviour of squarefree polynomials of degree n with coefficients in a restricted subset of $\mathbb{F}_q$.
There already exist several results regarding quantum unique ergodicity for modular forms on $SL(2, \mathbb{Z}) \backslash \mathcal{H}$ some of which have been generalized to the setting of Hilbert modular forms. We will talk about quantum unique ergodicity for shrinking sets in the context of Hilbert modular forms. Considering measures originating from Eisenstein series we shall look at the methods described by Matthew P. Young for the classical setting and see how they still apply in our setting. As a PhD student, I am conducting this research under the guidance of Nicole Raulf and Didier Lesesvre.
We discuss algebraicity results for $L$-functions associated to automorphic forms on quaternionic exceptional groups. Quaternionic modular forms (QMFs) are automorphic forms on these groups whose real components lie in the quaternionic discrete series. QMFs have a robust theory of Fourier coefficients developed by Gross-Wallach, Gan-Gross-Savin, and Pollack. Recently Pollack established that the space of cuspidal QMFs has a basis whose Fourier coefficients are algebraic numbers. We combine this theory with Rankin-Selberg integrals due to Hundley and Gurevich-Segal, along with our recent work on exceptional Eisenstein series, to establish algebraicity results that give evidence for Deligne's conjecture on the critical values of $L$-functions. Along the way, we develop exceptional differential operators to mirror the classical picture of Maass-Shimura operators and nearly holomorphic modular forms.
We show that a totally degenerate limit of discrete series representation admits a choice of $n$ cohomology group that is nonvanishing at a canonically defined degree. We then show that the combinatorial complexes used by Soergel to compute these cohomology groups satisfies Serre duality. We conclude that this produces two n cohomology groups, each for the TDLDS of $U(n+1)$ and $U(n)$, which are nonvanishing at the same degree, suggesting Gan Gross Prasad type branching laws for the TDLDS of unitary groups of any rank.
Subconvexity estimation is one of the most important and challenging problem in the theory of analytic number theory and $L$-functions. In my talk, following the estimations and ideas in Michel-Venkatesh and Hu-Michel-Nelson, we establish a spectral reciprocity formula which links twisted first moment of different $L$-functions. As an application, we can give an explicit hybrid subconvexity bound for triple product $L$-functions in the level aspect, allowing joint ramifications and conductor dropping range.
Let $A(m,n)$ be normalised Fourier coefficients of $F$, a Hecke Maass cusp form for $SL(3,Z)$. In this talk, we will discuss an upper bound for the sum \[\sum_{h\sim H} \sum{n\sim N} A(n) A(n+h)\] which is non-trivial in the range $N^{1/6+\varepsilon}<H<N^{1/2}$. This improves upon the best known range of $H$ ($H>N^{1/4}$ by Dasgupta-Leung-Young) for which cancellation can be shown in the above shifted convolution sum. This is an ongoing project with Ritwik Pal.
Let $K/\mathbb{Q}$ be a number field with algebraic closure $\overline{K}$ and $X/K$ a smooth geometrically connected projective curve. The Section Conjecture in Anabelian Geometry, states that if $X/K$ is of genus $g\geq 2$, then all sections of the exact sequence
\[1 \to \pi_1(X_{\overline{K}})\to \pi_1(X)\to \pi_1(\mathrm{Spec}(K))=\mathrm{Gal}_K \to 1\]
are induced by rational points $x \in X(K)$. In this talk, we will discuss how the now proven Serre's Conjecture for two-dimensional residual Galois representations of $\mathrm{Gal}_{\mathbb{Q}}$ relates to the (still open) Section Conjecture for $K=\mathbb{Q}$.
The recent joint work of Ben-Zvi, Sakellaridis and Venkatesh (BZSV) proposed a duality of the relative Langlands problem though the concept of hyperspherical data $(G, H, \iota, S)$. This talk will focus on the construction of its dual data and the joint work of Wan, Zhang, and myself on the exceptional case though $G = F_4$.
In the first part of this talk, we present a brief overview of Siegel modular forms of degree $2$ and the associated Dirichlet series. Mainly, we will talk about the character twist of the Spinor zeta function
\[Z_{F}(s, \chi): = \sum_{n \ge 1} \frac{\lambda_{F}(n)\chi(n)}{n^{s}}\]
associated to a Siegel eigenform $F \in S_{k}(\Gamma_{2})$ and its analytic property. Here, $\lambda_{F}(n)$ denotes the $n$-th Siegel Eigenvalue of $F$. In the second part, we will survey about (Partial) Generalized Riemann hypothesis for $L$-functions associated to modular forms, which says that there are infinitely many zeros on the critical line. In particular, we show that the twisted spinor zeta function $Z_{F}(s, \chi)$ has infinitely many zeros on the critical line for a quadratic character. We also try to look at some applications. This is ongoing work with Manish K. Pandey.
I’ll explain how Whittaker functions can be obtained via $q$-deformation of the Macdonald polynomials at $t = 0$ and sketch how the Bump-Stade identity on the Whittaker transform follows from this deformation.
