Related papers: An equidistribution theorem for holomorphic Siegel…

Motivated by the study of invariant rings of finite groups on the first Weyl algebras $A_{1}$ (\cite{AHV}) and finding interesting families of new noetherian rings, a class of algebras similar to $U(sl_{2})$ were introduced and studied by…

We prove an equidistribution of signs for the Fourier coefficients of Hilbert modular forms of half-integral weight. Our study focuses on certain subfamilies of coefficients that are accessible via the Shimura correspondence. This is a…

We prove several results on the distribution of values of $L$-functions at the edge of the critical strip, by constructing and studying a large class of random Euler products. Among new applications, we study families of symmetric power…

We study the geometric properties of holomorphic distributions of totally null $m$-planes on a $(2m+\epsilon)$-dimensional complex Riemannian manifold $(\mathcal{M}, \bm{g})$, where $\epsilon \in {0,1}$ and $m \geq 2$. In particular, given…

We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let $G$ be a reductive group over a number field $F$ which admits discrete series representations at…

We introduce a framework within which a large class of joint equidistribution problems can be studied and resolved with effective error terms. This involves proving a higher dimensional and $\mu$-analogue of the Erd\"{o}s-Tur\'{a}n…

We construct an Euler system attached to general-type cohomological cuspidal automorphic representations of $\mathrm{GSp}(4)$ twisted by a Groessencharacter of an imaginary quadratic field. We then use this to bound strict Selmer groups…

Recently Ono, Saad and the second author \cite{KHN} initiated a study of value distribution of certain families of Gaussian hypergeometric functions over large finite fields. They investigated two families of Gaussian hypergeometric…

We study the Iwasawa main conjecture for quadratic Hilbert modular forms over the p-cyclotomic tower. Using an Euler system in the cohomology of Siegel modular varieties, we prove the "Kato divisibility" of the Iwasawa main conjecture under…

Let $J$ be an almost complex structure on a 4-dimensional and unimodular Lie algebra $\mathfrak{g}$. We show that there exists a symplectic form taming $J$ if and only if there is a symplectic form compatible with $J$. We also introduce…

Let G be any of the complex classical groups GL(n), SO(2n+1), Sp(2n), O(2n), let g denote the Lie algebra of G, and let Z(g) denote the subalgebra of G-invariants in the universal enveloping algebra U(g). We derive a Taylor-type expansion…

Let $X$ be a complex $K3$ surface with an effective action of a group $G$ which preserves the holomorphic symplectic form. Let $$ Z_{X,G}(q) = \sum_{n=0}^{\infty} e\left(\operatorname{Hilb}^{n}(X)^{G} \right)\, q^{n-1} $$ be the generating…

We prove that Hecke eigenvalues for any Hilbert and Siegel modular forms are algebraic integers. Our method does not rely on cohomologicality nor Galois representations. We apply the integrality of Hecke eigenvalues for Hilbert modular…

We prove that if $F$ is a non-zero (possibly non-cuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many non-zero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and…

Let $G$ be a semisimple Lie group with discrete series. We use maps $K_0(C^*_rG)\to \mathbb{C}$ defined by orbital integrals to recover group theoretic information about $G$, including information contained in $K$-theory classes not…

Let G be a split adjoint semisimple group over Q and K a maximal compact subgroup of the real points G(R). We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of G(R)/K.…

We study the behavior of zeros and mass of holomorphic Hecke cusp forms on $SL_2(\mathbb Z) \backslash \mathbb H$ at small scales. In particular, we examine the distribution of the zeros within hyperbolic balls whose radii shrink…

We prove an equidistribution result about Hecke orbits on the Picard group of Shimura curves coming from definite quaternion algebras over function fields. In particular, we show the equidistribution of Hecke orbits of supersingular…

We consider elliptic second order partial differential operators with Lipschitz continuous leading order coefficients on finite cubes and the whole Euclidean space. We prove quantitative sampling and equidistribution theorems for…

The coset $G$-space of a finite group and a subgroup is a fundamental module of study of Schur and others around 1930; for example, its endomorphism algebra is a Hecke algebra of double cosets. We study and review its generalization $Q$ to…