相关论文: CM values of higher automorphic Green functions fo…

We prove classification results for the cuspidal automorphic algebraic representations of ${\rm GL}_n$ over $\mathbb{Q}$ ($n$ arbitrary) of small prime conductor and small motivic weight, in the spirit of the works of Chenevier, Lannes and…

We prove optimal annealed decay estimates on the derivative and mixed second derivative of the elliptic Green functions on $\mathbb{R}^d$ for random stationary measurable coefficients that satisfy a certain logarithmic Sobolev inequality…

Lusztig's algorithm of computing generalized Green functions of reductive groups involves an ambiguity of certain scalars. In this paper, for reductive groups of classical type with arbitrary characteristic, we determine those scalars…

In this paper we define a new type of modular object and construct explicit examples of such functions. Our functions are closely related to cusp forms constructed by Zagier which played an important role in the construction by Kohnen and…

We present a method for accurate evaluation of the Green function $G(\omega,r_1,...,r_d)$ at any real frequency $\omega$ and any lattice vector $(r_1,...,r_d)$ for a $d$-dimensional hypercubic lattice that may have anisotropic couplings…

We construct a family of special cycle classes on the regular integral model of an orthogonal Shimura variety, and show that these cycle classes appear as Fourier coefficients of a Siegel modular form. Passing to the generic fiber of the…

Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic $2$ and let $M$ be an indecomposable $kG$-module which affords a non-degenerate $G$-invariant symmetric bilinear form. We introduce the symmetric…

The characters of the (total) Springer representations are identified with the Green functions by Kazhdan [Israel J. Math. {\bf 28} (1977)], and the latter are identified with Hall-Littlewood's $Q$-functions by Green [Trans. Amer. Math.…

In this article, we study the Chow group of the motive associated to a tempered global $L$-packet $\pi$ of unitary groups of even rank with respect to a CM extension, whose global root number is $-1$. We show that, under some restrictions…

Under a largeness assumption on the size of the residue field, we give an explicit description of the positive-depth Deligne--Lusztig induction of unramified elliptic pairs $(T,\theta)$. When $\theta$ is regular, we show that positive-depth…

We show the existence of "Zagier duality" between vector valued harmonic weak Maass forms and vector valued weakly holomorphic modular forms of integral weight. This duality phenomenon arises naturally in the context of harmonic weak Maass…

In this paper we introduce an elliptic analog of the Bloch-Suslin complex and prove that it (essentially) computes the weight two parts of the groups $K_2(E)$ and $K_1(E)$ for an elliptic curve $E$ over an arbitrary field $k$. Combining…

In this article we prove a version of Kolyvagin's conjecture for modular forms at non-ordinary primes. In particular, we generalize the work of Wang on a converse to a higher weight Gross-Zagier-Kolyvagin theorem in order to prove the…

The goal of this paper is to prove a formula expressing the modular height of a quaternionic Shimura curve over a totally real number field in terms of the logarithmic derivative of the Dedekind zeta function of the totally real number…

This paper establishes an arithmetic intersection formula for central L-derivatives in higher weights.We prove that for a general cusp form (extending the previous result for newforms), the derivative is represented by the global height…

Quaternionic modular forms on $\mathsf{G}_2$ carry a surprisingly rich arithmetic structure. For example, they have a theory of Fourier expansions where the Fourier coefficients are indexed by totally real cubic rings. For quaternionic…

We study the Neumann Green's function for second order parabolic systems in divergence form with time-dependent measurable coefficients in a cylindrical domain $\mathcal{Q}=\Omega\times (-\infty,\infty)$, where $\Omega\subset \mathbb{R}^n$…

We prove the equivalence of a class of generalised Schur partition functions $\mathcal Z_G(q;\alpha)$ of 4d $\mathcal N=2$ superconformal gauge theories to contour integral representations of vector-valued modular forms of the type that…

In this survey, we present recent results of the authors about non-meromorphic modular objects known as polar harmonic Maass forms. These include the computation of Fourier coefficients of meromorphic modular forms and relations between…

We present a characteristic initial value approach to calculating the Green function of the Regge-Wheeler and Zerilli equations. We combine well-known numerical methods with newly derived initial data to obtain a scheme which can in…