Related papers: Multiplicative Hecke operators and their applicati…

We prove that amongst all real quadratic fields and all spaces of Hilbert modular forms of full level and of weight $2$ or greater, the product of two Hecke eigenforms is not a Hecke eigenform except for finitely many real quadratic fields…

Let $U(p)$ denote the Atkin operator of prime index $p$. Honda and Kaneko proved infinite families of congruences of the form $f|U(p) \equiv 0 \pmod{p}$ for weakly holomorphic modular forms of low weight and level and primes $p$ in certain…

Matrix representations of Hecke operators on classical holomorphical cusp forms and the corresponding period polynomials are well known. In this article we derive representations of Hecke operators for vector valued period functions for the…

In this article, we determine the trace of some Hecke operators on the spaces of level one automorphic forms on the special orthogonal groups of the euclidean lattices $\mathrm{E}_7$, $\mathrm{E}_8$ and $\mathrm{E}_8\oplus \mathrm{A}_1$,…

We study tensors on Lie groupoids suitably compatible with the groupoid structure, called {\em multiplicative}. Our main result gives a complete description of these objects only in terms of infinitesimal data. Special cases include the…

Let $\mathscr{S}_k^+(\cn,\Phi)$ denote the space generated by Hilbert modular newforms (over a fixed totally real field $K$) of weight $k$, level $\cn$ and Hecke character $\Phi$. We show how to decompose $\mathscr{S}_k^+(\cn,\Phi)$ into…

We study Hecke operators on moduli spaces of ramified $G$-bundles using the combinatorial language of Hecke graphs. We introduce a general notion of $\mathcal H$-ramification in the spirit of parahoric ramification, which depends on a…

We prove for L-function attached to an automorphic cusp form for the Hecke congruence group $\Gamma_0(D)$, which is also an eigenfunction of all the Hecke operators, that a positive proportion of its non-trivial zeros lie on the critical…

We describe a computational approach to the verification of Maeda's conjecture for the Hecke operator T2 on the space of cusp forms of level one. We provide experimental evidence for all weights less than 12000, as well as some applications…

In this paper we introduce a new subspace of Jacobi forms of higher degree via certain relations among Fourier coefficients. We prove that this space can also be characterized by duality properties of certain distinguished embedded Hecke…

We study Hecke operators associated with curves over a non-archimedean local field $K$ and over the rings $O/{\mathfrak m}^N$, where $O\subset K$ is the ring of integers. Our main result is commutativity of a certain "small" local Hecke…

We correct a mistake in \cite{St} leading to erroneous formulas in Theorems 5.2 and 5.4. As an immediate corollary of a formula in \cite{BCJ} we give a formula, which relates the Hecke operators $T(p^2)\circ T(p^{2l-2})$, $T(p^{2l})$ and…

We present a theory of reduction of binary quadratic forms with coefficients in Z[lambda], where lambda is the minimal translation in a Hecke group. We generalize from the modular group Gamma(1) = SL(2,Z) to the Hecke groups and make…

The $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator was introduced as an element of the $O_{\ell+1}$-spherical Hecke algebra associated with the Gelfand pair $O_{\ell+1}\subset GL_{\ell+1}(\mathbb{R})$. It was specified by the property to…

In this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of k-th invariant differentials…

We give a short proof of the trace formula for Hecke operators on modular forms for the modular group, using the action of Hecke operators on the space of period polynomials.

The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z)-j(\tau)$ given in terms of the Hecke system of $\operatorname{SL}_2(\mathbb Z)$-modular functions $j_n(\tau)$. It is prominent in…

We investigate the derived Hecke action on the cohomology of an arithmetic manifold associated to the multiplicative group over a number field. The degree one part of the action is proved to be non-vanishing modulo $p$ under mild…

We study the action of the derived Hecke algebra in the setting of dihedral weight one forms, and prove a conjecture of the second- and fourth- named authors relating this action to certain Stark units associated to the symmetric square…

Let $G$ denote a connected reductive group over a nonarchimedean local field $F$ of residue characteristic $p$, and let $\mathcal{C}$ denote an algebraically closed field of characteristic $\ell \neq p$. If $\rho$ is an irreducible, smooth…