Related papers: Harder's conjecture I

Let $F \in S_{k_1}(\Gamma^{(2)}(N_1))$ and $G \in S_{k_2}(\Gamma^{(2)}(N_2))$ be two Siegel cusp forms over the congruence subgroups $\Gamma^{(2)}(N_1)$ and $\Gamma^{(2)}(N_2)$ respectively. Assume that they are Hecke eigenforms in…

We formulate a series of conjectures on the stable tensor product of irreducible representations of symmetric groups, which are closely related to the reduced Kronecker coefficients. These conjectures are certain generalizations of…

We use the theta lift to study the multiplicity with which certain automorphic representations of cohomological type occur in a family of congruence covers of an arithmetic manifold. When the family of covers is a so-called `p-adic…

In this paper, we prove a decomposition formula for the Kazhdan-Lusztig basis of affine Hecke algebras of rank 2 with positive weight function. Then we discuss some applications of this kind of decomposition to Lusztig's conjectures P1-P15.

By adapting the work of Kudla and Millson we obtain a lifting of cuspidal cohomology classes for the symmetric space associated to GO(V) for an indefinite rational quadratic space V of even dimension to holomorphic Siegel modular forms on…

We give a different and simpler proof of a slightly modified (and weaker) variant of a recent theorem of Shelukhin extending Franks' "two-or-infinitely-many" theorem to Hamiltonian diffeomorphisms in higher dimensions and establishing a…

Let T_k denote the Hecke algebra acting on newforms of weight k and level N. We prove that the power of p dividing the index of T_k inside its normalisation grows at least linearly with k (for fixed N), answering a question of Serre. We…

Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture {\mf for symmetric convex bodies}. Our contributions include, in particular, simple self-contained proofs of their two key statements.…

In this paper, first we give a weak version of Ikeda's conjecture on the period of the Ikeda-Miyawaki lift. Next, we confirm this conjecture rigorously in some cases.

The existence of a "Plastikstufe" for a contact structure implies the Weinstein conjecture for all supporting contact forms.

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of…

Let $p$ be an odd prime. Given an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-D_K})$ where $p$ splits with $D_K>3$, and a $p$-ordinary newform $f \in S_k(\Gamma_0(N))$ such that $N$ verifies the Heegner hypothesis relative to $K$, we…

We conjecture that the set of all Hilbert functions of (artinian) level algebras enjoys a very natural form of regularity, which we call the {\em Interval Conjecture} (IC): If, for some positive integer $\alpha $, $(1,h_1,...,h_i,...,h_e)$…

We discuss several conjectures about derived equivalent varieties, defined over fields of arbitrary characteristics, and implications among them. In particular we show that the (conjectural) derived invariance of the Hasse-Weil Zeta…

Motivated by the study of periods of automorphic forms and relative trace formulae, we develop the theory of descent necessary to study orbital integrals arising in the fundamental lemma for a general class of symmetric spaces over a…

In this short note, we observe that the techniques of our recent work "Pseudo-modularity and Iwasawa theory" can be used to provide a new proof of some of the residually reducible modularity lifting results of Skinner and Wiles. In these…

For the anticyclotomic p-adic Rankin--Selberg L-function attached to a fixed Hecke eigenform and an imaginary quadratic field we introduce the second p-adic variable by considering Hida families of Hecke eigenforms parametrized by the…

Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose restrictions on the cohomology algebra of a smooth compact K\"ahler manifold or on the…

In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains, and Mahler's conjecture on the volume product of centrally symmetric…

We prove analogues of the fundamental theorem of algebraic K-theory for the second and third homology of SL_2 over an infinite field k. The statements involve Milnor-Witt K-theory and scissors congruence groups. We use these results to…