Related papers: Harder's conjecture I

We prove an analog of the Deligne conjecture for prestacks. We show that given a prestack $\mathbb A$, its Gerstenhaber--Schack complex $\mathbf{C}_{\mathsf{GS}}(\mathbb A)$ is naturally an $E_2$-algebra. This structure generalises both the…

Let S be a smooth projective surface equipped with a line bundle H. Lehn's conjecture is a formula for the top Segre class of the tautological bundle associated to H on the Hilbert scheme of points of S. Voisin has recently reduced Lehn's…

Let f be a modular form with complex multiplication. If f has critical slope, then Coleman's classicality theorem implies that there is a p-adic overconvergent generalized Hecke eigenform with the same Hecke eigenvalues as f. We give a…

In this paper we present several finite families of congruences between cusp forms and Eisenstein series of higher weights at powers of prime ideals. We formulate a conjecture which describes properties of the prime ideals and their…

Given a pair of distinct unitary cuspidal automorphic representations for GL(n) over a number field, let S denote the set of finite places at which the automorphic representations are unramified and their associated Hecke eigenvalues…

We formulate and discuss a conjecture which would extend a classical inequality of Bernstein.

Given a group, we construct a fundamental additive functor on its orbit category. We prove that any isomorphism conjecture valid for this fundamental isomorphism functor holds for all additive functors, like K-theory, cyclic homology,…

In this paper, we prove: Let A be a nonnegative primitive tensor with order m and dimension n. Then its primitive degree R(A)\leq (n-1)^2+1, and the upper bound is sharp. This confirms a conjecture of Shao [7].

We prove a conjecture of Rudnick and Sarnak on the mass equidistribution of Hecke eigenforms. This builds upon independent work of the authors see arxiv.org:math/0809.1640 and arxiv.org:math/0809.1635.

In [5] I.P. Goulden, D.M. Jackson, and R. Vakil formulated a conjecture relating certain Hurwitz numbers (enumerating ramified coverings of the sphere) to the intersection theory on a conjectural Picard variety. We are going to use their…

We work out instances of a general conjecture on congruences between Hecke eigenvalues of induced and cuspidal automorphic representations of a reductive group, modulo divisors of certain critical L-values, in the case that the group is a…

We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as $\mathrm{GSp}_2(\mathbb{A})$, $\mathrm{SO}(4,3)(\mathbb{\mathbb{A}})$ and $\mathrm{SO}(5,4)(\mathbb{A})$, where the…

In this paper we formulate a conjecture which partially generalizes the Gross-Kohnen-Zagier theorem to higher weight modular forms. For f in S_k(N) satisfying certain conditions, we construct a map from the Heegner points of level N to a…

Given two Siegel eigenforms of different weights, we determine explicit sets of Hecke eigenvalues for the two forms that must be distinct. In degree two, and under some additional conditions, we determine explicit sets of Fourier…

In this preprint we prove that any finite slope modular form fits into a p-adic family of modular forms which is indexed by the weight. Here, the term p-adic family means that p-adic congruences between weights entail certain p-adic…

We specialize the Eichler-Selberg trace formula to obtain trace formulas for the prime-to-level Hecke action on cusp forms for certain congruence groups of arbitrary level. As a consequence, we determine the asymptotic in the prime p of the…

We introduce in this paper the notion of Hodge similarities of transcendental lattices of hyperk\"ahler manifolds and investigate the Hodge conjecture for these Hodge morphisms. Studying K3 surfaces with a symplectic automorphism, we prove…

We prove that for a Hecke cuspform $f\in S_k(\Gamma_0(N),\chi)$ and a prime $l>\max\{k,6\}$ such that $l\nmid N$, there exists an infinite family $\{k_r\}_{r\geq 1}\subseteq\mathbb{Z}$ such that for each $k_r$, there is a cusp form…

A survey of direct and inverse type results for row sequences of Pad\'e and Hermite-Pad\'e approximation is given. A conjecture is posed on an inverse type result for type II Hermite-Pad\'e approximation when it is known that the sequence…

In this article, we prove the $p$-adic Kazhdan-Lusztig hypothesis for $\mathrm{GL}_n(F)$. While the approach via graded affine Hecke algebras due to recent work of Solleveld leads to more general results, this article serves to completes…