The first nontrivial zeroes of the Riemann ζ function are ≈ 1 2 ± 14.13472i. We investigate the question of whether or not any other Lfunction has a higher lowest zero. To do so we try to quantify the notion that the L-function of a “small” automorphic representation (i.e. one with small level and archimedean type) does not have small zeroes, and vice-versa. We prove that many types of automorp...

In this paper, we present a refinement of the global Langlands correspondence for discrete symplectic motives of rank 2n over Q. To such a motive Langlands conjecturally associates a generic, automorphic representation π of the split orthogonal group SO2n+1 over Q, which appears with multiplicity one in the cuspidal spectrum. Using the local theory of generic representations of odd orthogonal g...

Abstract We study the Picard–Lefschetz formula for Siegel modular threefolds of paramodular level and prove weight-monodromy conjecture its middle degree inner cohomology. give some applications to Langlands programme: using Rapoport-Zink uniformisation supersingular locus special fiber, we construct a geometric Jacquet–Langlands correspondence between $${\text {GSp}}_4$$ <mml:math xmlns:mml="h...

A strong impetus for this paper came, at least for the first author, from a question of Avner Ash, asking whether one can construct non-self-dual, nonmonomial cuspidal cohomology classes for suitable congruence subgroups Γ of SL(n,Z), say for n = 6. Such a construction, in special examples, has been known for some time for n = 3 (see [2, 34, 35, 36]); it is of course not possible for n = 2. One...

We investigate the homology of a congruence subgroup Γ SL3(Z) with coefficients in Steinberg modules St(Q3) and St(E3), where E is real quadratic field are Q. By Borel-Serre duality, H0(Γ,St(Q3)) isomorphic to H3(Γ,Q). Taking image connecting homomorphism H1(Γ,St(E3)/St(Q3))→H0(Γ,St(Q3)), followed by isomorphism, we obtain naturally defined Hecke-stable subspace H(Γ,E) conjecture that independe...

We investigate singularities of all parallel surfaces to a given regular surface. In generic context, the types of singularities of parallel surfaces are cuspidal edge, swallowtail, cuspidal lips, cuspidal beaks, cuspidal butterfly and 3-dimensional D4 singularities. We give criteria for these singularities types in terms of differential geometry (Theorem 3.4 and 3.5).

Abstract Let p and $\ell $ be primes such that $p&gt; 3$ $p \mid \ell -1$ k an even integer. We use deformation theory of pseudo-representations to study the completion Hecke algebra acting on space cuspidal modular forms weight level $\Gamma _0(\ell )$ at maximal Eisenstein ideal containing . give a necessary sufficient condition for $\mathbb {Z}_p$ -rank this greater than $1$ in terms vanishi...