On the GL(2n) eigenvariety: branching laws, Shalika families and $p$adic $L$functions
Abstract
In this paper, we prove that a GL(2n)eigenvariety is etale over the (pure) weight space at noncritical Shalika points, and construct multivariabled $p$adic $L$functions varying over the resulting Shalika components. Our constructions hold in tame level 1 and Iwahori level at $p$, and give $p$adic variation of $L$values (of regular algebraic cuspidal automorphic representations, or RACARs, of GL(2n) admitting Shalika models) over the whole pure weight space. In the case of GL(4), these results have been used by Loeffler and Zerbes to prove cases of the BlochKato conjecture for GSp(4). Our main innovations are: a) the introduction and systematic study of `Shalika refinements' of local representations of GL(2n), evaluating their attached local twisted zeta integrals; and b) the $p$adic interpolation of representationtheoretic branching laws for GL(n)$\times$GL(n) inside GL(2n). Using (b), we give a construction of manyvariabled $p$adic functionals on the overconvergent cohomology groups for GL(2n), interpolating the zeta integrals of (a). We exploit the resulting nonvanishing of these functionals to prove our main arithmetic applications.
 Publication:

arXiv eprints
 Pub Date:
 November 2022
 DOI:
 10.48550/arXiv.2211.08126
 arXiv:
 arXiv:2211.08126
 Bibcode:
 2022arXiv221108126B
 Keywords:

 Mathematics  Number Theory;
 Primary 11F33;
 11F67;
 Secondary 11R23;
 11G22
 EPrint:
 57 pages. Comments welcome