Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of p -adic L -functions
- PDF / 1,012,449 Bytes
- 79 Pages / 439.37 x 666.142 pts Page_size
- 22 Downloads / 174 Views
Mathematische Annalen
Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of p-adic L-functions Zheng Liu1 · Giovanni Rosso2 Received: 19 April 2019 / Revised: 1 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study the derivative of the standard p-adic L-function associated with a P-ordinary Siegel modular form (for P a parabolic subgroup of GL(n)) when it presents a semistable trivial zero. This implies part of Greenberg’s conjecture on the order and leading coefficient of p-adic L-functions at such trivial zero. We use the method of Greenberg– Stevens. For the construction of the improved p-adic L-function we develop Hida theory for non-cuspidal Siegel modular forms. Mathematics Subject Classification Primary 11F46; Secondary 11F33 · 11R23 · 11S40
Contents 1 2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-cuspidal Hida theory . . . . . . . . . . . . . . . . . . . . . . . 2.1 Compactifications of Siegel varieties . . . . . . . . . . . . . . 2.2 The Igusa tower over the ordinary locus and p-adic forms . . . 2.3 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Mumford construction . . . . . . . . . . . . . . . . . . . 2.5 The fiber of the push-forward to the minimal compactification S P,r S P,r −1 2.6 The quotient Vm,l /Vm,l . . . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Communicated by Wei Zhang.
B
Giovanni Rosso [email protected] https://sites.google.com/site/gvnros/ Zheng Liu [email protected] https://math.ucsb.edu/~zliu
1
Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, USA
2
Departments of Mathematics and Statistics, Concordia University, Montreal, QC, Canada
123
Z. Liu, G. Rosso 2.7 The space V S P,r , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 The q-expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 The U Pp -operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Hida families of p-adic Siegel modular forms vanishing along strata with cusp labels of rank > r 3 p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Generalities on standard L-functions for symplectic groups . . . . . . . . . . . . . . . . . . . 3.2 The doubling method for symplectic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The modified Euler factor at p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The choices of local sections for the Siegel Eisenstein series . . . . .
Data Loading...
