JournalsjemsVol. 24, No. 10pp. 3439–3503

pp-adic LL-functions of Hilbert cusp forms and the trivial zero conjecture

  • Daniel Barrera

    Universidad de Santiago de Chile, Chile
  • Mladen Dimitrov

    Université de Lille, France
  • Andrei Jorza

    University of Notre Dame, USA
$p$-adic $L$-functions of Hilbert cusp forms and the trivial zero conjecture cover
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Abstract

We prove a strong form of the trivial zero conjecture at the central point for the pp-adic LL-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of GL2\operatorname{GL}_2 over a totally real field, which is Iwahori spherical at places above pp.

In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the pp-adic LL-function of a nearly finite slope family and on improved pp-adic LL-functions that we construct using automorphic symbols and overconvergent cohomology.

For higher order zeros we develop a conceptually new approach studying the variation of the root number in partial families and establishing the vanishing of many Taylor coefficients of the pp-adic LL-function of the family.

Cite this article

Daniel Barrera, Mladen Dimitrov, Andrei Jorza, pp-adic LL-functions of Hilbert cusp forms and the trivial zero conjecture. J. Eur. Math. Soc. 24 (2022), no. 10, pp. 3439–3503

DOI 10.4171/JEMS/1165