-adic -functions of Hilbert cusp forms and the trivial zero conjecture
Daniel Barrera
Universidad de Santiago de Chile, ChileMladen Dimitrov
Université de Lille, FranceAndrei Jorza
University of Notre Dame, USA
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Abstract
We prove a strong form of the trivial zero conjecture at the central point for the -adic -function of a non-critically refined self-dual cohomological cuspidal automorphic representation of over a totally real field, which is Iwahori spherical at places above .
In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the -adic -function of a nearly finite slope family and on improved -adic -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 -adic -function of the family.
Cite this article
Daniel Barrera, Mladen Dimitrov, Andrei Jorza, -adic -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