Abstract
We prove a strong form of the trivial zero conjecture at the central point for the $p$-adic $L$-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of $\\operatorname{GL}\_2$ over a totally real field, which is Iwahori spherical at places above $p$. In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the $p$-adic $L$-function of a nearly finite slope family and on improved $p$-adic $L$-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 $p$-adic $L$-function of the family.
Highlights
The complex analytic L-function of an algebraic cuspidal automorphic representation π on a reductive group over a number field F lies at the heart of the Langlands program, and the relationship between its analytic properties, namely the order of vanishing at critical points, and the arithmetic of the conjecturally attached p-adic representation Vπ of the absolute Galois group GF is the content of the famous Bloch-Kato conjectures
This article is devoted to proving the Trivial Zero Conjecture at the central point s = 0, with precise interpolation factors and with the Greenberg-Benois arithmetic L -invariant, for the Galois representation Vπ(1) attached to a unitary self-dual cuspidal automorphic representation π of GL2 over a totally real field F having an arbitrary cohomological weight
Following [50] we introduce certain modules that will be later used to define overconvergent sheaves on the Hilbert modular variety
Summary
The complex analytic L-function of an algebraic cuspidal automorphic representation π on a reductive group over a number field F lies at the heart of the Langlands program, and the relationship between its analytic properties, namely the order of vanishing at critical points, and the arithmetic of the conjecturally attached p-adic representation Vπ of the absolute Galois group GF is the content of the famous Bloch-Kato conjectures. This article is devoted to proving the Trivial Zero Conjecture at the central point s = 0, with precise interpolation factors and with the Greenberg-Benois arithmetic L -invariant, for the Galois representation Vπ(1) attached to a unitary self-dual cuspidal automorphic representation π of GL2 over a totally real field F having an arbitrary cohomological weight. This allows us to establish a number of relations between the Taylor coefficients of Lp(x1, . The formula that we show is true even when the archimedean L-function vanishes at the central point, implying that the order of vanishing of the p-adic L-function is at least e + 1
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