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Extremal p -Adic L-Functions

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  • Santiago Molina

    (Departament de Matemàtica Aplicada, Campus Nord, UPC, 08034 Barcelona, Spain)

Abstract

In this note, we propose a new construction of cyclotomic p -adic L-functions that are attached to classical modular cuspidal eigenforms. This allows for us to cover most known cases to date and provides a method which is amenable to generalizations to automorphic forms on arbitrary groups. In the classical setting of GL 2 over Q , this allows for us to construct the p -adic L-function in the so far uncovered extremal case, which arises under the unlikely hypothesis that p -th Hecke polynomial has a double root. Although Tate’s conjecture implies that this case should never take place for GL 2 / Q , the obvious generalization does exist in nature for Hilbert cusp forms over totally real number fields of even degree, and this article proposes a method that should adapt to this setting. We further study the admissibility and the interpolation properties of these extremal p-adic L-functions L p ext ( f , s ) , and relate L p ext ( f , s ) to the two-variable p -adic L-function interpolating cyclotomic p -adic L-functions along a Coleman family.

Suggested Citation

  • Santiago Molina, 2021. "Extremal p -Adic L-Functions," Mathematics, MDPI, vol. 9(3), pages 1-26, January.
  • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:234-:d:486727
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    Keywords

    p -adic L-functions; Coleman families;

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