Extremal p -Adic L-Functions

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.