On the order of vanishing of the cyclotomic p-adic L-function

TL;DR

This paper investigates the properties of p-adic L-functions attached to newforms, establishing their non-vanishing on p-adic units and finiteness of vanishing order at p-adic integers for primes greater than 3.

Contribution

It proves the non-vanishing of the p-adic L-function on Z_p and the finiteness of the order of vanishing at p-adic integers for primes p > 3.

Findings

p-adic L-function is not identically zero on Z_p

Order of vanishing at p-adic integers is finite for p > 3

Results apply to newforms for Gamma_0(N) of even weight k

Abstract

For a newform for Gamma_0(N) of even weight k, we prove that its attached p-adic L-function is not identically zero on the group Z_p of the p-adic units. If p >3, we prove that the order of vanishing at any p-adic integer is finite.