# Consequences of functional equations for pairs of p-adic L-functions

## TL;DR

This paper explores the implications of functional equations of p-adic L-functions for elliptic curves at supersingular primes, revealing relationships between their leading terms, parity of vanishing orders, and invariance under twists.

## Contribution

It establishes new theoretical consequences of functional equations for p-adic L-functions, including parity results and invariance properties, advancing understanding in Iwasawa theory.

## Key findings

- Relationship between leading and sub-leading terms of p-adic L-functions
- Parity result for orders of vanishing at supersingular primes
- Invariance of Iwasawa invariants under conjugate twists

## Abstract

We prove consequences of functional equations of p-adic L-functions for elliptic curves at supersingular primes p. The results include a relationship between the leading and sub-leading terms (for which we use ideas of Wuthrich and Bianchi), a parity result of orders of vanishing, and invariance of Iwasaswa invariants under conjugate twists of the p-adic L-functions.