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On computing the Newton polygons of plus and minus $p$-adic $L$-functions

Sai Sanjeev Balakrishnan, Antonio Lei and Bharathwaj Palvannan

Vol. 18 (2025), No. 3, 401–416
Abstract

This article is a computational extension of [Forum Math. Sigma. 7 (2019), art. id. e25]. Let p 3 be a prime number and E an elliptic curve with good supersingular reduction at p and ap(E) = 0. We study the computation of the Newton polygons of Pollack’s plus and minus p-adic L-functions attached to E. This allows us to furnish new examples where Assumption GCD in the aforementioned article holds. Furthermore, we rectify two imprecisions in the previous article.

Keywords
Iwasawa theory, elliptic curves, supersingular primes
Mathematical Subject Classification
Primary: 11R23
Secondary: 11F67, 11G05
Supplementary material

Sage code for the computations

Milestones
Received: 31 January 2023
Revised: 16 October 2023
Accepted: 26 March 2024
Published: 28 April 2025

Communicated by Bjorn Poonen
Authors
Sai Sanjeev Balakrishnan
Department of Mathematics
University of California Berkeley
Berkeley
California
United States
Antonio Lei
Department of Mathematics and Statistics
University of Ottawa
Ottawa, ON
Canada
Bharathwaj Palvannan
Department of Mathematics
Indian Institute of Science
Bangalore
India