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

Balakrishnan, Sai Sanjeev; Lei, Antonio; Palvannan, Bharathwaj

doi:10.2140/involve.2025.18.401

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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

DOI: 10.2140/involve.2025.18.401

Abstract

This article is a computational extension of [Forum Math. Sigma. 7 (2019), art. id. e25]. Let $p \geq 3$ be a prime number and $E ∕ ℚ$ an elliptic curve with good supersingular reduction at $p$ and $a_{p} (E) = 0$ . 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.

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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

Vol. 18, No. 3, 2025