An introduction to p-adic L-functions

Rodrigues Jacinto, Joaquín; Williams, Chris

Abstract

These expository notes introduce p-adic L-functions and the foundations of Iwasawa theory. We focus on Kubota Leopoldt's p-adic analogue of the Riemann zeta function, which we describe in three di erent ways. We rst present a measure-theoretic (analytic) p-adic interpolation of special values of the Riemann zeta function. Next, we describe Coleman's (arithmetic) construction via cyclotomic units. Finally, we examine Iwasawa's (algebraic) construction via Galois modules over the Iwasawa algebra.

The Iwasawa Main conjecture, now a theorem due to Mazur and Wiles, says that these constructions agree. We will state the conjecture precisely, and give a proof when p is a Vandiver prime (which conjecturally covers every prime). Throughout, we discuss generalisations of these constructions and their connections to modern research directions in number theory.

Citation

Rodrigues Jacinto, J., & Williams, C. (2025). An introduction to p-adic L-functions. Essential Number Theory, 4(1), 101–216. https://doi.org/10.2140/ent.2025.4.101

• Journal Article Type: Article
• Acceptance Date: Dec 10, 2024
• Online Publication Date: Apr 3, 2025
• Publication Date: Apr 3, 2025
• Deposit Date: Mar 6, 2025
• Publicly Available Date: Mar 6, 2025
• Journal: Essential Number Theory
• Print ISSN: 2834-4626
• Electronic ISSN: 2834-4634
• Publisher: Mathematical Sciences Publishers
• Peer Reviewed: Peer Reviewed
• Volume: 4
• Issue: 1
• Pages: 101–216
• DOI: https://doi.org/10.2140/ent.2025.4.101
• Public URL: https://nottingham-repository.worktribe.com/output/46191422
• Publisher URL: https://msp.org/ent/2025/4-1/p03.xhtml
• Additional Information: First published in Essential Number Theory in Vol. 4 2025, No. 1 published by Mathematical Sciences Publishers; ; © 2025 The Authors, under license to MSP (Mathematical Sciences Publishers)