Computing $p$-adic $L$-functions of totally real number fields

Abstract

Abstract. We prove new explicit formulas for the p-adic L-functions of totally real number fields and show how these formulas can be used to compute values and representations of p-adic L-functions.

Computing 𝑝-adic L-functions of totally real fields

“…Remark 3.6. The reader may wonder how this general version of the algorithm compares to the work of Roblot [Rob15]. We make a few comments in the following cases:…”

“…Such p-adic L-functions were constructed in the 1970's independently by Barsky and Cassou-Noguès [ Bar78,CN79] based on the explicit formula for zeta values of Shintani [Shi76] and by Serre and Deligne-Ribet [Ser73, DR80] using Hilbert modular forms and an idea of Siegel [Sie68] going back to Hecke [Hec24,Satz 3]. An algorithm for computing via the approach of Cassou-Noguès was developed by Roblot 1 [Rob15]. Our algorithm follows the approach of Serre and Siegel, and its computational e ciency rests upon a method for computing with p-adic spaces of modular forms developed in previous work by the authors.…”

On -invariants attached to cyclic cubic number fields

“…Lastly, a new paper of Roblot [15] describes an explicit method for computing special values of Shintani p-adic L-functions over totally real number fields, by using the cone decomposition into partial zeta functions developed by Pi. Cassou-Noguès.…”

Overconvergent modular forms and their explicit arithmetic

“…We closely mirror the approach to p-adic L-functions in §2 developed in Serre [Ser73] and Deligne and Ribet [DR80], which is rooted in an idea that goes back to Hecke [Hec24] and Siegel [Sie68]. It should be noted that an alternative approach towards p-adic L-functions of Barsky and Cassou-Noguès [Bar78,CN79] based on the explicit formula for zeta values of Shintani [Shi76] was recently used to develop an algorithm for their computation by Roblot [Rob15]. Instead, here we take an approach using diagonal restrictions of Eisenstein series and p-adic interpolation, similar to that of Cohen [Coh76] and Cartier and Roy [CR72].…”