Computing p-adic L-functions of totally real fields

Abstract

We describe an algorithm for computing p-adic L-functions of characters of totally real elds. Such p-adic L-functions were constructed in the 1970’s independently by Barsky and CassouNoguès [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 Roblot1 [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. The idea of our method is simple. In Serre’s approach, the value of the p-adic L-function of a totally real eld of degree d at a non-positive integer 1 − k is interpreted as the constant term of a classical modular form of weight dk obtained by diagonally restricting a Hilbert Eisenstein series. For small values of k these constants can be computed easily using an idea of Siegel, which goes back to Hecke. To compute the p-adic L-function at arbitrary points in its domain, to some nite p-adic precision, we use a method for computing p-adically with modular forms in larger weight developed in [Lau11, Von15]. We compute the required constant term in very large weight indirectly, by nding su ciently many of its higher Fourier coe cients and using linear algebra to deduce the unknown constant term. Thus our approach is an algorithmic incarnation of Serre’s approach to p-adic L-functions of totally real elds [Ser73], obtaining p-adic congruences between the constant terms of modular forms by studying their higher Fourier coe cients.