Modeling λ‐invariants by p‐adic random matrices

Abstract

How often is the p-adic -invariant of an imaginary quadratic field equal to m? We model this by the statistics of random p-adic matrices, and test these predictions numerically.

On -invariants attached to cyclic cubic number fields

“…The techniques we exploit here are quite different from those employed in [1,4,6,16,17]. The key ingredient is to utilise the p-adic approximations developed by the first author in [2,3], which seem well suited to resolving problems of type (I) and (II) (see previous page), relatively quickly.…”

“…+ b 0 to a reasonable accuracy, which in turn requires us to compute the initial coefficients occurring in the Taylor series for F χ,β (X) about X = 0 (see [4,Proposition 5.3]). To date, the methods employed to work out these Taylor series coefficients have either d. delbourgo and q. chao involved expanding L p (s, χω 1+β ) as a series in s − 1 and then using some protracted linear algebra [1,6], or have involved computing the algebraic values ζ(1 − n, χ) then applying interpolation [4]. Here we propose an alternative approach.…”

“…Note that the approach of Ellenberg, Jain and Venkatesh [4,Proposition 5.3] implies that if one knows the coefficients c j (F χ,β ) of the power series F χ,β (X) up to an accuracy of p K+1−j , then one can compute the first few coefficients a 0 , a 1 , . .…”

“…Of course, the Iwasawa Main Conjecture (proved by Mazur and Wiles [13,19]) relates these zeros to the Λ-module structure of certain towers of ideal class groups. Recently Ellenberg, Jain and Venkatesh [4] have studied the connection between the way in which the number of zeros varies over a family of quadratic fields and the statistics predicted by the corresponding p-adic random matrices. One therefore expects that the techniques developed here can be used to study an analogous problem for a family of cubic twists instead.…”

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

“…There have already appeared many works on the computation of p-adic L-functions, starting with Iwasawa-Sims [20] in 1965 (although they are not explicitly mentioned in the paper) to the more recent computational study of their zeroes by Ernvall-Metsänkylä [16,17] in the mid-1990's and the current work of Ellenberg-Jain-Venkatesh [15] that provides a conjectural model for the behavior of the λ-invariant of p-adic L-functions in terms of properties of p-adic random matrices. However, most of these articles deal only with L-functions over Q or that can be written as a product of such L-functions.…”