An introduction to p-adic L-functions — Lean blueprint

1.1. Part I — The Kubota–Leopoldt p-adic L-function🔗

The goal of Part I is a single analytic object \zeta_p and a single interpolation theorem.

  • What a p-adic L-function should be (§2). The Riemann zeta function \zeta(s) has rational special values \zeta(1-n) = -B_n/n. After removing the Euler factor at p, the values (1 - p^{n-1})\zeta(1-n) satisfy Kummer's congruences modulo powers of p, which is exactly the statement that they extend to a continuous (indeed analytic) function on \Zp — the p-adic L-function. This chapter fixes the interpolation property we are aiming for.

  • Measures and the Iwasawa algebra (§3). A p-adic measure on \Zp is a bounded \Cp-linear functional on continuous functions; the Mahler transform identifies measures with power series, giving the ring isomorphism \Lam := \Zp[[\Zpx]] \cong \Zp[[T]] (the Iwasawa algebra). This is the foundational chapter: measures, the Iwasawa algebra, Mahler transforms, the measure-theoretic toolbox, pseudo-measures, and locally analytic distributions.

  • Construction of \zeta_p (§4). Following Kubota–Leopoldt, one writes down, for a coprime to p, an explicit measure \mu_a on \Zp, restricts it to \Zpx, and rescales away the dependence on a to obtain the pseudo-measure whose Mellin transform is the Kubota–Leopoldt p-adic L-function.

  • Interpolation at Dirichlet characters (§5). Integrating \zeta_p against a Dirichlet character \chi of p-power conductor — and then against tame conductors — recovers, up to explicit Euler and Gauss-sum factors, the special values L(\chi, 1-n) of the complex Dirichlet L-function. The Mellin transform turns measures on \Zpx into analytic functions on \Zp.

  • The value at s=1 (§6) and the residue at s=1 (§7). The p-adic analytic class number formula: \zeta_p has a simple pole at s=1 whose residue is the p-adic analogue of 1 - 1/p, mirroring the complex residue of \zeta.

  • The p-adic family of Eisenstein series (§8). The constant terms of a p-adic family of Eisenstein series are values of \zeta_p, foreshadowing the modular-forms picture of Part II.