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Katz type p-adic L-functions for primes p non-split in the CM field

Fabrizio Andreatta, Adrian Iovita

Abstract

For every triple F,K,p where F is a classical elliptic eigenform, K is a quadratic imaginary field and p> 3 is a prime integer which is not split in K, we attach a p-adic L function which interpolates the algebraic parts of the special values of the complex L-functions of F twisted by certain algebraic Hecke characters of K. This construction extends a classical construction of N. Katz, for F an Eisenstein series and of Bertolini-Darmon-Prasana, for F a cuspform, when p is split in K. Moreover we prove a Kronecker limit formula, respectively p-adic Gross-Zagier formulae for our newly defined p-adic L-functions.

Katz type p-adic L-functions for primes p non-split in the CM field

Abstract

For every triple F,K,p where F is a classical elliptic eigenform, K is a quadratic imaginary field and p> 3 is a prime integer which is not split in K, we attach a p-adic L function which interpolates the algebraic parts of the special values of the complex L-functions of F twisted by certain algebraic Hecke characters of K. This construction extends a classical construction of N. Katz, for F an Eisenstein series and of Bertolini-Darmon-Prasana, for F a cuspform, when p is split in K. Moreover we prove a Kronecker limit formula, respectively p-adic Gross-Zagier formulae for our newly defined p-adic L-functions.

Paper Structure

This paper contains 38 sections, 35 theorems, 160 equations.

Table of Contents

  1. Introduction.
  2. Classical $L$-values.
  3. Algebraic Hecke characters.
  4. Algebraic Hecke characters and their $p$-adic avatars.
  5. Classical values of $L(F, \chi, s)$.
  6. The case $F=E_{k,\epsilon}$ is an Eisenstein series.
  7. The case $F=f$ is a cuspform.
  8. Conclusions.
  9. $p$-adic families of Hecke characters.
  10. Preliminaries on CM-elliptic curves.
  11. Canonical subgroups for CM elliptic curves. The inert case.
  12. Canonical subgroups for CM elliptic curves. The ramified case.
  13. A technical lemma
  14. Adelic description of CM elliptic curves.
  15. Vector bundles with marked sections. The sheaves ${\mathbb W}_k$.
  16. ...and 23 more sections

Key Result

Lemma 2.2

The group ${{\cal H}}(c,{\mathfrak{N}})$ is finite and ${{\cal H}}(c,{\mathfrak{N}}) \cong I(c,{\mathfrak{N}}_c)/P(c,{\mathfrak{N}}_c)$. In particular, for $N=1$ we have ${{\cal H}}(c,1) \cong {\mathrm{Pic}}({\cal O}_c)$ the group of invertible fractional ${\cal O}_c$-ideals. For general $N$ the nat

Theorems & Definitions (91)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Definition 2.7
  • ...and 81 more