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Genus theory of p-adic pseudo-measures -- Tame kernels and abelian p-ramification

Georges Gras

TL;DR

The paper develops a genus-theoretic framework for $p$-adic pseudo-measures $L_g$ to connect $p$-adic $L$-values with arithmetic invariants of real abelian fields, notably tame kernels $K_2({\mathbb Z}_K)$ and abelian $p$-ramification groups ${\mathcal T}_{K,p}$. It uses Stickelberger elements, their twists, and Mellin transforms to build $L_g$-adic measures whose specializations yield annihilation results and valuations linking $K_2$-groups to $L_p$-values, via Birch– Tate-type formulas and reflection theorems. The key contributions include explicit results for $p=2,3$ (Theorem A), a proof of the Deng–Li conjecture for a real quadratic family (Theorem B), and a general lower bound for $\mathrm{rk}_pK_2({\mathbb Z}_K)$ in cyclic $p$-extensions with $p\ge 5$ (Theorem C). The work provides computational methods (PARI/GP) and tables, illustrating deep links between $p$-adic analytic data and concrete arithmetic invariants, with potential extensions to higher $K$-groups and broader base fields.

Abstract

We consider, for real abelian fields K, the Birch--Tate formula linking the tame kernel \#K\_2(Z\_K) to $ζ$\_K(-1); we compare, for quadratic and cyclic cubic fields with p=2,3, \#K\_2(\BZ\_K)[p^$\infty$] to the order of the torsion group T\_{K, p} of abelian p-ramification theory given by the residue of $ζ$\_{K, p}(s) at s=1. This is done via the ``genus theory'' of p-adic pseudo-measures, inaugurated in the 1970/80's and the fact that T\_{K, p} only depends on the p-class group and on the normalized p-adic regulator of K (Theorem A). We apply this to prove a conjecture of Deng--Li giving the structures of K\_2(Z\_K)[2^$\infty$] for an interesting family of real quadratic fields (Theorem B). Then, for p>3, we give a lower bound of the p-rank of K\_2(\BZ\_K) in cyclic p-extensions (Theorem C). Complements, PARI programs and tables are given in an Appendix.

Genus theory of p-adic pseudo-measures -- Tame kernels and abelian p-ramification

TL;DR

The paper develops a genus-theoretic framework for -adic pseudo-measures to connect -adic -values with arithmetic invariants of real abelian fields, notably tame kernels and abelian -ramification groups . It uses Stickelberger elements, their twists, and Mellin transforms to build -adic measures whose specializations yield annihilation results and valuations linking -groups to -values, via Birch– Tate-type formulas and reflection theorems. The key contributions include explicit results for (Theorem A), a proof of the Deng–Li conjecture for a real quadratic family (Theorem B), and a general lower bound for in cyclic -extensions with (Theorem C). The work provides computational methods (PARI/GP) and tables, illustrating deep links between -adic analytic data and concrete arithmetic invariants, with potential extensions to higher -groups and broader base fields.

Abstract

We consider, for real abelian fields K, the Birch--Tate formula linking the tame kernel \#K\_2(Z\_K) to \_K(-1); we compare, for quadratic and cyclic cubic fields with p=2,3, \#K\_2(\BZ\_K)[p^] to the order of the torsion group T\_{K, p} of abelian p-ramification theory given by the residue of \_{K, p}(s) at s=1. This is done via the ``genus theory'' of p-adic pseudo-measures, inaugurated in the 1970/80's and the fact that T\_{K, p} only depends on the p-class group and on the normalized p-adic regulator of K (Theorem A). We apply this to prove a conjecture of Deng--Li giving the structures of K\_2(Z\_K)[2^] for an interesting family of real quadratic fields (Theorem B). Then, for p>3, we give a lower bound of the p-rank of K\_2(\BZ\_K) in cyclic p-extensions (Theorem C). Complements, PARI programs and tables are given in an Appendix.
Paper Structure (51 sections, 15 theorems, 66 equations)

This paper contains 51 sections, 15 theorems, 66 equations.

Table of Contents

  1. Introduction and prerequisites
  2. Notations
  3. Overview of the method
  4. Torsion group of abelian Lg-ramification
  5. Tame and regular Hilbert's kernels
  6. First reflection theorem
  7. Second reflection theorem
  8. Arithmetic v.s. analytic properties
  9. Main results of the article
  10. Computation of Lg
  11. Birch-Tate formula and Lg-functions
  12. Formulas for quadratic and cubic fields
  13. Definition of a Lg-adic pseudo-measure Lg
  14. The Stickelberger elements
  15. Norms of the Stickelberger elements
  16. ...and 36 more sections

Key Result

Theorem 4.1

Let $c$ be any odd integer, prime to $p$ and to the conductor $f$ of $K$. For all $n$ large enough, let $f_n$ be the conductor of $L_n = K(\mu_{qp^n})$, and for all $a \in [1, f_n]$, prime to $f_n$, let $a'_{c}$ be the unique integer in $[1, f_n]$ such that $a'_{c} \cdot c-a = \lambda^n_a(c) f_n$,

Theorems & Definitions (24)

  • Remark 1.1
  • Theorem 4.1
  • Corollary 4.2
  • Corollary 4.3
  • Theorem 4.4
  • Theorem 4.5
  • Corollary 4.6
  • Theorem 4.7
  • Lemma 5.1
  • proof
  • ...and 14 more