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On the Bernoulli--Hurwitz periods

Luochen Zhao

TL;DR

This work develops a comprehensive $p$-adic framework for Bernoulli--Hurwitz numbers attached to CM elliptic curves with a split prime, constructing a $p$-adic measure $\mu_{\mathrm{BH}}$ from elliptic functions and proving that its periods are special values of a weight-one Eisenstein family. By connecting these periods to Katz’s $p$-adic Eisenstein measure, the authors obtain explicit interpolation formulas for the Bernoulli--Hurwitz $p$-adic zeta function $\zeta_p^{\mathrm{BH}}(s)$ across ordinary CM and non-CM elliptic curves, including a $p$-adic first limit formula. The strategy hinges on modularity of $q$-expansions and the appearance of weight-one Hasse-type invariants, enabling congruence-based propagation of period relations to special values while avoiding elliptic units. The results illuminate the interpolation behavior of $p$-adic zeta functions in the CM setting and suggest new connections to Gross-type factorizations and CM Chowla–Selberg-type formulas, with potential extensions to broader CM contexts via Katz’s Eisenstein machinery. Overall, the paper provides a modularity-driven, arithmetic interpolation framework for Bernoulli--Hurwitz measures and their $p$-adic zeta functions for ordinary elliptic curves.

Abstract

Let $E$ be an elliptic curve having CM by the ring of integers of an imaginary quadratic field $K$ in which $p$ splits. Following Lichtenbaum, the Bernoulli--Hurwitz numbers of $E$ (i.e., values of Eisenstein series evaluated at $E$ up to normalization) admit integral representations given by a $p$-adic measure constructed from an elliptic function. We show that the periods of this measure are in fact special values of a family of weight one Eisenstein series at the CM curve $E$ equipped with certain level data, and explicitly relate it to Katz's one-variable $p$-adic Eisenstein measure, whereby we derive period formulas of the Bernoulli--Hurwitz measure attached to any ordinary elliptic curve $\mathcal{E}$ defined over a local field. Moreover, by exploiting the modularity of these periods, and thanks to the existence of abundant weight one Hasse-type invariants, we present a novel approach to the interpolation property of the Bernoulli--Hurwitz $p$-adic zeta functions of the ordinary elliptic curve $\mathcal{E}$, and obtain a $p$-adic Kronecker's first limit formula.

On the Bernoulli--Hurwitz periods

TL;DR

This work develops a comprehensive -adic framework for Bernoulli--Hurwitz numbers attached to CM elliptic curves with a split prime, constructing a -adic measure from elliptic functions and proving that its periods are special values of a weight-one Eisenstein family. By connecting these periods to Katz’s -adic Eisenstein measure, the authors obtain explicit interpolation formulas for the Bernoulli--Hurwitz -adic zeta function across ordinary CM and non-CM elliptic curves, including a -adic first limit formula. The strategy hinges on modularity of -expansions and the appearance of weight-one Hasse-type invariants, enabling congruence-based propagation of period relations to special values while avoiding elliptic units. The results illuminate the interpolation behavior of -adic zeta functions in the CM setting and suggest new connections to Gross-type factorizations and CM Chowla–Selberg-type formulas, with potential extensions to broader CM contexts via Katz’s Eisenstein machinery. Overall, the paper provides a modularity-driven, arithmetic interpolation framework for Bernoulli--Hurwitz measures and their -adic zeta functions for ordinary elliptic curves.

Abstract

Let be an elliptic curve having CM by the ring of integers of an imaginary quadratic field in which splits. Following Lichtenbaum, the Bernoulli--Hurwitz numbers of (i.e., values of Eisenstein series evaluated at up to normalization) admit integral representations given by a -adic measure constructed from an elliptic function. We show that the periods of this measure are in fact special values of a family of weight one Eisenstein series at the CM curve equipped with certain level data, and explicitly relate it to Katz's one-variable -adic Eisenstein measure, whereby we derive period formulas of the Bernoulli--Hurwitz measure attached to any ordinary elliptic curve defined over a local field. Moreover, by exploiting the modularity of these periods, and thanks to the existence of abundant weight one Hasse-type invariants, we present a novel approach to the interpolation property of the Bernoulli--Hurwitz -adic zeta functions of the ordinary elliptic curve , and obtain a -adic Kronecker's first limit formula.
Paper Structure (30 sections, 43 theorems, 238 equations)

This paper contains 30 sections, 43 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization
  4. Notation
  5. Conventions of modular forms
  6. Acknowledgment
  7. Construction of the measure
  8. Some elliptic functions
  9. A change-of-variable diagram
  10. The measure $\mu_{c,N}$
  11. Computations of the periods
  12. Reduction to $\mathbf{C}/L$
  13. Reduction to $\mathbf{C}^\times/q^{\mathbf{Z}}$
  14. Explicit period formulas
  15. Modularity of the $q$-expansions
  16. ...and 15 more sections

Key Result

Theorem A

Let $n\in \mathbf{Z}_{\ge 0}$ and $a\in \mathbf{Z}_p$. Write $L = \Omega_\infty\mathfrak{a}$ for some $\Omega_\infty\in \mathbf{C}^\times$ and fractional ideal $\mathfrak{a}\subset K$ prime to $\mathfrak{p}$, with $\mathfrak{a} = \mathbf{Z}+\mathbf{Z}\varsigma$ for some $\mathrm{im}(\varsigma)>0$. where the right hand side is valued in $F_\mathfrak{P}(E[\bar{\mathfrak{p}}^n])$. Moreover, the con

Theorems & Definitions (93)

  • Theorem A: Theorem \ref{['thm:epf']}
  • Remark 1.1
  • Remark 1.2
  • Theorem B
  • Theorem C
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • proof
  • ...and 83 more