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A note on Shintani's invariants

Bora Yalkinoglu

TL;DR

The paper tackles the problem of expressing Shintani's invariants $X(f)$, originally defined via the double sine function, in terms of $q$-Pochhammer symbols for real quadratic fields. Building on Yamamoto's observation and using a discretized modular-geodesic framework guided by minus continued fractions and Chebyshev polynomials, the authors derive a main theorem that rewrites $X(f)$ as a limit of $q$-Pochhammer ratios with a single base parameter $q$. They provide explicit decomposition data for principal ideals, obtain recursion formulas for the cone data $(x_k,y_k)$, and illustrate with concrete numerical examples that recover known Shintani values without relying on the double sine directly. A key contribution is the reduction to a single-$q$ parameter, which clarifies the archimedean perspective on Shintani invariants and sets the stage for potential connections with $p$-adic approaches to Hilbert's 12th problem. The work thus advances a unified $q$-analytic formulation of Shintani invariants for real quadratic fields and clarifies the structural role of modular-geodesic discretization in their computation.

Abstract

Shintani's celebrated invariants are conjectured to generate abelian extensions of real quadratic number fields, offering a potential solution to Hilbert's 12th problem in that setting. In this note, we derive new expressions for Shintani's invariants by generalizing an observation of Yamamoto, who showed that these invariants - originally formulated using the double sine function - can be expressed in terms of the q-Pochhammer symbol.

A note on Shintani's invariants

TL;DR

The paper tackles the problem of expressing Shintani's invariants , originally defined via the double sine function, in terms of -Pochhammer symbols for real quadratic fields. Building on Yamamoto's observation and using a discretized modular-geodesic framework guided by minus continued fractions and Chebyshev polynomials, the authors derive a main theorem that rewrites as a limit of -Pochhammer ratios with a single base parameter . They provide explicit decomposition data for principal ideals, obtain recursion formulas for the cone data , and illustrate with concrete numerical examples that recover known Shintani values without relying on the double sine directly. A key contribution is the reduction to a single- parameter, which clarifies the archimedean perspective on Shintani invariants and sets the stage for potential connections with -adic approaches to Hilbert's 12th problem. The work thus advances a unified -analytic formulation of Shintani invariants for real quadratic fields and clarifies the structural role of modular-geodesic discretization in their computation.

Abstract

Shintani's celebrated invariants are conjectured to generate abelian extensions of real quadratic number fields, offering a potential solution to Hilbert's 12th problem in that setting. In this note, we derive new expressions for Shintani's invariants by generalizing an observation of Yamamoto, who showed that these invariants - originally formulated using the double sine function - can be expressed in terms of the q-Pochhammer symbol.
Paper Structure (13 sections, 16 theorems, 94 equations)

This paper contains 13 sections, 16 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Shintani's invariant
  4. Shintani's invariant
  5. Explicit decomposition data for principal ideals
  6. Examples
  7. The $q$-Pochhammer function and Shintani's invariant
  8. Results of Shintani and Yamamoto
  9. Modular geodesics
  10. A new formula for Shintani's invariants
  11. Examples
  12. Chebyshev polynomials of the first and second kind
  13. A result about $g(p)$

Key Result

Theorem 1.1

Let $K=\mathbb Q(\sqrt{d})$ be a real quadratic number field with positive fundamental unit $\varepsilon = \frac{a+b\sqrt{d}}{2} \in \mathcal{O}_{K,+}^\times$, with $a,b \in \mathbb N$, such that $\langle 1,\varepsilon \rangle_\mathbb Z = \mathcal{O} _K$. Let $\mathfrak f = (u+v\sqrt{d}) \in I_K$ be where is the $q$-Pochhammer symbol and $\tau_n = \frac{T_{n+1}(a)+i b \sqrt{d}}{T_n(a)}$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.1
  • Remark 2.3
  • ...and 33 more