Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Shintani--Faddeev modular cocycle: Stark units from $q$-Pochhammer ratios

Gene S. Kopp

Abstract

We give a new interpretation of Stark units associated to real quadratic fields as real multiplication values of a modular cocycle. The cocycle of interest is a meromorphic factor describing the modular transformations of the $q$-Pochhammer symbol and is related to the Shintani--Barnes double sine function and the Faddeev quantum dilogarithm. We prove a refinement of Shintani's Kronecker limit formula that relates square roots of Stark class invariants to real multiplication values of the cocycle, which are cohomological invariants.

The Shintani--Faddeev modular cocycle: Stark units from $q$-Pochhammer ratios

Abstract

We give a new interpretation of Stark units associated to real quadratic fields as real multiplication values of a modular cocycle. The cocycle of interest is a meromorphic factor describing the modular transformations of the -Pochhammer symbol and is related to the Shintani--Barnes double sine function and the Faddeev quantum dilogarithm. We prove a refinement of Shintani's Kronecker limit formula that relates square roots of Stark class invariants to real multiplication values of the cocycle, which are cohomological invariants.

Paper Structure

This paper contains 62 sections, 84 theorems, 427 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Shintani--Faddeev Jacobi and modular cocycles
  3. Partial zeta functions and the Stark conjectures
  4. Eta-multiplier and theta-multiplier characters
  5. Main result: a limit formula
  6. Conditional results and conjectures: algebraicity
  7. Prior work
  8. Applications and future work
  9. Structure of this paper
  10. List of notation
  11. Preliminaries on $q$-Pochhammer symbols and half-integral weight modular forms
  12. $\mathrm{SL}_2$ and the standard symplectic form
  13. Fractional linear transformations
  14. The covering groups $\widetilde{\mathrm{SL}_2(\mathbb{R})}$ and $\mathrm{Mp}_2(\mathbb{R})$
  15. The $q$-Pochhammer symbol and variants
  16. ...and 47 more sections

Key Result

Theorem 1.1

Let $\mathcal{O}$ be an order in a real quadratic field $F \subset \mathbb{R}$, with Galois conjugation map $x \mapsto x'$, and let $\mathfrak{m}$ be a nonzero $\mathcal{O}$-ideal. Let $\mathfrak{A} \in \overline{\mathop{\mathrm{Clm}}\nolimits}^\flat_{\mathfrak{m}\infty_2}(\mathcal{O}) \setminus \ov with $A \left(\right) = \lambda \left(\right)$ for $\lambda > 1$. Let $n = \frac{2}{\left| \phi^{-

Figures (1)

  • Figure 1: The top plot compares $y=\left| \varpi_{\left(04/5\right) }\!\left(\sqrt{3}+i e^{-6\log(2+\sqrt{3})t}\right) \right|$ and $y=\mu^t$ for $\mu = e^{-\frac{7\pi i}{20}}\sqrt{\nu}$ and $\nu$ as in \ref{['eq:nunumber']}. The middle and bottom plots show graphs of $y=\left| \mu^{-t} \,\varpi_{\left(04/5\right) }\!\left(\sqrt{3}+i e^{-6\log(2+\sqrt{3})t}\right) \right|$ and $y=\arg\!\left(\mu^{-t} \,\varpi_{\left(04/5\right) }\!\left(\sqrt{3}+i e^{-6\log(2+\sqrt{3})t}\right)\right)$, respectively.

Theorems & Definitions (215)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Theorem 1.3
  • Conjecture 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 205 more