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Regulators on some abelian coverings of $\mathbb{P}^1$ minus $n+2$ points

Yusuke Nemoto, Takuya Yamauchi

TL;DR

The paper develops explicit motivic cohomology elements on abelian coverings of $P^1$ minus points via Milnor $K_2$-symbols on generalized Fermat curves and their quotient superelliptic curves, and proves their regulators are non-trivial by expressing them through Appell-Lauricella hypergeometric functions. It then provides a concrete regulator computation leading to explicit series representations, and validates the Beilinson conjecture numerically for several low-genus cases over $Q$ and a quadratic extension, linking regulator values to derivatives of $L$-functions with precise rational factors. The work demonstrates how covering constructions and hypergeometric expressions yield computable, integral motivic elements with regulators that match predicted $L$-values, enhancing understanding of regulator maps in the Beilinson framework. Overall, the results offer new, verifiable instances of Beilinson's conjecture for families of curves and illustrate the practical utility of hypergeometric functions in regulator computations.

Abstract

In this paper, we construct certain rational or integral elements in the motivic cohomology of superelliptic curves which are quotient curves of abelian coverings of $\mathbb{P}^1$ minus $n+2$ points, and prove that these elements are non-trivial by expressing their regulators in terms of Appell-Lauricella hypergeometric functions. We also check that such elements are integral under a mild assumption. We also give various numerical examples for the Beilinson conjecture on special values of $L$-functions of the superelliptic curves by using hypergeometric expressions.

Regulators on some abelian coverings of $\mathbb{P}^1$ minus $n+2$ points

TL;DR

The paper develops explicit motivic cohomology elements on abelian coverings of minus points via Milnor -symbols on generalized Fermat curves and their quotient superelliptic curves, and proves their regulators are non-trivial by expressing them through Appell-Lauricella hypergeometric functions. It then provides a concrete regulator computation leading to explicit series representations, and validates the Beilinson conjecture numerically for several low-genus cases over and a quadratic extension, linking regulator values to derivatives of -functions with precise rational factors. The work demonstrates how covering constructions and hypergeometric expressions yield computable, integral motivic elements with regulators that match predicted -values, enhancing understanding of regulator maps in the Beilinson framework. Overall, the results offer new, verifiable instances of Beilinson's conjecture for families of curves and illustrate the practical utility of hypergeometric functions in regulator computations.

Abstract

In this paper, we construct certain rational or integral elements in the motivic cohomology of superelliptic curves which are quotient curves of abelian coverings of minus points, and prove that these elements are non-trivial by expressing their regulators in terms of Appell-Lauricella hypergeometric functions. We also check that such elements are integral under a mild assumption. We also give various numerical examples for the Beilinson conjecture on special values of -functions of the superelliptic curves by using hypergeometric expressions.
Paper Structure (11 sections, 9 theorems, 106 equations)

This paper contains 11 sections, 9 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preparation
  3. Appell-Lauricella hypergeometric functions
  4. Beilinson conjecture
  5. Elements in motivic cohomology
  6. Computation of regulators
  7. Numerical verification of the Beilinson conjecture
  8. The case of elliptic curves over $\mathbb{Q}$
  9. The case of a hyperelliptic curve of genus $2$ over $\mathbb{Q}$
  10. The case of a superelliptic curve of genus $3$ over $\mathbb{Q}$
  11. Beilinson conjecture over the quadratic field

Key Result

Theorem 1.1

(Theorem mot, Proposition intE, Corollary intX) Keep the notation as above. Assume all roots of the polynomial are roots of unity. Then, $\xi_{X_{N, n}}$ (resp. $\{1-Y, X\}$) is an element in the motivic rational cohomology $H^2_{\mathscr{M}} (X_{N, n}, \mathbb{Q}(2))\subset K_2^M(k_{X_{N,n}}(X_{N, n}))$ (resp. $H^2_{\mathscr{M}} (E_{N, n}, \mathbb{Q}(2))\subset K_2^M(k_{E_{N,n}}(E_{N, n}))$. F

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2: Theorem \ref{['main:1']}
  • Definition 2.1: cf. Slater
  • Proposition 2.2: KS
  • Proposition 2.3
  • Conjecture 2.4: Beilinson, DJZNekovar
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 6 more