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Tropical Weil's reciprocity law and Weil's pairing

Nikita Kalinin, Matthew Magin

TL;DR

This work introduces a tropical analogue of Weil's reciprocity law and uses it to derive a transparent combinatorial perspective on the classical reciprocity. It defines a tropical Weil symbol and proves a tropical reciprocity law on tropical curves, then constructs a tropical Weil pairing on divisors of degree zero via tropical harmonic functions. By relating the tropical pairing to a classical analogue involving real-normalized differentials, the paper illuminates connections between tropical geometry and classical complex/number-theoretic theories and suggests directions for detropicalization and extensions to,q quasi-meromorphic settings. Overall, the results provide a concrete, combinatorial framework for Weil-type reciprocity in the tropical setting and a natural pairing on degree-zero divisors with potential arithmetic interpretations.

Abstract

The Weil reciprocity law asserts that given two meromorphic functions $f, g$ on a compact complex curve, the product of the values of $f$ over the roots and poles of $g$ is equal to the product of the values of $g$ over the roots and poles of $f$. We state and prove a tropical version of this reciprocity; the tropical ideas lead to yet another transparent ``combinatorial'' proof of the classical Weil reciprocity law. Then, we construct a tropical Weil pairing on the set of divisors of degree zero.

Tropical Weil's reciprocity law and Weil's pairing

TL;DR

This work introduces a tropical analogue of Weil's reciprocity law and uses it to derive a transparent combinatorial perspective on the classical reciprocity. It defines a tropical Weil symbol and proves a tropical reciprocity law on tropical curves, then constructs a tropical Weil pairing on divisors of degree zero via tropical harmonic functions. By relating the tropical pairing to a classical analogue involving real-normalized differentials, the paper illuminates connections between tropical geometry and classical complex/number-theoretic theories and suggests directions for detropicalization and extensions to,q quasi-meromorphic settings. Overall, the results provide a concrete, combinatorial framework for Weil-type reciprocity in the tropical setting and a natural pairing on degree-zero divisors with potential arithmetic interpretations.

Abstract

The Weil reciprocity law asserts that given two meromorphic functions on a compact complex curve, the product of the values of over the roots and poles of is equal to the product of the values of over the roots and poles of . We state and prove a tropical version of this reciprocity; the tropical ideas lead to yet another transparent ``combinatorial'' proof of the classical Weil reciprocity law. Then, we construct a tropical Weil pairing on the set of divisors of degree zero.
Paper Structure (4 sections, 7 theorems, 26 equations, 2 figures)

This paper contains 4 sections, 7 theorems, 26 equations, 2 figures.

Table of Contents

  1. Weil's reciprocity law
  2. Tropical Weil's reciprocity law
  3. Combinatorial proof of Weil's reciprocity law
  4. Tropical Weil pairing

Key Result

Theorem 1

Let $f, g$ be non-zero meromorphic functions on the compact Riemann surface $S$ with no common zeros or poles. Then

Figures (2)

  • Figure 1: A Riemann surface $S$ shrinks to the tropical curve $\Gamma$.
  • Figure 2: Cutting the Riemann surface $S$ into the cylinders.

Theorems & Definitions (20)

  • Theorem 1: weil2009oeuvres
  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Theorem 2: Tropical Weil's reciprocity law
  • proof
  • Definition 4
  • Theorem 3: Tropical Weil's reciprocity law
  • Proposition 1
  • ...and 10 more