# Harer-Zagier generating functions, the Redfield-Polya cycle index and Cohen semilinear congruences

## TL;DR

This paper explores the connection between Harer-Zagier generating functions, necklace polynomials, and Cohen semilinear congruences, revealing how Taylor expansions relate to solutions of these congruences in the context of moduli spaces.

## Contribution

It establishes a novel link between generating functions for moduli space Euler characteristics and solutions to Cohen semilinear congruences, advancing understanding in algebraic topology and number theory.

## Key findings

- Taylor expansions depend on solutions of Cohen semilinear congruences
- Harer-Zagier generating functions involve n-necklace polynomials
- New connections between algebraic topology and number theory

## Abstract

Harer-Zagier generating functions for Euler characteristics of moduli spaces of curves contain $n$-necklace polynomials. Taylor expansions for these polynomials depend on numbers of solutions of Cohen semilinear congruences.