The locus of Riemann surfaces of genus $2(p-1)$ with $4p$ automorphisms

TL;DR

The paper classifies the locus $\\mathscr{M}_{g_p}^{4p}$ of genus $g_p=2(p-1)$ Riemann surfaces with automorphism group of order $4p$, proving it has complex dimension $1$ and consists of two 1-dimensional families plus a finite collection of quasiplatonic surfaces whose counts depend on $pmod 3$. It provides explicit models $X_p,Y_p,Z_p$ realizing all possibilities for maximal $4p$-order automorphism actions and establishes the full automorphism groups for these surfaces. It then derives isogeny decompositions of their Jacobians, including a Poincaré decomposition for $JX_p$, and exhibits CM/RM phenomena: $JX_p$ has complex multiplication by $Q(oldsymbol{ u_p})$, $JY_p$ real multiplication by $Q(oldsymbol{ u_p}+oldsymbol{ u_p}^{-1})$, and $JZ_p$ factors with CM by multiple fields. Collectively, these results connect automorphism-group data to explicit curve models, Jacobian decompositions, and special multiplication structures, enriching the understanding of loci with prescribed symmetries in moduli spaces.

Abstract

Let $p \geqslant 5$ be a prime number. In this article we provide a complete and explicit description of the locus formed by the compact Riemann surfaces of genus $2(p-1)$ that are endowed with a group of automorphisms of order $4p$. In addition, we provide isogeny decompositions of the corresponding Jacobian varieties and study if the most symmetric ones admit complex and real multiplication.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Proposition 1
• Theorem 3
• Corollary 1
• Theorem 4
• Corollary 2
• Theorem 5
• Theorem 6
• Corollary 3