Maximal curves over finite fields and a modular isogeny

TL;DR

This work constructs curves of genus $7$ and $12$ over $\mathbb F_{11^5}$ that attain the Hasse-Weil-Serre bound, by leveraging quotients of modular curves of Borel-Cartan type and a Chen-type isogeny decomposition. The authors generalize isogeny relations to the full family of Atkin-Lehner quotients, enabling precise point-counts via Hecke eigenforms and newforms data, and they implement a DLMS23-based algorithm to obtain explicit equations. They produce two explicit maximal curves with equations: a canonical genus $7$ model for $X(6,7)/\langle W_6,W_7\rangle$ and a genus $12$ model for $X(156,1)/\langle W_{13}\rangle$, both reaching the Serre bound over $\mathbb F_{11^5}$. In addition, the paper reports 36 curves with improved lower bounds for $\#X(\mathbb F_q)$ across various $(g,q)$, and documents many other maximal quotients, including non-isomorphic examples with the same real Weil polynomial. The methods and data advance explicit constructions of high-point curves, with potential applications to coding theory and finite-field arithmetic.

Abstract

We prove the existence of curves of genus $7$ and $12$ over the field with $11^5$ elements, reaching the Hasse-Weil-Serre upper bound. These curves are quotients of modular curves and we give explicit equations. We compute the number of points of many quotient modular curves in the same family without providing equations. For various pairs (genus, finite field) we find new records for the largest known number of points. In other instances we find quotient modular curves that are maximal, matching already known results. To perform these computations, we provide a generalization of Chen's isogeny result.

Theorems & Definitions (17)

• Theorem 2
• proof
• Example 5
• Example 6
• Example 8
• Example 9
• Lemma A.2
• proof
• Lemma A.4
• proof