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.
Dose, V., Maria Lido, G., Mercuri, P., Stirpe, C. (2026). Maximal curves over finite fields and a modular isogeny. FINITE FIELDS AND THEIR APPLICATIONS, 113 [10.1016/j.ffa.2026.102831].
Maximal curves over finite fields and a modular isogeny
Pietro Mercuri
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2026-08-01
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.
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Dose V., Lido G., Mercuri P., Stirpe C. - Maximal curves over finite fields and a modular isogeny (2026).pdf
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