An algorithm based on Ribet's isogeny computes point counts for Shimura curves, yielding 116 record curves, automorphism results for 9288 of 10609 curves, and a tetragonal classification up to 32 exceptions each.
Maximal curves over finite fields and a modular isogeny
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
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.
fields
math.NT 1years
2025 1verdicts
CONDITIONAL 1representative citing papers
citing papers explorer
-
Point counts, automorphisms, and gonalities of Shimura curves
An algorithm based on Ribet's isogeny computes point counts for Shimura curves, yielding 116 record curves, automorphism results for 9288 of 10609 curves, and a tetragonal classification up to 32 exceptions each.