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Primes dividing invariants of CM Picard curves. (English) Zbl 07184219
Summary: We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based, not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.

MSC:
14H45 Special algebraic curves and curves of low genus
14K22 Complex multiplication and abelian varieties
11H06 Lattices and convex bodies (number-theoretic aspects)
14G50 Applications to coding theory and cryptography of arithmetic geometry
14H40 Jacobians, Prym varieties
14Q05 Computational aspects of algebraic curves
Software:
genus3; SageMath
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