[Paper Review] Primes dividing invariants of CM Picard curves

Kilicer, P., Elisa Lorenzo García+1 more|arXiv (Cornell University)|Jan 1, 2019

Algebraic Geometry and Number Theory31 references10 citations

TL;DR

This paper provides a sharp, explicit bound on the primes dividing the denominators of invariants for genus-3 Picard curves with complex multiplication (CM), using a novel approach based on a specific type of good reduction rather than bad reduction. By analyzing the endomorphism structure of the Jacobian modulo such primes, the authors derive a small, computable set S of primes that can appear in denominators, enabling practical construction of CM Picard curves. The bound is significantly tighter than previous methods and is proven to be effective for explicit computations.

ABSTRACT

We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Our proof is simpler than the previous proofs for genus 2 and 3 and, unlike previous bounds for genus 3, our bounds are sharp enough for use in class polynomial computation.

Motivation & Objective

• To determine the primes that can divide the denominators of invariants for Picard curves of genus 3 with complex multiplication.
• To overcome the limitations of prior bounds based on bad reduction, which yield overly large and impractical estimates.
• To develop a method that yields a small, computable set S of primes such that all denominators are S-units, suitable for explicit construction of CM curves.
• To provide a conjectural bound on the exponents in the denominator, enabling full computation of class polynomials.

Proposed method

• Introduce a new set of absolute invariants j1, j2, j3 for Picard curves, which have better arithmetic properties than previous invariants.
• Use a refined reduction theory: instead of relying on bad reduction, focus on a specific type of good reduction where the Jacobian's endomorphism ring modulo p contains the full order of the CM field.
• Apply the theory of stable reduction and minimal regular models to analyze the special fiber of the Néron model, ensuring that the reduction type forces commutativity of endomorphism algebra elements.
• Use the embedding of the CM order into the endomorphism algebra of the reduced Jacobian to derive a contradiction for large primes, proving that only small primes can divide the denominators.
• Leverage the explicit structure of the reduction to prove commutativity of quaternion algebra elements directly, avoiding the need for asymptotic arguments.
• Implement the bounds in SageMath and provide numerical examples to validate the practicality of the method.

Experimental results

Research questions

• RQ1 Which primes can divide the denominators of invariants for genus-3 Picard curves with complex multiplication?
• RQ2 Can a bound on these primes be derived that is small and computable, suitable for explicit construction of CM curves?
• RQ3 Does focusing on a specific type of good reduction yield sharper bounds than traditional methods based on bad reduction?
• RQ4 Can the exponents in the denominator be bounded in a way that allows full computation of class polynomials?
• RQ5 Why are the denominators of the Koike-Weng invariants smaller than those of other invariants, despite lacking a theoretical bound?

Key findings

• The paper proves that the primes dividing the denominators of the invariants j1 and j2 are exactly the primes in a small, explicitly computable set S, which depends only on the CM field and the order.
• For the invariant j1, the denominator is an S-unit, and the set S is bounded by the primes of good reduction with a specific ramification and residue field structure.
• The bound is sharp enough to be used in explicit constructions of Picard curves with CM, unlike previous bounds which were too large.
• The authors provide a conjectural bound on the exponents in the denominator, which, if true, would allow full computation of class polynomials.
• Numerical examples show that the denominators of j1 and j2 are significantly smaller than those of discriminant-based invariants, and comparable to or smaller than those of the Koike-Weng invariants.
• The SageMath implementation of the bounds is publicly available, enabling practical use in cryptographic and arithmetic geometry applications.