[Paper Review] On the paramodularity of typical abelian surfaces (and reduction of G-covariant bilinear forms)

Armand Brumer, Ariel Pacetti+4 more|arXiv (Cornell University)|May 28, 2018

Advanced Algebra and Geometry51 references17 citations

TL;DR

This paper establishes the first known case of paramodularity for a typical abelian surface over Q—specifically, the Jacobian of a genus 2 curve of conductor 277—by rigorously verifying that its L-function matches that of a non-Gritsenko lift Siegel paramodular form of weight 2 and level 277. Using an extended Faltings–Serre method, the authors compute Hecke eigenvalues via specialization of Siegel modular forms to modular curves and prove Galois representation equivalence modulo 2 and for a finite set of primes, completing the modularity verification for this case and extending it to infinitely many twists.

ABSTRACT

Generalizing the method of Faltings-Serre, we rigorously verify that certain abelian surfaces without extra endomorphisms are paramodular. To compute the required Hecke eigenvalues, we develop a method of specialization of Siegel paramodular forms to modular curves. In the appendix, Serre proves a result extending his work on the reduction of G-invariant bilinear forms modulo primes to the case of G-covariant forms.

Motivation & Objective

• To prove the paramodular conjecture for a typical abelian surface over Q with minimal endomorphisms (End(A) = Z), specifically for conductor 277.
• To develop a method for computing Hecke eigenvalues of Siegel paramodular forms by specializing them to modular curves.
• To extend the Faltings–Serre method to verify Galois representation equivalence between the 2-adic Tate module of an abelian surface and the Galois representation attached to a Siegel modular form.
• To establish paramodularity for two additional isogeny classes of abelian surfaces of conductors 353 and 587, and to show that infinitely many quadratic twists are also paramodular.

Proposed method

• Apply an extended Faltings–Serre method to compare 2-adic Galois representations attached to the abelian surface and a Siegel paramodular form.
• Verify mod 2 Galois equivalence by computing the residual image of the 2-torsion field of the abelian surface and matching it with the mod 2 reduction of the Hecke eigenvalues of the Siegel form.
• Specialize the Siegel paramodular form to modular curves to compute its Hecke eigenvalues efficiently, using rational functions in Gritsenko lifts.
• Use group-theoretic and Galois-theoretic analysis of GSp4(F2) ≃ S6 to classify possible Galois images and eliminate spurious candidates.
• Compute traces of Frobenius for a finite, effectively computable set of primes (e.g., p ≤ 43 for N = 277) to confirm representation equivalence.
• Leverage Serre’s appendix on reduction of G-covariant bilinear forms to ensure the existence of a compatible residual form on the semisimplification of the mod 2 representation.

Experimental results

Research questions

• RQ1 Does the Jacobian of the genus 2 curve y² + (x³ + x² + x + 1)y = −x² − x over Q of conductor 277 satisfy the paramodular conjecture?
• RQ2 Can Hecke eigenvalues of Siegel paramodular forms be computed effectively via specialization to modular curves?
• RQ3 Is the mod 2 Galois representation of a typical abelian surface isomorphic to that of a non-Gritsenko lift Siegel modular form of the same level and weight?
• RQ4 Can the Faltings–Serre method be extended to prove modularity for abelian surfaces without extra endomorphisms?
• RQ5 Are infinitely many quadratic twists of the abelian surface of conductor 277 also paramodular?

Key findings

• The L-function of the Jacobian A of the genus 2 curve y² + (x³ + x² + x + 1)y = −x² − x over Q of conductor 277 matches the spin L-function of the unique non-Gritsenko lift Siegel paramodular form of weight 2 and level 277.
• The mod 2 Galois representations of A and the Siegel form are isomorphic and absolutely irreducible, with image isomorphic to S5(b) inside GSp4(F2) ≃ S6.
• For conductor 277, equality of traces of Frobenius holds for all primes p ≤ 43, confirming Galois representation equivalence.
• The method successfully verifies paramodularity for two additional isogeny classes: conductors 353 and 587, with trace matching verified for primes up to 41 and 43 respectively.
• The paper establishes paramodularity for infinitely many quadratic twists of the conductor 277 surface, using local compatibility of L-functions under twisting.
• The appendix by Serre generalizes the reduction of G-invariant forms to G-covariant forms, proving that ε-covariant nondegenerate bilinear forms on K-vector spaces descend to nondegenerate ε-covariant forms on the semisimplification of the mod π reduction.