Gross vectors modulo 2 and elliptic curves of prime conductor

Abstract

Let p > 3 be a prime, and let S_p denote the geometric isomorphism classes of supersingular elliptic curves in characteristic p whose j-invariants lie in F_p. For each negative fundamental discriminant -D for which p is inert in Q(sqrt(-D)), let m_i(D), i in S_p, be the integral coefficients of the corresponding Gross vector. We prove that the vectors (m_i(D) mod 2)_{i in S_p} span F_2^{S_p}. The key step reduces the parity of the representation numbers of Gross's ternary lattices to representation by rank-two sublattices perpendicular to Frobenius. Using Ibukiyama's explicit maximal orders, the resulting primitive binary forms are identified with those occurring in the Xiao--Zhou--Deng--Qu parametrization of supersingular elliptic curves over F_p. Class field theory and Chebotarev's theorem then allow the individual supersingular coordinates to be isolated. As a consequence, if E/Q has prime conductor p and positive Mordell--Weil rank, then every coefficient of its Brandt eigenvector indexed by S_p is even, proving a conjecture of Kazalicki and Kohen. Thus an odd coefficient at a rational supersingular class is an algebraic certificate of rank 0. Combining this parity theorem with formulas of Mestre and Gross--Kudla, we also prove that the modular degree of every positive-rank elliptic curve of prime conductor and root number +1 is divisible by 4. Consequently, Watkins' conjecture holds for all such curves of rank 2.

Disclosure

“pean Union through the European Regional Devel- opment Fund – Competitiveness and Cohesion Programme 2021–2027. During the development of this work, the authors used Aleph, a mathematical research as- sistant developed by Cantab Pi, and ChatGPT (OpenAI) for literature searches, exploratory work on proof strategies, and assistance with drafting and editorial review. The authors indepen- dently checked all mathematical statements, proofs, computations, and citations and take full r”

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2026-08-23T05:13:47.716829Z

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