Gross vectors modulo 2 and elliptic curves of prime conductor

Technical Summary: Gross Vectors Modulo 2 and Elliptic Curves of Prime Conductor

Problem Statement
The paper addresses the parity properties of the coefficients of Brandt eigenvectors associated with supersingular elliptic curves over finite fields $\mathbb{F}_p$. Specifically, let $p > 3$ be a prime and $S_p$ denote the set of geometric isomorphism classes of supersingular elliptic curves whose $j$-invariants lie in $\mathbb{F}_p$. For an elliptic curve $E/\mathbb{Q}$ of prime conductor $p$, the Jacquet–Langlands correspondence associates a primitive integral Brandt eigenvector $v_E = \sum c_i e_i$. The central problem is to determine the parity of the coefficients $c_i$ for $i \in S_p$ when $E$ has positive Mordell–Weil rank.

This inquiry stems from the study of divisor polynomials and a conjecture by Kazalicki and Kohen, which posits that if $E$ has positive rank and root number $+1$, then every coefficient $c_i$ indexed by a rational supersingular class ($i \in S_p$) must be even. While previous results established this under restrictive conditions (positive discriminant, no rational 2-torsion), the general case remained open. Furthermore, this parity question is linked to Watkins' conjecture regarding the divisibility of the modular degree $m_E$ by powers of 2 based on the rank of $E$.

Methodology
The authors employ a multi-step strategy combining arithmetic geometry, quaternion algebras, and class field theory:

1. Reduction to Gross Vectors: The problem is translated from the coefficients $c_i$ of the Brandt eigenvector to the parity of coefficients $m_i(D)$ of "Gross vectors" (CM vectors) associated with negative fundamental discriminants $-D$ where $p$ is inert. Using the orthogonality of Gross vectors with respect to the Brandt eigenvector of a positive-rank curve, the authors show that the vector of parities $(c_i \mod 2)_{i \in S_p}$ must lie in the orthogonal complement of the space spanned by the Gross parity rows.
2. Frobenius Reduction: A key technical step reduces the parity of the representation numbers of Gross's ternary lattices $L_i$ to the parity of representation numbers of rank-two sublattices $L_i^\perp$ perpendicular to the Frobenius endomorphism $\pi_i$. This reduction relies on the action of Frobenius conjugation, which groups vectors outside the perpendicular lattice into orbits of size four, rendering their contribution even modulo 2.
3. Identification with Binary Forms: Using Ibukiyama's explicit maximal orders in the quaternion algebra $B_{p,\infty}$, the authors identify the perpendicular lattices $L_i^\perp$ with specific binary quadratic forms. These forms fall into two branches based on their discriminants: $-16p$ and $-p$ (the latter occurring only when $p \equiv 3 \pmod 4$).
4. Class Field Theory and Chebotarev: The authors utilize the Xiao–Zhou–Deng–Qu parametrization to map supersingular classes to inverse orbits of form classes. They then apply Chebotarev's Density Theorem in the corresponding ring class fields to construct specific admissible discriminants (primes and semiprimes) that isolate individual supersingular coordinates. By choosing primes that split in specific ways within the class groups of quadratic orders, they demonstrate that the Gross parity rows can generate any standard basis vector in the space $\mathbb{F}_2^{S_p}$.

Key Contributions and Results

• The Gross-Row Spanning Theorem (Theorem 1.1): The primary result proves that the vectors of Gross coefficients modulo 2, indexed by $S_p$, span the entire vector space $\mathbb{F}_2^{S_p}$. This is achieved by explicitly constructing Gross rows that isolate each coordinate, handling exceptional $j$-invariants ($0$ and $1728$) separately and treating the $-16p$ and $-p$ discriminant branches using distinct prime selection strategies.
• Resolution of the Kazalicki–Kohen Parity Conjecture (Corollary 1.2): As a direct consequence of the spanning theorem, the authors prove that if an elliptic curve $E/\mathbb{Q}$ has prime conductor $p$ and positive Mordell–Weil rank, then every coefficient $c_i$ of its Brandt eigenvector for $i \in S_p$ is even. This removes previous restrictions on the discriminant and the existence of rational 2-torsion points. The paper notes that this provides a one-sided criterion: an odd coefficient at a rational supersingular class certifies that the rank is zero.
• Divisibility of the Modular Degree (Corollary 1.3 and Theorem 6.1): Combining the parity result with Mestre's norm formula and the Gross–Kudla cubic identity, the authors prove that for any elliptic curve of prime conductor $p$, positive rank, and root number $+1$, the modular degree $m_E$ is divisible by 4.
• Verification of Watkins' Conjecture for Rank 2: The result confirms Watkins' conjecture ($2^{\text{rank}} \mid m_E$) for all elliptic curves of prime conductor and rank 2 with root number $+1$.

Significance
The paper claims to settle a specific parity conjecture regarding Brandt eigenvectors, providing a robust algebraic certificate for rank 0 based on the parity of supersingular coefficients. By establishing the spanning property of Gross vectors modulo 2, the work bridges the gap between the arithmetic of supersingular curves and the analytic properties of modular forms (specifically the vanishing of $L$-functions at the central point). The application to Watkins' conjecture extends the known cases of this conjecture to all prime-conductor curves of rank 2 with root number $+1$, removing prior discriminant constraints. The methodology demonstrates the power of reducing ternary representation problems to binary forms via Frobenius symmetry and utilizing explicit maximal order models to control representation parities via class field theory.