模2的格罗斯向量与素导体的椭圆曲线

中文摘要 AI 辅助

设p>3为素数，S_p表示特征p下j不变量属于F_p的超奇异椭圆曲线的几何同构类集合。对每个负基本判别式-D，若p在Q(√(-D))中是惰性的，则令m_i(D)（i∈S_p）为对应格罗斯（Gross）向量的整数系数。我们证明向量(m_i(D) mod 2)_{i∈S_p}张成F_2^{S_p}。关键步骤是将格罗斯三元格的表示数的奇偶性，归约为与弗罗贝尼乌斯（Frobenius）垂直的秩二子格的表示问题。利用 Ibukiyama 的显式极大序，得到的本原二元型可与Xiao–Zhou–Deng–Qu对F_p上超奇异椭圆曲线的参数化中出现的二元型等同。随后，类域论与切博塔廖夫（Chebotarev）定理可分离出单个超奇异坐标。由此，若E/Q具有素导体p且 Mordell–Weil 秩为正，则其以S_p为索引的布兰特（Brandt）特征向量的每个系数均为偶数，证明了 Kazalicki 和 Kohen 的一个猜想。因此，有理超奇异类处的奇系数是秩为0的代数证明。结合该奇偶性定理与 Mestre 及 Gross–Kudla 的公式，我们还证明：每个具有素导体且根数为+1的正秩椭圆曲线的模次数均被4整除。进而，Watkins 猜想对所有此类秩为2的曲线成立。

英文摘要

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.

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18 pages