AI 中文总结
本文通过射影核计算椭圆双覆盖的高阶 Aharonov 不变量，将其分解为 Eisenstein 项与 Eisenstein–Kronecker 函数，并揭示其与 Hasse 不变量的联系及约化性质。
AI 中文摘要
我们通过计算每个椭圆双覆盖的射影核，确定了其至少二阶的 Aharonov 不变量。在至少三阶时，该公式将不变量分解为一个常数 Eisenstein 项与一个在乘以二后求值的 Eisenstein–Kronecker 函数。该公式确定了分歧主部分、挠点特化以及同源迹，包括同源核中二阶挠点的修正。它还通过每个二次投影实现了相同的内蕴 de Rham 张量：在更高阶时，这是 Eisenstein 截面的像，而二阶时则给出 Hodge 线的经典 Weierstrass 补集。一个独立的微分计算表明，阶为 $p-1$ 的 Bernoulli 归一化不变量约化为定义导子的第 $p$ 次迭代与第一次迭代的商。对于椭圆不变量导子，该商是 Hasse 不变量，与其可分轨迹上的有理函数无关。万有双覆盖提供了一个全局特化，在二阶挠点之外是整的，其约化在该轨迹上正则延拓。经典的 Eisenstein 零点定理和超奇异除子同余式随后描述了精确性轨迹及其约化。
英文摘要
We determine the Aharonov invariants of order at least two of every elliptic double cover by evaluating its projective kernel. In orders at least three, the formula separates the invariant into a constant Eisenstein term and an Eisenstein--Kronecker function evaluated under multiplication by two. The formula determines the ramification principal parts, torsion specializations, and isogeny traces, including the correction from two-torsion in an isogeny kernel. It also realizes the same intrinsic de Rham tensor through every degree-two projection: in higher orders this is the image of an Eisenstein section, while order two gives the classical Weierstrass complement to the Hodge line. A separate differential calculation shows that the Bernoulli-normalized invariant of order $p-1$ reduces to the quotient of the $p$th and first iterates of the defining derivation. For an elliptic invariant derivation this quotient is the Hasse invariant, independently of the rational function on its separable locus. The universal double cover provides a global specialization, integral away from two-torsion, whose reduction extends regularly across that locus. Classical Eisenstein zero theorems and the supersingular divisor congruence then describe the exactness loci and their reduction.
Comments23 pages
