Higher Schwarzians of Elliptic Double Covers and Eisenstein--Kronecker Functions

RESUMO

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.