, Let ? = 8? + 7 in the Siegel case, and ? = 4 Tr K/Q (?) + 7 in the Hilbert case. Then, given ? P and ? i(P ) at precision O(z ? )
, It is enough to recover the rational fractions s and p; afterwards, q and r can be deduced from the equation of C ? . First, assume that P is a Weierstrass point of C. Then s, p are invariant under the hyperelliptic involution. Therefore we have to recover univariate rational fractions in u of degree d ? 2? (resp. d ? Tr(?)). This can be done in quasi-linear time from their power series expansion up
, Since u has valuation 2 in z, we need to compute ? P at precision O(z 4d+1 )
, We now have to compute rational fractions of degree d ? 4? + 3 (resp. d ? 2 Tr(?) + 3)
, Compute at most 4 candidates for the tangent matrix of the isogeny ? using Proposition 4.26 in the Siegel case
, Choose a base point P on C such that ? P is of generic type, and compute the power series ? P and ? i(P ) up to precision O z 8?+7
Let U ? A 2 (k) be the open set consisting of abelian surfaces A such that Aut(A) ? {±1} ,
A), j(A ? ) be their Igusa invariants. Assume that A and A ? are ?-isogenous over k, and that the subvariety of A 3 × A 3 cut out by the modular equations ? ?,i for 1 ? i ? 3 is normal at (j(A), j(A ? )). Then, given j(A) and j(A ? ), Algorithm 6.1 succeeds and returns 1. a field extension k ? /k of degree dividing 8, 2 ,
, the rational representation (s, p, q, r) ? k ? (u, v) 4 of an ?-isogeny ? : Jac(C) ?
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