Tenth Algorithmic Number Theory Symposium ANTSX

On the evaluation of modular polynomials
Andrew V. Sutherland
Abstract: We present two algorithms that, given a prime l and an elliptic curve E/F_q, directly compute the polynomial Phi_l(j(E),Y) in F_q[Y] whose roots are the jinvariants of the elliptic curves that are lisogenous to E. We do not assume that the modular polynomial Phi_l(X,Y) is given. The algorithms may be adapted to handle other types of modular polynomials, and we consider applications to point counting and the computation of endomorphism rings. We demonstrate the practical efficiency of the algorithms by setting a new pointcounting record, modulo a prime q with more than 5,000 decimal digits, and by evaluating a modular polynomial of level l = 100,019.
Files available: paper (PDF), slides
© 201112 Kiran S. Kedlaya (with thanks to Pierrick Gaudry and Emmanuel ThomÃ©)
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