An explicit height bound for the classical modular polynomial

For a prime m, let Phi_m be the classical modular polynomial, and let h(Phi_m) denote its logarithmic height. By specializing a theorem of Cohen, we prove that h(Phi_m) <= 6 m log m + 16 m + 14 sqrt m log m. As a corollary, we find that h(Phi_m) <= 6 m log m + 18 m also holds. A table of h(Phi_m) values is provided for m <= 3607.Comment: Minor correction to the constants in Theorem 1 and Corollary 9. To appear in the Ramanujan Journal. 17 pages

A p-adic algorithm to compute the Hilbert class polynomial

Abstract. Classicaly, the Hilbert class polynomial P ∆ ∈ Z[X] of an imaginary quadratic discriminant ∆ is computed using complex analytic techniques. In 2002, Couveignes and Henocq [5] suggested a p-adic algorithm to compute P∆. Unlike the complex analytic method, it does not suffer from problems caused by rounding errors. In this paper we complete the outline given in [5] and we prove that, if the Generalized Riemann Hypothesis holds true, the expected runtime of the p-adic algorithm is eO(|∆|). We illustrate the algorithm by computing the polynomial P−639 using a 643-adic algorithm. 1

AWARDS

Thesis: Constructing elliptic curves of prescribed order; advisor: P. Stevenhagen. available a

CONSTRUCTING SUPERSINGULAR ELLIPTIC CURVES

Abstract. We give an algorithm that constructs, on input of a prime power q and an integer t, a supersingular elliptic curve over Fq with trace of Frobenius t in case such a curve exists. If GRH holds true, the expected run time of our algorithm is eO((log q) 3). We illustrate the algorithm by showing how to construct supersingular curves of prime order. 1

Elliptic curves with a given number of points

We present a non-archimedean method to construct, given an integer N ≥ 1, a finite field Fq and an elliptic curve E/Fq such that E(Fq) has order N