René Schoof

Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p

In this paper we present a deterministic algorithm to compute the number of F^-points of an elliptic curve that is defined over a finite field Fv and which is given by a Weierstrass equation. The algorithm takes 0(log9 q) elementary operations. As an application wc give an algorithm to compute square roots mod p. For fixed .i e Z, it takes 0(log9p) elementary operations to compute fx mod p. 1. Introduction. In this paper we present an algorithm to compute the number of F(/-points of an elliptic curve defined over a finite field F , which is given by a Weierstrass equation. We restrict ourselves to the case where the characteristic of F^ is not 2 or 3. The algorithm is deterministic, does not depend on any unproved hypotheses and takes 0(log9 0. If one applies fast multiplication techniques, the algorithm will take 0((|x|1/2log p)6+f) elementary operations for any e > 0. Let £ be an elliptic curve defined over the prime field Fp and let an affine model of it be given by a Weierstrass equation Y2 = X3 + AX + B (A,BeFp). An explicit formula for the number of F^-points on £ is given by

Nonsingular plane cubic curves over finite fields

Abstract We determine the number of projectively inequivalent nonsingular plane cubic curves over a finite field F q with a fixed number of points defined over F q . We count these curves by counting elliptic curves over F q together with a rational point which is annihilated by 3, up to a certain equivalence relation.

Hecke operators and the weight distributions of certain codes

We obtain the weight distributions of the Melas and Zetterberg codes and the double error correcting quadratic Goppa codes in terms of the traces of certain Hecke operators acting on spaces of cusp forms for the congruence subgroupΓ1(4) ⊂ SL2(Z). The result is obtained from a description of the weight distributions of the dual codes in terms of class numbers of binary quadratic forms and a combination of the Eichler Selberg Trace Formula with the MacWilliams identities.

Families of curves and weight distributions of codes

In this expository paper we show how one can, in a uniform way, calculate the weight distributions of some well-known binary cyclic codes. The codes are related to certain families of curves, and the weight distributions are related to the distribution of the number of rational points on the curves.

Algebraic curves over F2 with many rational points

Abstract A smooth, projective, absolutely irreducible curve of genus 19 over F 2 admitting an infinite S -class field tower is presented. Here S is a set of four F 2 -rational points on the curve. This is shown to imply that A (2) = limsup # X ( F 2 )/ g(X) ≥ 4/(19 − 1) ≈ 0.222. Here the limit is taken over curves X over F 2 of genus g ( X ) → ∞.

Effectivity of Arakelov divisors and the theta divisor of a number field

We introduce the notion of an effective Arakelov divisor for a number field and the arithmetical analogue of the dimension of the space of sections of a line bundle. We study the analogue of the theta divisor for a number field.

Weight formulas for ternary Melas codes

In this paper we derive a formula for the frequencies of the weights in ternary Melas codes and we illustrate this formula by computing a table of examples.

Class numbers of real cyclotomic fields of prime conductor

The class numbers hl+ of the real cyclotomic fields Q(ζl + + ζl+-1) are notoriously hard to compute. Indeed, the number hl+ is not known for a single prime l ≥ 71. In this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic fields Q(ζl + ζl-1) for the primes l > 10,000. It is quite likely that these subgroups are in fact equal to the class groups themselves, but there is at present no hope of proving this rigorously. In the last section of the paper we argue -on the basis of the Cohen-Lenstra heuristics-that the probability that our table is actually a table of class numbers hl+, is at least 98%.

The exponents of the groups of points on the reductions of an elliptic curve

Let E denote an elliptic curve over Q without complex multiplication. It is shown that the exponents of the groups E(F p ) grow at least as fast as \( \frac{{\sqrt {P} \log \;p}}{{{{(\log \;\log \;p)}^2}}} \).

Minus class groups of the fields of the l -th roots of unity

We show that for any prime number l > 2 the minus class group of the field of the l-th roots of unity Q p (ζl) admits a finite free resolution of length 1 as a module over the ring Z[G]/(1 + i). Here i denotes complex conjugation in G = Gal(Q p (ζl)/Q p ) ≅ (Z/lZ)*. Moreover, for the primes l < 509 we show that the minus class group is cyclic as a module over this ring. For these primes we also determine the structure of the minus class group.