René Schoof

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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 subgroup T,(4)cSL,(Z). The result is obtained from a description of the weight distributions of the dual codes in terms of class numbers(More)
The class numbers h+l of the real cyclotomic fields Q(ζl + ζ −1 l ) are notoriously hard to compute. Indeed, the number h+l 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 + ζ −1 l ) for the primes l < 10, 000. It is quite likely that these(More)
Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of the set of reduced Arakelov divisors. As an application we(More)
A smooth, projective, absolutely irreducible curve of genus 19 over F2 admitting an infinite S-class field tower is presented. Here S is a set of four F2-rational points on the curve. This is shown to imply that A(2) = limsup#X(F2)/g(X) ≥ 4/(19 − 1) ≈ 0.222. Here the limit is taken over curves X over F2 of genus g(X)→∞.