Computing Riemann–Roch Spaces in Algebraic Function Fields and Related Topics

Abstract

We develop a simple and efficient algorithm to compute Riemann-Roch spaces of divisors in general algebraic function fields which does not use the Brill-Noether method of adjoints or any series expansions. The basic idea also leads to an elementary proof of the Riemann-Roch theorem. We describe the connection to the geometry of numbers of algebraic function fields and develop a notion and algorithm for divisor reduction. An important application is to compute in the divisor class group of an algebraic function field.

Computing functions on Jacobians and their quotients

“…This is classically achieved by blowing up or using series expansions, but none of these methods is fully satisfactory: the complexity of blowing up is not well understood in the worst cases, and computing series expansions is only possible when the characteristic of K is zero or large enough. Work by Hess [17], using general normalization algorithms, provides a satisfactory algorithm in general. Possible references for these algorithms are Hess [17], Makdisi [19], Diem [11], or the quick account at the beginning of [9].…”

“…Work by Hess [17], using general normalization algorithms, provides a satisfactory algorithm in general. Possible references for these algorithms are Hess [17], Makdisi [19], Diem [11], or the quick account at the beginning of [9].…”

“…After base change toK it becomes linearly equivalent to a translate of lW −(g−1)o . Since every translate of W −(g−1)o is ample [29, Chapter II, § 6] and l 3 we deduce [29,Chapter III,§ 17] that E is very ample. We now apply Lemma 1 to the hyperplane section E. Its dimension is n = g − 1 and its degree d is…”

The GHS Attack Revisited

“…Such a divisor decomposes uniquely into a sum of places of certain degrees and multiplicities just like the case of rational integers and prime factorizations, and smoothness probabilities hold. Computing these divisor class representatives can be done by reduction techniques as described in [14], and this leads also to a way of computing in the divisor class group of C 0 which generalizes the Cantor method for hyperelliptic curves. We remark that for hyperelliptic curves addition takes O(g for some constant c > 1, and we expect this to be essentially true for q = p because of possible alternating signs of the trace terms.…”

Reduced normal form of local integral bases

“…A relevant concept is that of reduced basis of a lattice with respect to the given length function. W. M. Schmidt used reduced bases of integral closures of certain subrings of function fields of curves over finite fields, as a crutial tool for the design of algorithms to compute bases of the Riemann-Roch spaces attached to divisors of the curve [12,13,8,2].…”