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Glasgow Mathematical Journal (1)

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A DIOPHANTINE FROBENIUS PROBLEM RELATED TO RIEMANN SURFACES
CORMAC O'SULLIVAN, ANTHONY WEAVER
Journal:

Glasgow Mathematical Journal / Volume 53 / Issue 3 / September 2011

Published online by Cambridge University Press:

10 March 2011, pp. 501-522

Print publication:

September 2011
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We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair (p, q), 2 < p < q, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of compact Riemann surfaces which can be realised as semi-regular pq-fold coverings of surfaces of lower genus. In this context, the Frobenius number is (up to an additive translation) the largest genus in which no surface is such a covering. In many cases it is also the largest genus in which no surface admits an automorphism of order pq. The general t-dimensional Frobenius problem (t ≥ 3) is NP-hard, and it may be that our restricted problem retains this property.