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Appendix to Dinesh S. Thakur's ‘Behaviour of Function Field Gauss Sums at Infinity’

Appendix to Dinesh S. Thakur's ‘Behaviour of Function Field Gauss Sums at Infinity’ APPENDIX TO DINESH S. THAKUR'S 'BEHAVIOUR OF FUNCTION FIELD GAUSS SUMS AT INFINITY' JOSE FELIPE VOLOCH The notion of exceptional prime is introduced in Definition 1.4 of the main paper. We characterize exceptional primes, show that they always exist when the class number is bigger than one, and make other remarks about them. We shall use a more geometric language. Let X be an algebraic curve of genus g, defined over a finite field, and P a rational point of X, which will play the role of oo in the main paper. Recall that there is a 1-1 correspondence between fractional ideals of the ring of functions on X holomorphic away from P and divisors on Zwith support disjoint from P. Proofs of results on orders of linear systems used below can be found in [14]. Denote, for a divisor D, L(D) = {xeK:(x) + D^ 0} and by /(£>) its dimension over the field of constants. The gaps of an ideal defined by a divisor D are the integers for which L{nP—D) = L((n — l)P—D). Indeed, this means that any function on the ideal with degree at most n has degree at most n— 1. Recall that http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Appendix to Dinesh S. Thakur's ‘Behaviour of Function Field Gauss Sums at Infinity’

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/25.5.425
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Abstract

APPENDIX TO DINESH S. THAKUR'S 'BEHAVIOUR OF FUNCTION FIELD GAUSS SUMS AT INFINITY' JOSE FELIPE VOLOCH The notion of exceptional prime is introduced in Definition 1.4 of the main paper. We characterize exceptional primes, show that they always exist when the class number is bigger than one, and make other remarks about them. We shall use a more geometric language. Let X be an algebraic curve of genus g, defined over a finite field, and P a rational point of X, which will play the role of oo in the main paper. Recall that there is a 1-1 correspondence between fractional ideals of the ring of functions on X holomorphic away from P and divisors on Zwith support disjoint from P. Proofs of results on orders of linear systems used below can be found in [14]. Denote, for a divisor D, L(D) = {xeK:(x) + D^ 0} and by /(£>) its dimension over the field of constants. The gaps of an ideal defined by a divisor D are the integers for which L{nP—D) = L((n — l)P—D). Indeed, this means that any function on the ideal with degree at most n has degree at most n— 1. Recall that

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Sep 1, 1993

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