TY - JOUR
T1 - On modular functions in characteristic p
AU - Winnie Li, Wen Ching
N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.
PY - 1978/12
Y1 - 1978/12
N2 - Let k - Fq(T) be a function field of one variable over a finite field F. For a nonzero polynomial A E Fq[T] one can define the modular group r(A). In this paper, we continue a theme introduced by Weil, and study the A-harmonic modular functions for r(^4). The main purpose of this paper is to give a natural definition of A-harmonic Eisenstein series for r(A) so that we obtain a decomposition theory of λ-harmonic modular functions, analogous to the classical results of Hecke. That is, we prove Modular Functions = Eisenstein Series 0 Cusp Functions. Moreover, the dimension of the space generated by λ-harmonic Eisenstein series for T(/t) is equal to the number of cusps of T(j4), and so is independent of λ. For the definition of A-harmonic Eisenstein series and the proof of decomposition theory, we consider two cases: (i) λ ±2Vq and (ii) λ = ±lVq, separately. Case (i) is treated in the usual way. Case (ii), being a “degenerate’* case, is more interesting and requires more complicated analysis.
AB - Let k - Fq(T) be a function field of one variable over a finite field F. For a nonzero polynomial A E Fq[T] one can define the modular group r(A). In this paper, we continue a theme introduced by Weil, and study the A-harmonic modular functions for r(^4). The main purpose of this paper is to give a natural definition of A-harmonic Eisenstein series for r(A) so that we obtain a decomposition theory of λ-harmonic modular functions, analogous to the classical results of Hecke. That is, we prove Modular Functions = Eisenstein Series 0 Cusp Functions. Moreover, the dimension of the space generated by λ-harmonic Eisenstein series for T(/t) is equal to the number of cusps of T(j4), and so is independent of λ. For the definition of A-harmonic Eisenstein series and the proof of decomposition theory, we consider two cases: (i) λ ±2Vq and (ii) λ = ±lVq, separately. Case (i) is treated in the usual way. Case (ii), being a “degenerate’* case, is more interesting and requires more complicated analysis.
UR - http://www.scopus.com/inward/record.url?scp=84968490687&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84968490687&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-1978-0515538-9
DO - 10.1090/S0002-9947-1978-0515538-9
M3 - Article
AN - SCOPUS:84968490687
SN - 0002-9947
VL - 246
SP - 231
EP - 259
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
ER -