### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 231-259 |

Number of pages | 29 |

Journal | Transactions of the American Mathematical Society |

Volume | 246 |

DOIs | |

Publication status | Published - Jan 1 1978 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics