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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Supercuspidal representations and Poincaré series over function fields
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by Daniel Bump and Shuzo Takahashi PDF
Trans. Amer. Math. Soc. 340 (1993), 395-413 Request permission


In this paper, we will give a new construction of certain cusp forms on $GL(2)$ over a rational function field. The forms which we construct are analogs of holomorphic modular forms, in that the local representations at the infinite place are in the discrete series. The novelty of our approach is that we are able to give a very explicit construction of these forms as certain ’Poincaré series.’ We will also study the exponential sums which arise in the Fourier expansions of these Poincaré series.
  • E.-U. Gekeler, Automorphe Formen über $\textbf {F}_q(T)$ mit kleinem Führer, Abh. Math. Sem. Univ. Hamburg 55 (1985), 111–146 (German). MR 831522, DOI 10.1007/BF02941492
  • H. Jacquet and R. P. Langlands, Automorphic forms on $\textrm {GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654, DOI 10.1007/BFb0058988
  • P. Kutzko, On the supercuspidal representations of $GL(2)$, I and II, Amer, J. Math. 100 (1978), 43-60 and 705-716.
  • Ilya Piatetski-Shapiro, Complex representations of $\textrm {GL}(2,\,K)$ for finite fields $K$, Contemporary Mathematics, vol. 16, American Mathematical Society, Providence, R.I., 1983. MR 696772
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 340 (1993), 395-413
  • MSC: Primary 11F12; Secondary 11R58
  • DOI:
  • MathSciNet review: 1152320