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TIMS Taipei Number Theory Seminar (2017 Fall)

On Eisenstein series, theta series, and Green\'s functions over function fields

Prof. Fu-Tsun Wei

2017 - 09 - 29 (Fri.)
13:00 - 14:00
103, Mathematics Research Center Building (ori. New Math. Bldg.)

In this talk, I will discuss connections between Eisenstein series, theta series, and Green\'s functions in the function field context. First, we may write the mirabolic Eisenstein series of GL(4) as the Mellin transform of the corresponding theta series. Using the Jacquet-Langlands-Shimizu correspondence, we obtain an alternative way to get the \"doubling method\" integral representation of Godement-Jacquet L-functions in the GL(2) case. On the other hand, an adelic formulation of the Green\'s function on Bruhat-Tits trees enables us to express the mirabolic Eisenstein series in terms of Green\'s functions. Finally, as in the classical case, the Fourier coefficients of the Green\'s functions (with respect to one variable) gives us Poincare series on GL(2) (with respect to another variable).