Fall 2021 Automorphic
forms on GL(2)
This is an introductory course for automorphic
representations and associated L-functions on GL(2).
The goal is to understand the analytic properties of L-functions via representation theory.
(I) Local theory of representation on GL(2)
(1) Weil representations on SL(2)
(2) Explicit construction of representations on GL(2) as quotients of Weil representations
(3) Construction of Whittaker models and Kirillov models
(4) Classification of admissible irreducible representations on GL(2)
(II) Eisenstein series on GL(2) and applications
(1) Intertwining operators
(2) Explicit computation of Fourier coefficients and constant terms
(3) Non-vanishing of the Riemann zeta function at 1+it, t\not =0
(4) Langlands' computation of Tamagawa number of SL(2)
(III) Theta liftng
(1) Local Jacquet-Langlands correspondence
(2) Jacquet-Langlands-Shimizu lift
(IV) Theory of L-functions
(1) L-functions for GL(2) and converse theroems
(2) L-functions for GL(2)xGL(2): Rankin-Selberg convolution
(3) Quadratic base change
The first meeting: September 24,
Place: Room 201 Astro-Math Buliding
Jacquet-Langlands. Automorphic forms on GL(2), part I, LNM 114.
Jacquet. Automorhic forms on GL(2), part II, LNM 278.
Shimizu, Hideo. Theta series and automorphic forms on GL(2), J. Math. Soc. Japan, vol 24 p.638-683.
Prerequisites: Complex analysis and linear algebra