Professor: Alex Kontorovich

Office Hours: Tuesdays 1-2 pm

Suggested readings:

Bump (Automorphic Forms and Representations), Goldfeld-Hundley (Automorphic Representations and L-functions), Koblitz (P-adic Numbers, P-adic Analysis, and Zeta-Functions)

	WEEK LECTURE 1 VIDEO LECTURE 2 VIDEO COMMENTS/TOPICS HOMEWORK (from Goldfeld-Hundley) Week 1 01/17, 01/20 Lecture 1: Diophantus, p-adic numbers, Ostrowski's Theorem Lecture 2: p-adic numbers, units, topology, local compactness 1.1 - 1.11, due 1/27 Week 2 01/24, 01/27 Lecture 3: p-adic Haar measure, multiplicative Haar measure, additive character, Fourier transform Lecture 4: p-adic Fourier transform/inverse, adeles, discreteness of Q, Chinese Remainder Thm Week 3 01/31, 02/03 Lecture 5: Fundamental Domain for the action of rationals on adeles, weak/strong approximation, ideles and their topology, adelic character Lecture 6: Fundamental Domain for the multiplicative action of non-zero rationals on ideles, adelic character is periodic on the rationals, factorizable adelic Schwartz functions and their Fourier transforms Week 4 02/7, 02/10 Lecture 7: Classical Poisson summation, Mellin transform, Theta function, Riemann zeta function, analytic continuation and functional equation Lecture 8: Adelic Poisson summation, Theta function, Mellin transform, Euler product Read Artin's 1943 paper on Global Fields Read Tate's 1950 thesis Homework: Bump #3.1.2(a), 3.1.7 due 2/24 Week 5 02/14, 02/17 Lecture 9: Adelic Mellin transform of theta function, analytic continuation and functional equation for zeta, Dirichlet characters and L-function Lecture 10: Crash course in Representation Theory, unitary representations, regular representation, Schur's Lemma, characters Week 6 02/21, 02/24 Lecture 11: Decomposition of L^2(G) intro irreducibles, application to counting, finite Fourier transform, extension of Dirichlet characters to real-values Lecture 12: Twisted, dilated Poisson summation, analytic continuation and functional equation for Dirichlet L-functions, adelic automorphic representations Week 7 02/28, 03/03 Lecture 13: Adelic automorphic representations of GL(1), Hecke characters as Dirichlet characters, L-functions Lecture 14: Adelic L-function, functional equation, analytic continuation, primitive Dirichlet character, Conrey label Week 8 03/7, 03/10 Lecture 15: Local Zeta-functions, independence on choice of test function, functional equation, evaluation of local gamma factors, Rankin-Selberg convolutions Lecture 16: Fun with Modular Forms, Basel Problem, Partition Function, Kepler Conjecture, Hales, Viazovska, Elliptic Curves, Modularity Conjecture, Fermat's Last Theorem, Wiles/Taylor-Wiles, Moster Group, Moonshine, Borcherds, Ramanujan Conjectures, Deligne See here for Romik's proof of Viazovska's theorem without computers. Goldfeld-Hundley Chap 2 #2.1 -- 2.10 Due 3/31 Week 9 03/14, 03/17 SPRING BREAK Week 10 03/21, 03/24 Lecture 17: Action of SL(2,R) on upper half plane, projective line, series expansion, classification of motions, fundamental domain for SL(2,Z) Lecture 18: Dirichlet domains, Riemannian geometry, unit tangent bundle, geodesics, invariant measure, Iwasawa decomposition, L^2 space, hyperbolic Laplace operator, Maass forms, modular forms, Petersson inner product Week 11 03/28, 03/31 Lecture 19: Ramanujan conjecture, Langlands-Satake parameters, Rankin-Selberg L-function, subconvexity, Langlands functoriality, cuspidal modular forms as Petersson square-integrable functions, GL(2,Q_p), Iwasawa decomposition Lecture 20: Iwasawa decomposition for GL(2,Q_p) and GL(2,A), Strong Approximation in SL(2,A), fundamental domain for GL(2,Q) acting on GL(2,A) Week 12 04/04, 04/07 Lecture 21: Fundamental domain for GL(2,Q) acting on GL(2,A), Hecke L-function attached to modular form, Maass form Lecture 22: Mellin transform, inversion, convolution, K-Bessel function, Hecke operators Week 13 04/11, 04/14 Lecture 23: Hecke operators, Hecke relations, Euler product, Ramanujan Conjecture, Hecke bound on Fourier coefficients, Selberg 1/4 Conjecture, Holds for SL(2,Z) Lecture 24: Selberg 1/4 for SL(2,Z), Level 1 Converse theorem, Leven N Functional Equation Week 14 04/18, 04/21 Lecture 25: Twisted functional equation on Gamma_0(N) Lecture 26: Weil's Converse Theorem for Gamma_0(N) Week 15 04/25, 04/28 Lecture 27: Tangent space at identity, Lie algebra, exponential map, linear differential operators, Lie bracket, Jacobi identity, universal enveloping algebra, its center commutes with the regular representation, center is generated by Casimir, Casimir is Laplacian on K-isotypic vectors Lecture 28: Automorphic representations, K-isotypic components, Gelfand-Graev-PS Higher Rank L-functions, e.g., Exterior Square