Lectures on automorphic L-functions

On tensor product $L$-functions and Langlands functoriality

‎In the spirit of the Langlands proposal on  Beyond Endoscopy ‎we discuss the explicit relation between the Langlands functorial transfers and automorphic $L$-functions‎. ‎It is well-known that the poles of the $L$-functions have deep impact to the Langlands functoriality‎. ‎Our discussion also‎ ‎includes the meaning of the central value of the tensor product $L$-functions in terms of the Langl...

Automorphic forms , L - functions and number theory ( March 12 – 16 ) Three Introductory lectures

Langlands picture of automorphic forms and L - functions

Recall the lectures in the first two weeks. We discussed Tate’s thesis, i.e. automorphic forms of GL1, which is a generalization of Hecke’s work in 1910’s. Now suppose f ∈ Sk(Γ) is an Hecke eigenform with normalized 1-st Fourier coefficient a1 = 1. Ramanujan-Deligne theorem says ∣∣a(p)p− k−1 2 ∣∣ 6 2, for any p. Let cp = a(p)p− k−1 2 = αp + α−1 p . The Dirichlet series associated f , which is n...

Review of “lectures on Automorphic L-functions" by James Cogdell, Henry Kim and Ram Murty

The Langlands Functoriality Conjecture is the number theorist’s “grand unified theory". It describes and elucidates the platonic realm of modular forms, L-functions, and motives—a world arguably no less real, in its richness, than the physicist’s universe of elusive particles and far-flung galaxies. The timely volume under review (referred to henceforth as CKM) is made up of three distinct cont...

Modular Forms and - adic Representations ∗

the circumstance that L-functions can be introduced not only in the context of automorphic forms, with which he has had some experience, but also in the context of diophantine geometry. That this circumstance can be the source of deep problems was, I believe, first perceived by E. Artin. He was, to be sure, concerned with forms on GL(1) and with varieties of dimension 0. This remains the only c...