Galois representations, automorphic forms and L-functions

Start Date
End Date
CIRM Luminy 
Meeting Type
Contact Name
Jaclyn Lang 


The relationship between L-functions, automorphic forms and Galois representations is a central theme in Number Theory. Some of the most intriguing aspects of this relationship are the conjectural links between analytic properties of L-functions and arithmetic invariants of varieties over number fields. These links, known as Bloch–Kato conjectures, can be seen as a vast generalization of the class number formula for Dedekind zeta-functions and the conjecture of Birchand Swinnerton-Dyer for elliptic curves. The only known way to attack these conjectures is to relate the underlying geometric objects to automorphic forms using the Taylor–Wiles method and to study special values of L-functions via Iwasawa theory. The latter approach requires an extensive development of the theory of p-adic L-functions. From this point of view, the p-adic analogs of the Bloch–Kato conjecture include the p-adic Beilinson Conjectures, formulated by Perrin-Riou, and the Main Conjectures of Iwasawa theory. The last two decades have seen spectacular progress in the use of p-adic methods to solve new instances of the Bloch-Kato Conjectures. The method of Euler systems, initially invented by Kolyvagin and Kato in the context of modular forms, was partially extended to some new situations (Rankin–Selberg L-functions, triple products, symplectic groups) and applied to Gross-Stark conjectures. The generalization of the Ribet–Wiles approach to some unitary groups lead to the proof of the Greenberg Main Conjecture for a large class of modular forms. The theory of Selmer complexes generalized and simplified algebraic aspects of the theory, and found impressive appli-cations to the Parity Conjecture. Classical constructions of p-adic L-functions were extended to algebraic groups of higher ranks. Also the realm of interaction between automorphic forms and Galois representations is in full bloom, and recently established R = T theorems have led to proofs of generalizations of Serre’s Modularity Conjecture and the Sato-Tate Conjecture. The aim of the conference is to bring together leading experts and emerging scientists, to report on the latest developments, to foster scientific exchanges, to initiate new interaction and collaboration, and thereby contribute to the scientific advancement of the field.


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