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The goals of this program are to promote high-quality research in Number Theory in Barcelona, as well as to contribute to the training of researchers in this and related areas. It will combine research conferences, courses, and instructional workshops. In particular, it will bring together worldwide experts in the field with the aim to foster advances in the research projects of the involved BGSMath members and groups. Namely we plan to focus on the following ground-breaking lines of research:

Euler systems and the conjectures of Birch and Swinnerton-Dyer and Bloch-Kato. In order to celebrate mathematics in the new millennium, the Clay Mathematics Institute established seven $ 1.000.000 Prize Problems. The Prizes were conceived to record some of the most important challenges with which mathematicians were grappling at the turn of the second millennium. One of these is the conjecture of Birch and Swinnerton-Dyer (BSD), widely open since the 1960’s, together with other cornerstones in mathematics like the Riemann hypothesis, Hodge conjecture, P vs NP problem, Navier-Stokes equation, Yang-Mills and Mass gap, and the Poincare conjecture. The Birch and Swinnerton-Dyer conjecture stands as the tip of the iceberg formed by the vast conjectural program of Beilinson, Bloch and Kato, and all the attempts taken so far to proving it exploit the deep connections between Shimura varieties, Galois representations and automorphic forms. Hence the conjecture can actually be stated in a much more general context, including the twist of E by irreducible Artin Galois representations of the absolute Galois group GK of K. The generalization of BSD formulated by Bloch and Kato applies to arbitrary motives arising from higher-dimensional varieties.

Experience shows that it might be more natural and fruitful to stare at the conjecture from this broader perspective, and this becomes apparent in this project, where the full picture is exploited in order to derive a neat strategy for proving new instances of the original BSD and solving important related problems.

During the research program hosted by the BGSmath we plan to develop innovative and unconventional strategies for proving groundbreaking results towards the resolution of the conjecture of Birch and Swinnerton-Dyer on elliptic curves over number fields and their generalizations by Bloch and Kato (BK) to arbitrary motives associated to higher-dimensional algebraic varieties over global fields. Moreover, we hope to exploit our background and experience on this subject in order to apply our methods and techniques for establishing bridges with our areas and prove important results concerning related questions.

Arithmetic Langlands Program: advances in reciprocity and functoriality. The Langlands Program is considered to be one of cornerstones of modern arithmetic geometry. It predicts a precise relation between automorphic forms on the one hand and arithmetic varieties and their Galois representations on the other. Particular cases of this relationship are the celebrated Shimura-Taniyama-Weil conjecture, proved by Wiles and Taylor-Wiles as the key step in their proof of Fermat’s Last Theorem, and Serre’s Modularity Conjecture, now a theorem thanks to the groundbreaking works of Khare, Wintenberger, and Dieulefait.

During the six-weeks program we plan to achieve fundamental results linking automorphic forms and Galois representations, with special emphasis on Langlands functoriality results, base change and modularity over totally real number fields.