By Gebhard Böckle,David Burns,David Goss,Dinesh Thakur,Fabien Trihan,Douglas Ulmer,Francesc Bars,Ignazio Longhi

This quantity collects the texts of 5 classes given within the mathematics Geometry examine Programme 2009-2010 on the CRM Barcelona. them all care for attribute p worldwide fields; the typical topic round which they're founded is the mathematics of L-functions (and different exact functions), investigated in a number of features. 3 classes study probably the most vital contemporary rules within the confident attribute concept chanced on via Goss (a box in tumultuous improvement, that is seeing a couple of superb advances): they hide respectively crystals over functionality fields (with a few purposes to L-functions of t-motives), gamma and zeta features in attribute p, and the binomial theorem. the opposite are fascinated about themes toward the classical idea of abelian forms over quantity fields: they provide respectively an intensive creation to the mathematics of Jacobians over functionality fields (including the present prestige of the BSD conjecture and its geometric analogues, and the development of Mordell-Weil teams of excessive rank) and a state-of-the-art survey of Geometric Iwasawa idea explaining the hot proofs of varied models of the most Conjecture, within the commutative and non-commutative settings.

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