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Understanding the conjecture of Birch and Swinnerton-Dyer

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To every elliptic curve over a global field one can associate a regular proper model which is geometrically a two-dimensional object and which reveals more underlying structures and dualities than its generic fibre. Unlike the classical adelic structure on one-dimensional arithmetic schemes, there are two adelic structures on arithmetic surfaces: one is more suitable for geometry and another is more suitable for analysis and arithmetic. The two-dimensional adelic analysis studies the zeta function of the surface lifting it to a zeta integral using the second adelic structure. Its main theorem reduces the study of analytic properties of the zeta integral to those of a boundary term which is an integral over the weak boundary of adelic spaces of the second type. To study the latter one uses the symbol map from K_1 of the first adelic structure and K_1 of the second adelic structure to K_2 of the first adelic structure. The (known in some partial cases but not really understood) equality of the analytic and arithmetic ranks becomes much more transparent and natural in the language of the two adelic structures on the surface and their interplay. Moreover, the adelic approach includes a potential to explain the finiteness of the Brauer-Grothendieck group of the surface and hence of Shah.

This talk is part of the Number Theory Seminar series.

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