Benedict Gross recently gave a series of lectures here at the University of Virginia on things related to the Birch and Swinnerton-Dyer Conjecture. One of the recent notable developments he mentioned is the work of Yun and Zhang. It is about the function field analogue but they obtain information about the full Taylor series of the $L$-function. The paper has recently been accepted at Annals of Mathematics.
Abdelmalek Abdesselam
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