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On the tower of fields conjecture of Gross

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Mahesh Kakde

On the tower of fields conjecture of Gross

The tower of fields conjecture formulated by Gross is an integral refinement of the Gross—Stark conjecture. In the rank one case it relates the Brumer—Stark unit in a CM extension of a totally real field with the Stickelberger element of a larger extension. This gives important information about the Brumer—Stark unit that we do not get from the Brumer—Stark conjecture. The higher rank case gives information about the Rubin—Brumer—Stark element. After presenting the formulation of the tower of fields conjecture, I will present our reformulation using Fitting ideals. I will then sketch its proof. The rank one case is joint work with Samit Dasgupta whereas the general case is work in progress with Dasgupta and Micheal Spiess.