In 1980, Gross conjectured a formula for the expected leading term at s=0 of the p-adic L-function associated to characters of totally real number fields. The conjecture states a precise relation between this leading term and p-adic regulator of p-units in an abelian extension. In the talk I will present a precise formulation of the conjecture and describe its relevance for Hilbertâ€™s 12th problem. I will then sketch proof of this conjecture of Gross. This is a joint work with Samit Dasgupta and Kevin Ventullo.

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Last updated: 14 Feb 2019