2:00 pm Thursday, March 30, 2017
Algebra and Number Theory: L-functions and regulators by Samit Dasgupta (UCSC) in RLM 9.168
(Note that this colloquium-style talk is an interview for the FII position in Number Theory.) In this talk, I will introduce the concept of L-functions in number theory, mainly through examples. L-functions can be attached to Galois characters, elliptic curves, algebraic varieties, or in general “motives.” In every case, there is a conjectural formula for the values of these L-functions at certain integer points. Examples of these special value formulae include the famous conjectures of Stark, Birch—Swinnerton-Dyer, Beilinson, and Bloch—Kato. The conjectures typically equate special L-values to a product of three terms—an algebraic number, a regulator, and a period. The regulator is the determinant of a matrix whose entries are logarithmic heights. The conjectures have been proven in very few cases, with the higher rank settings (i.e. when the matrix has dimension greater than 1) appearing to be the most difficult. For the second half of the talk, I will discuss the analogs of these conjectures for p-adic L-functions. I will conclude with a description of my recent proof of the Gross—Stark conjecture (joint with Mahesh Kakde and Kevin Ventullo). This result gives an exact formula for the leading terms of Deligne—Ribet p-adic L-functions at 0 in terms of p-adic regulators of p-units. Time permitting, I will conclude with the statements of famous and mysterious open conjectures regarding the non-vanishing of certain p-adic regulators. Submitted by
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