Arithmetic Geometry and Number Theory RTG Seminar: On the Gross-Stark conjecture

Seminar | October 30 | 3:10-5 p.m. | 891 Evans Hall

 Samit Dasgupta, UCSC

 Department of Mathematics

$Seminar\, Format$:

The seminar consists of two 50-minute talks, a pre-talk (3:10-4:00) and an advanced talk (4:10-5:00), with a 10-minute break (4:00-4:10) between them. The advanced talk is a regular formal presentation about recent research results to general audiences in arithmetic geometry and number theory; the pre-talk (3:10-4:00) is to introduce some prerequisites or background for the advanced talk to audiences consisting of graduate students.

$Pre$-$talk$: Stark's conjectures (classical and $p$-adic)

In this talk we will state the classical Stark conjecture. Next we will qualitatively describe the two $p$-adic Stark conjectures (at $s=1$ and $s=0$), due to Serre-Solomon and Gross, respectively. If time permits, I will sketch "Ribet's Method", which is a technique to relate special values of L-functions to algebraic objects, and will be employed in the second talk.

$Advanced\, talk$: On the Gross-Stark conjecture

In 1980, Gross conjectured a formula for the expected leading term at $s=0$ of the Deligne–Ribet $p$-adic $L$-function associated to a totally even character ψ of a totally real field $F$. The conjecture states that after scaling by $L(\psi \omega ^{-1}, 0)$, this value is equal to a $p$-adic regulator of units in the abelian extension of $F$ cut out by $\psi \omega ^{-1}$. In this talk we describe a proof of Gross's conjecture. This is joint work with Mahesh Kakde and Kevin Ventullo. If time permits, we will briefly describe joint work with Michael Spiess on a refinement of Gross's conjecture that gives a formula for the characteristic polynomial of the regulator matrix. This refined conjecture is still open.