Arithmetic Geometry and Number Theory RTG Seminar: Tamagawa Numbers of Linear Algebraic Groups

Seminar | October 2 | 3:10-5 p.m. | 891 Evans Hall

 Zev Rosengarten, Stanford University

 Department of Mathematics

In 1981, Sansuc obtained a formula for Tamagawa numbers of reductive groups over number fields, modulo some then unknown results on the arithmetic of simply-connected groups which have since been proven, particularly Weil's conjecture on Tamagawa numbers over number fields. One easily deduces that this same formula holds for all linear algebraic groups over number fields. Sansuc's method still works to treat reductive groups in the function field setting, thanks to the recent resolution of Weil's conjecture in the function field setting by Lurie and Gaitsgory. However, due to the imperfection of function fields, the reductive case is very far from the general one; indeed, Sansuc's formula does not hold for all linear algebraic groups over function fields. We propose a modification of Sansuc's formula that recaptures it in the number field case and also gives a correct answer over number fields. We have proven this formula for all pseudo-reductive groups in characteristic greater than 3, as well as for all commutative groups (in any characteristic). The commutative case (which is essential even for the general pseudo-reductive case) is a corollary of a vast generalization of the Poitou-Tate nine-term exact sequence, from finite group schemes to arbitrary affine commutative group schemes of finite type.

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.