Zev Rosengarten

email: zevr@stanford.edu

Tate Duality in Positive Dimension: This manuscript generalizes classical Poitou-Tate duality (local duality, the nine-term exact sequence, etc.) from finite group schemes to arbitrary affine commutative group schemes of finite type. The motivation for doing so was to prove Conjecture 1.1.1 in the manuscript, which proposes a simple formula for Tamagawa numbers of connected linear algebraic groups over global fields, the new case being that of global function fields.

Note: I have found a counterexample to Conjecture 1.1.1 in the above manuscript. The example is a wound non-commutative two-dimensional unipotent group. I will write all of this up soon. I hope to find a suitable modification of Conjecture 1.1.1 that works in all cases (the problematic ones arising from non-commutative solvable groups).

Tamagawa Numbers of Pseudo-reductive Groups: This paper proves for all pseudo-reductive groups the Tamagawa number formula conjectured in the Tate duality manuscript above, except for a technical wrinkle in characteristics 2 and 3, namely, the equality of two natural measures on certain types of exotic pseudo-reductive groups that only show up in low characteristics. I hope to have this equality sorted out in the coming months.

Picard Groups of Linear Algebraic Groups: This paper proves various properties of Picard groups of linear algebraic groups. It studies the subgroup of primitive line bundles and proves that it is finite over every global function field. The proof of this finiteness rests crucially on the structure theory of pseudo-reductive groups developed by Conrad, Gabber, and Prasad. In the last section of the paper, various interesting counterexamples are constructed.

I expect the Tamagawa and Picard Groups papers above to be merged into some expanded paper at some point that proves the conjectured Tamagawa formula for all connected linear algebraic groups, rather than just the commutative and pseudo-reductive ones.Some undergrad papers (a couple of these are rather old, and I haven't bothered to improve the texing):

An Erdos-Turan Inequality For Compact Simply-Connected Semisimple Lie Groups

Yet Another Proof of Quadratic Reciprocity Note: Ehud de Shalit tells me that this proof was actually already found by Richard Swan.