Reed Leon Gordon-Sarney

Ph.D. Candidate, Emory Mathematics/Computer Science

Papers

(joint with V. Suresh, submitted) Totaro's Question on Zero-Cycles on Torsors (arXiv:1702.00516)

(to appear in Proceedings of the AMS) Totaro's Question for Adjoint Groups of Types A1 and A2n (arXiv:1701.03124)

(to appear in Transactions of the AMS) Totaro's Question for Tori of Low Rank (arXiv:1603.07718)


Galois Cohomology

For any variety X over a field k, we can define the arithmetic invariants

ind(X) := gcd{[L:k] : L/k is a finite separable field extension such that X(L) ≠ ∅}

min(X) := min{[L:k] : L/k is a finite separable field extension such that X(L) ≠ ∅}

The index ind(X) divides min(X) by construction. When are they equal? The question was originally raised by Serre in the '60s for torsors X under linear algebraic groups G in the ind(X) = 1 case and later generalized by Totaro for the higher index cases. Affirmative proofs for any groups G are extremely rare. I settled the conjecture when G is any torus of rank ≤ 2 over an arbitrary field and when G is an absolutely simple classical adjoint group of type A1 or A2n over a field of characteristic ≠ 2. Consequently, I proved that any regular variety X containing such a torsor as a dense open subset has a point after a degree ind(X) field extension. In particular, ind(X) = min(X) when X is a del Pezzo surface of degree 6, a toric variety.

Recently, V. Suresh and I have constructed the first known examples of torsors under linear algebraic groups giving a negative answer to this question, including examples of torsors under tori of all ranks ≥ 3.


The Brauer Group

In a 2014 paper, Uematsu showed using explicit cocycle computations that Br X = Br k where k is a field of characteristic 0 and X is the affine diagonal quadric over k determined by

x2 + by2 + cz2 + d = 0

for some b, c, dk*. With the guidance of our project leaders Jean-Louis Colliot-Thélène and Yonatan Harpaz at the 2015 Arizona Winter School, four colleagues and I found an alternate, cocycle-free proof of this theorem for k = ℂ(b,c,d) where b, c, and d are algebraically independent indeterminates over ℂ.


Space-Time Block Coding

Coding theorists and electrical engineers have been using division algebra-based codes in wireless communication since 1998. Certain invariants arising in algebraic number theory of division algebras and their orders (e.g., maximal orders) directly translate to desirable coding-theoretic properties. I explored tensor product constructions of maximal order-based codes from existing ones using class field theory.