Let k be a perfect field of virtual cohomological dimension ≤ 2. Let G be a connected linear algebraic group over k such that Gsc satisfies a Hasse principle over k. Let X be a principal homogeneous space under G over k. We show that if X admits a zero cycle of degree one, then X has a k-rational point. Introduction The following question of Serre [10, p. 192] is open in general. Q: Let k be a ...

Gauss’s 2F1 ( 1 2 1 2 1 | x ) hypergeometric function gives periods of elliptic curves in Legendre normal form. Certain truncations of this hypergeometric function give the Hasse invariants for these curves. Here we study another form, which we call the Clausen form, and we prove that certain truncations of 3F2 ( 1 2 1 2 1 2 1 1 | x ) and 2F1 ( 1 4 3 4 1 | x ) in Fp[x] are related to the charac...

We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of examples of violations of the Hasse principle which are due to the Brauer-Manin obstruction, subject to the c...

H. Hasse conjectured that all multiplicative relations between Gauss sums essentially follow from the Davenport-Hasse product formula and the norm relation for Gauss sums. While this is known to be false, very few counterexamples, now known as sign ambiguities, have been given. Here, we provide an explicit product formula giving an infinite class of new sign ambiguities and resolve the ambiguou...

Given any global field k of characteristic 2, we construct a Châtelet surface over k that fails to satisfy the Hasse principle. This failure is due to a Brauer-Manin obstruction. This construction extends a result of Poonen to characteristic 2, thereby showing that the étale-Brauer obstruction is insufficient to explain all failures of the Hasse principle over a global field of any characteristic.

For a homogeneous space with connected or abelian stabilizer of a connected linear algebraic group defined over a number field, a cohomological obstruction to the Hasse principle is defined in terms of Galois hypercohomology with coefficients in a complex of two abelian algebraic groups. This obstruction is proved to be the only obstruction to the Hasse principle. It is proved that up to sign t...

Given any global field k of characteristic 2, we construct a Châtelet surface over k which fails to satisfy the Hasse principle. This failure is due to a Brauer-Manin obstruction. This construction extends a result of Poonen to characteristic 2, thereby showing that the étale-Brauer obstruction is insufficient to explain all failures of the Hasse principle over a global field of any characteris...

We present a novel randomized algorithm to factor polynomials over a finite field Fq of odd characteristic using rank 2 Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial f ∈ Fq[x] to be factored) with respect to a random Drinfeld module φ with complex multiplication. Factors of f supported on prime ideals with supersin...

Maximization of submodular functions on a ground set is a NP-hard combinatorial optimization problem. Data correcting algorithms are among the several algorithms suggested for solving this problem exactly and approximately. From the point of view of Hasse diagrams data correcting algorithms use information belonging to only one level in the Hasse diagram adjacent to the level of the solution at...