The Cassels-Tate pairing on polarized abelian varieties

Let (A,\lambda) be a principally polarized abelian variety defined over a global field k, and let \Sha(A) be its Shafarevich-Tate group. Let \Sha(A)_\nd denote the quotient of \Sha(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing \Sha(A)_\nd \times \Sha(A)_\nd \rightarrow \Q/\Z. If A is an elliptic curve, then by a result of Cassels the pairing is alternating. But in general it is only antisymmetric. Using some new but equivalent definitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on \Sha(A)_\nd. These criteria are expressed in terms of an element c \in \Sha(A)_\nd that is canonically associated to the polarization \lambda. In the case that A is the Jacobian of some curve, a down-to-earth version of the result allows us to determine effectively whether \#\Sha(A) (if finite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperelliptic curves of even genus g \ge 2 over \Q have a Jacobian with nonsquare \#\Sha (if finite). For example, it appears that this density is about 13% for curves of genus 2. The proof makes use of a general result relating global and local densities; this result can be applied in other situations.Comment: 41 pages, published versio

Spin-orbit torque switching of synthetic antiferromagnets

We report that synthetic antiferromagnets (SAFs) can be efficiently switched by spin-orbit torques (SOTs) and the switching scheme does not obey the usual SOT switching rule. We show that both the positive and negative spin Hall angle (SHA)-like switching can be observed in Pt/SAF structures with only positive SHA, depending on the strength of applied in-plane fields. A new switching mechanism directly arising from the asymmetric domain expansion is proposed to explain the anomalous switching behaviors. Contrary to the macrospin-based switching model that the SOT switching direction is determined by the sign of SHA, the new switching mechanism suggests that the SOT switching direction is dominated by the field-modulated domain wall motion and can be reversed even with the same sign of SHA. The new switching mechanism is further confirmed by the domain wall motion measurements. The anomalous switching behaviors provide important insights for understanding SOT switching mechanisms and also offer novel features for applications.Comment: 40 pages, 14 figure

On some universal algebras associated to the category of Lie bialgebras

In our previous work (math/0008128), we studied the set Quant(K) of all universal quantization functors of Lie bialgebras over a field K of characteristic zero, compatible with duals and doubles. We showed that Quant(K) is canonically isomorphic to a product G_0(K) \times Sha(K), where G_0(K) is a universal group and Sha(K) is a quotient set of a set B(K) of families of Lie polynomials by the action of a group G(K). We prove here that G_0(K) is equal to the multiplicative group 1 + h K[[h]]. So Quant(K) is `as close as it can be' to Sha(K). We also show that the only universal derivations of Lie bialgebras are multiples of the composition of the bracket with the cobracket. Finally, we prove that the stabilizer of any element of B(K) is reduced to the 1-parameter subgroup of G(K) generated by the corresponding `square of the antipode'.Comment: expanded version, containing related result

Visualising Sha[2] in Abelian Surfaces

Given an elliptic curve E1 over a number field and an element s in its 2-Selmer group, we give two different ways to construct infinitely many Abelian surfaces A such that the homogeneous space representing s occurs as a fibre of A over another elliptic curve E2. We show that by comparing the 2-Selmer groups of E1, E2 and A, we can obtain information about Sha(E1/K)[2] and we give examples where we use this to obtain a sharp bound on the Mordell-Weil rank of an elliptic curve. As a tool, we give a precise description of the m-Selmer group of an Abelian surface A that is m-isogenous to a product of elliptic curves E1 x E2. One of the constructions can be applied iteratively to obtain information about Sha(E1/K)[2^n]. We give an example where we use this iterated application to exhibit an element of order 4 in Sha(E1/Q).Comment: 17 page