Shadow lines in the arithmetic of elliptic curves

Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K)\otimes Q_p, known as a shadow line. When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q)\otimes Qp. If, in addition, p splits in K/Q, then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. We conclude by illustrating these algorithms in a collection of examples.

Crocodiles in the regulatory swamp: navigating the dangers of outsourcing, SaaS and shadow IT

Corporates are entering the brave new world of the internet and digitization without much regard for the fine print of a growing regulation regime. More traditional outsourcing arrangements are already falling foul of the regulators as rules and supervision intensifies. Furthermore, ‘shadow IT’ is proliferating as the attractions of SaaS, mobile, cloud services, social media, and endless new ‘apps’ drive usage outside corporate IT. Initial cost-benefit analyses of the Cloud make such arrangements look immediately attractive but losing control of architecture, security, applications and deployment can have far reaching and damaging regulatory consequences. From research in financial services, this paper details the increasing body of regulations, their inherent risks for businesses and how the dangers can be pre-empted and managed. We then delineate a model for managing these risks specifically focused on investigating, strategizing and governing outsourcing arrangements and related regulatory obligations