Categorical structures enriched in a quantaloid: regular presheaves, regular semicategories

We study presheaves on semicategories enriched in a quantaloid: this gives rise to the notion of regular presheaf. A semicategory is regular when its representable presheaves are regular, and its regular presheaves then constitute an essential (co)localization of the category of all of its presheaves. The notion of regular semidistributor allows to establish the Morita equivalence of regular semicategories. Continuous orders and Omega-sets provide examples.Comment: 21 page

Convex powerdomains II

AbstractThe study of powerdomains defined as completions via Frink ideals is continued. It is shown how to represent directed ideals ofP(D)by certain compact subsets of the original domainD, and arbitrary Frink ideals by sets of such subsets. The operations union and big union are defined and their properties studied. Finally, some results on the relationship of this powerdomain to the classical Plotkin powerdomain are presented

Modularity of abelian surfaces over imaginary quadratic fields

Modular forms for GL(2) over an imaginary quadratic field K are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over K or an associated abelian surface with quaternionic multiplication over K. We give explicit evidence in the way of examples to support this conjecture in the latter case. Furthermore, the quaternionic surfaces given correspond to genuine Bianchi newforms, which answers a question posed by J. Cremona as to whether this phenomenon can happen