The Buzzard-Diamond-Jarvis conjecture for unitary groups

Let p > 2 be prime. We prove the weight part of Serre's conjecture for rank two unitary groups for mod p representations in the unramified case (that is, the Buzzard-Diamond-Jarvis conjecture for unitary groups), by proving that any Serre weight which occurs is a predicted weight. This completes the analysis begun in [BLGG11], which proved that all predicted Serre weights occur. Our methods are purely local, using the theory of (phi,Ghat)-modules to determine the possible reductions of certain two-dimensional crystalline representations.Comment: J. Amer. Math. Soc., to appear. Contains minor corrections from published versio

Congruences between Hilbert modular forms: constructing ordinary lifts

Under mild hypotheses, we prove that if F is a totally real field, k is the algebraic closure of the finite field with l elements and r : G_F --> GL_2(k) is irreducible and modular, then there is a finite solvable totally real extension F'/F such that r|_{G_F'} has a modular lift which is ordinary at each place dividing l. We deduce a similar result for r itself, under the assumption that at places v|l the representation r|_{G_F_v} is reducible. This allows us to deduce improvements to results in the literature on modularity lifting theorems for potentially Barsotti-Tate representations and the Buzzard-Diamond-Jarvis conjecture. The proof makes use of a novel lifting technique, going via rank 4 unitary groups.Comment: 48 page

The Sato-Tate conjecture for Hilbert modular forms

We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of \GL_2(\A_F), $F$ a totally real field, which are not of CM type. The argument is based on the potential automorphy techniques developed by Taylor et. al., but makes use of automorphy lifting theorems over ramified fields, together with a 'topological' argument with local deformation rings. In particular, we give a new proof of the conjecture for modular forms, which does not make use of potential automorphy theorems for non-ordinary $n$-dimensional Galois representations.Comment: 59 pages. Essentially final version, to appear in Journal of the AMS. This version does not incorporate any minor changes (e.g. typographical changes) made in proo

Factorization homology of enriched ∞\infty-categories

For an arbitrary symmetric monoidal $\infty$-category $V$, we define the factorization homology of $V$-enriched $\infty$-categories over (possibly stratified) 1-manifolds and study its basic properties. In the case that $V$ is \textit{cartesian} symmetric monoidal, by considering the circle and its self-covering maps we obtain a notion of \textit{unstable topological cyclic homology}, which we endow with an \textit{unstable cyclotomic trace map}. As we show in \cite{AMR-trace}, these induce their stable counterparts through linearization (in the sense of Goodwillie calculus)

SLS/BIALL Academic Law Library Survey 2015/2016

Survey report outlining the activities and funding of academic law libraries in the UK and Ireland in the academic year 2015/2016. The figures have been taken from the results of a survey questionnaire undertaken by Academic Services staff at the Institute of Advanced Legal Studies on behalf of the Society of Legal Scholars (SLS). The report is based on returns from 97 university and college libraries in the UK and Ireland (institutions offering either undergraduate, postgraduate or vocational courses) who responded to the survey conducted in March 2017. It is the only survey of its kind and provides data which academic law library managers use to bench-mark their own services and law course validation bodies note when appraising the provision of institutions seeking to run law courses. The report includes a summary of key findings, a compilation of the statistics, conclusions drawn from the figures and illustrative diagrams

Potentially crystalline lifts of certain prescribed types

We prove several results concerning the existence of potentially crystalline lifts with prescribed Hodge-Tate weights and inertial types of a given n-dimensional mod p representation of the absolute Galois group of K, where K/Q_p is a finite extension. Some of these results are proved by purely local methods, and are expected to be useful in the application of automorphy lifting theorems. The proofs of the other results are global, making use of automorphy lifting theorems.Comment: 22 pages; final version, to appear in Document

IALS@70: the growth of IALS Library and its development of digital initiatives for the UK legal community

As a way of placing the launch of IALS Digital into context, David Gee (Deputy Librarian, Institute of Advanced Legal Studies) outlines the growth of IALS Library over the past 70 years and its development of a wide range of digital initiatives over the past 30 years