Research

I am interested in the cohomology of arithmetic groups, arithmetic of automorphic forms, and the Langlands program. I especially focus on the cases where the underlying arithmetic manifolds have no visible links to algebraic geometry. I enjoy developing computer programs and carrying out numerical experiments to explore and gain insight. 

I have been involved in the construction of a database on Bianchi modular forms (that is, modular forms for GL(2) over imaginary quadratic fields) and related arithmetic data inside the L-functions and Modular Forms Database (LMFDB).

In another strand of my research, I study operator K-theory of C*-algebras associated to arithmetic groups from the point of view of number theory and automorphic forms. 

Here is my Google Scholar profile.

Here is the Sheffield Number Theory Group website.

PhD STUDENTS

Ciaran Schembri (2019) (first job: postdoc, Dartmouth College, USA)

Lewis Combes (2024) (first job: postdoc, University of Sydney, AUS)

Daniel Bassett (current)

PAPERS, PREPRINTS etc

Almost all of them are on the Arxiv.

• K-Theory and Representation Theory
  book edited by myself and Roger Plymen, LMS Textbook Series, in press

• Group C*-algebras, C*-correspondences and K-theory
  with Bram Mesland, 
  expository chapter in the book "K-Theory and Representation Theory''

• Equal rank local theta correspondence as a strong Morita equivalence 
  with Bram Mesland (VIDEO of a talk on this work geared towards operator algebraists)
  to appear in Selecta Mathematica

• Torsion in the homology of principal congruence subgroups of Bianchi groups 
  appendix to the paper "Congruence RFRS towers" by Ian Agol and Matthew Stover
  to appear in Ann. Inst. Fourier, Grenoble
  The CODE used in the appendix.

• Torsion homology growth for noncongruence subgroups of Bianchi groups 
  with Gonca Kızılaslan
  International Journal of Number Theory 16.4 (2020)

• A K-theoretic Selberg trace formula 
  with Bram Mesland and Hang Wang
  Proceedings of IWOTA 2018 (Ronald Douglas memorial volume) (2020)

• Hecke operators in KK-theory and the K-homology of Bianchi groups 
  with Bram Mesland
  Journal of Noncommutative Geometry 14.1 (2020)

• Hecke modules for arithmetic groups via bivariant K-theory 
  with Bram Mesland
  Annals of K-theory 3-4 (2018)

• Mod p Base Change transfer for GL(2) 
  with Andrew Jones
  Journal of the Ramanujan Mathematical Society 33.3 (2018) 

• On the asymptotic Fermat's Last Theorem over number fields 
  with Samir Siksek
  Commentarii Mathematici Helvetici  93.2 (2018)  

• Dimension formulae for spaces of lifted Bianchi modular forms 
  with Panagiotis Tsaknias
  (not intended for publication, companion to the paper below.)

• Torsion homology growth and cycle complexity of arithmetic manifolds 
  with Nicolas Bergeron and Akshay Venkatesh
  Duke Math Journal 165.9 (2016) 

• Uniformization of modular elliptic curves via p-adic periods 
  with Xevi Guitart and Marc Masdeu
  Journal of Algebra. 445 (2016) 

• Darmon points on elliptic curves over number fields of arbitrary signature 
  with Xevi Guitart and Marc Masdeu
  Proceedings of the LMS. (3) 111.2 (2015) 

• Theta lifts of Bianchi modular forms and applications to the Paramodularity Conjecture 
  with Tobias Berger, Lassina Dembélé and Ariel Pacetti
  Journal of the LMS. (2) 92.2 (2015) 

• Lower Bounds on the Dimension of the Cohomology of Bianchi Groups via Sczech Cocycles 
  with Seyfi Turkelli
  J. Théorie Nombres Bordeaux. 28.1 (2016)  

• Arithmetic aspects of Bianchi groups 
  In "Computations with Modular Forms", Eds: G.Boeckle, G.Wiese.
  Contributions in Mathematical and Computational Sciences, Volume 6, 2014, Springer-Verlag. 

• On level one cuspidal Bianchi modular forms 
  with Alexander Rahm
  LMS Journal of Computation and Mathematics. 16 (2013) 

• On the homology of non-arithmetic hyperbolic tetrahedral groups 
  International Journal of Number Theory. 8.2 (2012)
  The DATA discussed in the paper.

• On the integral cohomology of Bianchi groups 
  Experimental Mathematics. 20.4 (2011)
  The DATA discussed in the paper.

• Weight reduction for mod ℓ Bianchi modular forms 
  with Seyfi Turkelli
  Journal of Number Theory. 129 (2009)

• The nonexistence of certain representations of the absolute Galois group of quadratic fields 
  Proceedings of the AMS. 137 (2009)