Mathematics Department King's College, University of London

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Research in Number Theory

• Members of the group and current principal interests
• Research
• Number theory in London
• Postgraduate study
• Recent publications and preprints

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Members of the group and current principal interests Back to top Professor David Burns The Tamagawa number conjecture of Bloch and Kato (and its equivariant refinements); Non-commutative Iwasawa theory; Algebraic K-theory. Professor Colin Bushnell The representation theory of p-adic groups and its relations with Number Theory via the Langlands conjectures. Dr. David Solomon Stark's conjectures, their refinements and p-adic analogues.

Current research students
Alexis Cooper, Andrew Parker, Andrew Jones

Some recent research students and post-PhD positions
Manuel Breuning (2002-2004): EPSRC Postdoctoral Research Fellow, Nottingham University
Anthony Hayward (2001-2004): Accenture
Sey-Yoon Kim (1999-2002): Postdoctoral Fellow, McMaster University
Shaun Stevens (1995-1998): Postdoctoral Research Fellow, University of Orsay; Lecturer in Mathematics, University of East Anglia

Research	Back to top

King's College London has a strong tradition of research in number theory and this continues today with a particular emphasis on algebraic and representation-theoretic aspects of the subject. Staff are at the forefront of research in areas attracting international interest. Recent projects have involved collaboration with workers in France, Germany, Japan, the United States and South Korea.

Much of the work done at King's involves L- and zeta-functions in one guise or another, not only the complex versions of these functions attached to number fields but also their p-adic-valued analogues and similar functions attached to local fields. However, rather than study their quantitative and `analytic' properties per se, our interest centres on the extraordinary capacity of these functions for reflecting the fine algebraic structure of such objects as the unit groups, class groups, algebraic K-groups, Galois groups and matrix-groups that occur in the arithmetic of the local and global fields. Indeed, the last thirty years have seen a huge amount of interest by number theorists worldwide in exploiting these links in even more general contexts: for instance, the L- and zeta-functions associated to arithmo-geometric objects such as algebraic curves defined over (Galois extensions of) number fields. This has stimulated - and, in turn, been stimulated by - some far-reaching conjectures and while much remains mysterious, progress is slowly being made.

Number theory in London	Back to top

The King's College Number Theory Group is based in the Mathematics Department, Strand campus, King's College London.

London itself is perhaps the most active centre of number theory research in the United Kingdom, with several colleges of the University of London very strong in research in important areas of number theory and/or arithmetic algebraic geometry. This lively number-theoretic community makes for an active weekly Number Theory Seminar, which rotates between King's, University College and Imperial College. Every term, there is also a Study Group held weekly on a topic of general interest or current progress.

London is also blessed with the country's best mathematics library collections, notably the LMS collection in UCL library and also the British Library, to complement KCL's own extensive reference collection in the Maughan Library.

Postgraduate study	Back to top

The mathematics department has a lively postgraduate community. Information on the KCL PhD programme is here.

Recent publications and preprints	Back to top

More details can be found on the relevant personal homepages.

2004

• D. Solomon, Abelian conjectures of stark type in Z_p-extensions of totally real fields,
  in Stark's Conjectures: Recent Work and New Directions, Contemporary Mathematics 358, AMS (2004).
• D. Burns, Congruences between derivatives of abelian L-functions at s=0,
  preprint (2004).
  
• D. Solomon, On twisted Zeta-functions at s=0,
  manuscript submitted.
• D. Burns and A. Hayward, Explicit Units and the Equivariant Tamagawa Number Conjecture, II,
  manuscript submitted (2004).
  
• M. Breuning, Equivariant local epsilon constants and étale cohomology,
  to appear in J. London Math. Soc.
  
• A. Agboola and D. Burns, Twisted forms and relative K-theory,
  manuscript submitted (2004).
  
• D. Burns, On the values of equivariant Zeta functions of curves over finite fields,
  Documenta Math. 9 (2004) 357-399.
  
• X.-F. Roblot and D. Solomon, Verifying a p-adic abelian Stark conjecture at s=1,
  J. Number Theory 107 (2004), no. 1, 168-206.
• D. Burns and J. Lee, On the refined class number formula of Gross,
  J. Number Theory 107 (2004), 282-286.
• M. Breuning, On equivariant global epsilon constants for certain dihedral extensions,
  Math. Comp. 73 (2004), no. 246, 881-898.
• D. Burns, B. Köck and V. Snaith, Refined and l-adic Euler Characteristics of nearly-perfect complexes,
  J. Algebra 272 (2004), 247-272.
  
• D. Burns, Equivariant Whitehead torsion and refined Euler characteristics,
  CRM Proceedings and Lecture Notes 36 (2004) 35-59.
• A. Hayward, A class number formula for higher derivatives of abelian L-functions,
  Compos. Math. 140 (2004), 99-129.
  

2003

• D. Burns and C. Greither, Equivariant Weierstrass preparation and values of L-functions at negative integers,
  Documenta Math., Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 157-185.
• W. Bley and D. Burns, Equivariant epsilon constants, discriminants and etale cohomology,
  Proc. London Math. Soc. 87 (2003), 545-590.
• C. J. Bushnell and G. Henniart, Local tame lifting for GL(n). IV. Simple characters and base change,
  Proc. London Math. Soc. (3) 87 (2003), no. 2, 337-362.
• D. Burns, Equivariant Tamagawa numbers and refined abelian Stark conjectures,
  J. Math. Soc. Univ. Tokyo 10 (2003), 225-259.
• C. J. Bushnell and G. Henniart, Generalized Whittaker models and the Bernstein center,
  Amer. J. Math. 125 (2003), no. 3, 513-547.
• D. Burns and C. Greither, On the equivariant Tamagawa number conjecture for Tate motives,
  Inventiones Math. 153 (2003), 303-359.
• C. J. Bushnell and G. Henniart, Explicit unramified base change: GL(p) of a p-adic field,
  J. Number Theory 99 (2003), no. 1, 74-89.
• D. Burns and M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients II,
  Amer. J. Math. 125 (2003) 475-512.

2002

• D. Solomon, On p-adic abelian Stark conjectures at s=1,
  Ann. Inst. Fourier (Grenoble) 52 (2002), no. 2, 379-417.
• W. Bley and D. Burns, Étale cohomology and a generalisation of Hilbert's Theorem 132,
  Math. Zeit. 239 (2002), 1-25.
• C. J. Bushnell and G. Henniart, On the derived subgroups of certain unipotent subgroups of reductive groups over infinite fields,
  Transform. Groups 7 (2002), no. 3, 211-230.
• D. Solomon, Twisted zeta-functions and abelian Stark conjectures,
  J. Number Theory 94 (2002), no. 1, 10-48.
• C. J. Bushnell and G. Henniart, appendix to H. H. Kim, F. Shahidi, Functorial products for GL₂ x GL₃ and the symmetric cube for GL₂,
  Ann. of Math. (2) 155 (2002), no. 3, 837-893.

2001

• C. J. Bushnell and G. Henniart, The local Rankin-Selberg convolution for GL(n): divisibility of the conductor,
  Math. Ann. 321 (2001), no. 2, 455-461.
• A. Agboola and D. Burns, Grothendieck groups of bundles on varieties over finite fields,
  K Theory 23 (2001) 251-303.
• W. Bley and D. Burns, Explicit Units and the Equivariant Tamagawa Number Conjecture,
  Amer. J. Math. 123 (2001), 931-949.
• C. J. Bushnell and G. Henniart, Sur le comportement, par torsion, des facteurs epsilon de paires,
  Canad. J. Math. 53 (2001), no. 6, 1141-1173.
• S. Hu and D. Solomon, Properties of higher-dimensional Shintani generating functions and cocycles on PGL₃(Q),
  Proc. London Math. Soc. (3) 82 (2001), no.1, 64-88.
• D. Burns and M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients,
  Documenta Math. 6 (2001), 501-570.
• C. J. Bushnell, Representations of reductive p-adic groups: localization of Hecke algebras and applications,
  J. London Math. Soc. (2) 63 (2001), no. 2, 364-386.
• W. Bley and D. Burns, Equivariant Tamagawa Numbers, Fitting ideals and Iwasawa theory,
  Compos. Math. 126 213-247.
• C. J. Bushnell, P. C. Kutzko, Types in reductive p-adic groups: the Hecke algebra of a cover,
  Proc. Amer. Math. Soc. 129 (2001), no. 2, 601-607.
• D. Burns, Equivariant Tamagawa Numbers and Galois module theory,
  Compos. Math. 129 (2001), 203-237.

2000

• C. J. Bushnell and G. Henniart, Davenport-Hasse relations and an explicit Langlands correspondence. II. Twisting conjectures. Colloque International de Théorie des Nombres (Talence, 1999),
  J. Théor. Nombres Bordeaux 12 (2000), no. 2, 309-347.
• C. J. Bushnell and G. Henniart, Calculs de facteurs epsilon de paires sur un corps local. II,
  Compos. Math. 123 (2000), no. 1, 89-115.
• D. Burns, On the equivariant structure of ideals in abelian extensions of local fields (with an Appendix by W. Bley),
  Comm. Math. Helv. 75 (2000), 1-44.
• C. J. Bushnell and G. Henniart, Correspondance de Jacquet-Langlands explicite. II. Le cas de degré égal à la caractéristique résiduelle,
  Manuscripta Math. 102 (2000), no. 2, 211-225.
• C. J. Bushnell and G. Henniart, Davenport-Hasse relations and an explicit Langlands correspondence,
  J. Reine Angew. Math. 519 (2000), 171-199.

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