Lloyd James Peter Kilford

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BSc & ARCS, Imperial College, University of London, August 1999.
PhD & DIC, Imperial College, University of London, November 2002.
Bateman Research Instructor at the California Institute of Technology, September 2002 -- July 2004.
Visitor at the Max-Planck-Institut fuer Mathematik, August 2004 -- September 2004.
Fellow at Merton College, Oxford (and member of the Mathematical Institute and the Number Theory Group), September 2004 -- September 2007.
Research Fellow at the University of Bristol, September 2007 --.

I am interested in Number Theory; more specifically in such topics as (the arithmetic of) Modular Forms, Elliptic Curves and Arithmetic Algebraic Geometry -- especially the computational aspects of these subjects. I often use computer algebra packages like MAGMA or PARI/GP or SAGE in my work.

I am listed as one of the many contributors to SAGE 4.0.

At the moment I am working on Hecke operators acting on quaternionic modular forms, and considering certain non-Gorenstein Hecke algebras. I am also considering how best one should approximate certain Hecke operators acting on classical modular forms of level p, where X_0(p) has genus 1.

Here are some programs for computing the Hecke operator U_p on quaternionic modular forms.

Here is some code to calculate U_p acting on overconvergent p-adic modular forms, for p such that X_0(p) has genus 1 (in other words, p = 11, 17 or 19).

I taught the course Introduction to Modular Forms in Oxford, as a trial for an undergraduate course to run in 2005--2006, and as a graduate course again in 2006--2007.

I taught the Part C course Introduction to Modular Forms; the lectures were at noon on Tuesdays and Fridays in the Mathematics Institute in Oxford.

In my previous post at Caltech, I taught courses in graduate algebra, Galois theory, elliptic curves for undergraduates and number theory.

I have written a book, Modular Forms: A classical and computational introduction, available on Amazon. It is a graduate-level introduction to the theory of modular forms, starting with the construction of Eisenstein series and working up to the action of Hecke operators. It also covers the computational side of the theory, and shows how SAGE and MAGMA can be used to compute modular forms.

The cover was created using SAGE by Tomas Boothby; I am very grateful to him for his work.

• Counting Mod $\ell$ Solutions via Modular Forms, with Edray Goins, in preparation.
• On mod $p$ modular representations which are defined over $\F_p$: II, with Gabor Wiese, to appear in the Glasgow Mathematical Journal.
• Experimental finding of modular forms for noncongruence subgroups.
• Computing the action of the $U_p$ operator on $p$-adic overconvergent modular forms when $X_0(p)$ has genus~1, to appear in the Journal of Number Theory.
• On a $p$-adic extension of the Jacquet-Langlands correspondence to weight 1, submitted.
• Generating spaces of modular forms with $\eta$-quotients, JP journal of Algebra, Number Theory and Applications, Volume 8 No. 2 , 2007, pages 213--216.
• On the failure of the Gorenstein property for Hecke algebras of prime weight, with Gabor Wiese, math.NT/0612317, Experimental Mathematics, Volume 17, number 1, pages 37--52.
• On the slopes of the $U_5$ operator acting on overconvergent modular forms, math.NT/0606363, Journal de Theorie des Nombres de Bordeaux, Volume 20, number 1, 2008, pages 165--182.
• On mod $p$ modular representations which are defined over $\F_p$, math.NT/0606052, Glasnik Matematicki, Volume 43, number 1, 2008, pages 1--6.
• The 2-adic eigencurve at the boundary of weight space, with Kevin Buzzard, Compositio Mathematica, Volume 141(3), 2005, pages 605--619.
• A Mathematical Tourist in Oxford [about the Merton College astrolabes], Mathematics Today, Volume 41, Number 4, page 116.
• A Mathematical Tourist in Germany [about Gallo-Roman dodecahedra], Mathematics Today, Volume 40, Number 6, December 2004, page 204.
• Explicit computations of the slopes of the U_5 operator acting on overconvergent modular forms, Preprint number MPIM2004-91 of the Max-Planck-Institut preprint series.
• A generalization of a necessary and sufficient condition for primality due to Vantieghem, International Journal of Mathematics and Mathematical Sciences, Volume 70, pages 3889--3892, 2004.
• Computing with overconvergent modular forms, Communications in Computer Algebra (the SIGSAM Bulletin), Volume 38, Number 3, September 2004.
• Slopes of 2-adic overconvergent modular forms with small level. Mathematics Research Letters, Volume 11, Issue 6, November 2004, pages 723--739.
• Slopes of overconvergent modular forms, PhD thesis, Imperial College of Science, Technology and Medicine, October 2002. Available as DVI or PDF or PS.
• Some non-Gorenstein Hecke algebras attached to spaces of modular forms, Journal of Number Theory, Volume 97, No. 1, November 2002, pages 157--164.

• Chartered Mathematician.
• Member of the Institute for Mathematics and its Applications.
• Committee Member for the Younger Members' Group of the IMA.
• Committee Member for the West of England Branch of the IMA.
• Editor for the Young Mathematicians' Network.
• Reviewed two articles for The Handbook of Information Security (John Wiley & Sons), one for IMRN, one for LMS JCM.
• Reviewer for Zentralblatt.
• Served on Kimball Martin's thesis defence committee, May 2004.
• Member of the committee of the Bristol and Bath Branch of the British Association for the Advancement of Science.
• Co-organizer of the Workshop: Computations with Modular Forms at Bristol, held 17 --- 23 August 2008.

• ASCIIMathML - provides a nice way to write mathematics on web pages. See an example here.
• The Art Of Problem-Solving website has a very useful online application called the TeXer, which creates .gif images from LaTeX input. Here is an example:

This page last modified by L. J. P. Kilford
Tuesday, 09-June-2009 10:39:00 BST