Kevin Buzzard

Imperial College London | Imperial · Department of Mathematics

Computers have already changed the way that humans do mathematics: they enable us to compute efficiently. But will they soon be helping us to reason ? And will they one day start reasoning themselves? We give an overview of recent developments in neural networks, computer theorem provers, and large language models.

A proof is one of the most important concepts of mathematics. However, there is a striking difference between how a proof is defined in theory and how it is used in practice. This puts the unique status of mathematics as exact science into peril. Now may be the time to reconcile theory and practice, i.e. precision and intuition, through the advent...

We discuss the idea that computers might soon help mathematicians to prove theorems in areas where they have not previously been useful. Furthermore we argue that these same computer tools will also help us in the communication and teaching of mathematics.

We tell the story of how schemes were formalized in three different ways in the Lean theorem prover.

Perfectoid spaces are sophisticated objects in arithmetic geometry introduced by Peter Scholze in 2012. We formalised enough definitions and theorems in topology, algebra and geometry to define perfectoid spaces in the Lean theorem prover. This experiment confirms that a proof assistant can handle complexity in that direction, which is rather diffe...

We survey the progress (or lack thereof!) that has been made on some questions about the p-adic slopes of modular forms that were raised by the first author in Buzzard (Astérisque 298:1–15, 2005), discuss strategies for making further progress, and examine other related questions.

We report on a computation of holomorphic cuspidal modular forms of weight one and small level (currently level at most $1500$) and classification of them according to the projective image of their attached Artin representations. The data we have gathered, such as Fourier expansions and projective images of Hecke newforms and dimensions of space of...

We survey the progress (or lack thereof!) that has been made on some questions about the p-adic slopes of modular forms that were raised by the first author in [Buz05], discuss strategies for making further progress, and examine other related questions.

Automorphic forms and Galois representations have played a central role in the development of modern number theory, with the former coming to prominence via the celebrated Langlands program and Wiles' proof of Fermat's Last Theorem. This two-volume collection arose from the 94th LMS-EPSRC Durham Symposium on 'Automorphic Forms and Galois Representa...

We develop some of the foundations of affinoid pre-adic spaces without Noetherian or finiteness hypotheses. We give some explicit examples of non-adic affinoid pre-adic spaces (including a locally perfectoid one). On the positive side, we also show that if every affinoid subspace of an affinoid pre-adic space is uniform, then the structure presheaf...

We report on a systematic computation of weight one cuspidal eigenforms for the group Γ 1(N) in characteristic zero and in characteristic p>2. Perhaps the most surprising result was the existence of a mod 199 weight 1 cusp form of level 82 which does not lift to characteristic zero.

We explain a highly efficient algorithm for playing the simplest type of dots and boxes endgame optimally (by which we mean ``in such a way so as to maximise the number of boxes that you take''). The algorithm is sufficiently simple that it can be learnt and used in over-the-board games by humans. The types of endgames we solve come up commonly in...

We report on a systematic computation of weight one cuspidal eigenforms for the group $\Gamma_1(N)$ in characteristic zero and in characteristic $p>2$. Perhaps the most surprising result was the existence of a mod 199 weight~1 cusp form of level 82 which does not lift to characteristic zero.

We use the p-adic local Langlands correspondence for GL2(Qp) to explicitly compute the reduction modulo p of certain 2-dimensional crystalline representations of small slope, and give applications to modular forms. 1 Introduction. Let f = ∑ n≥1 anq n be a weight k cusp form for the group Γ1(N) ⊆ SL2(Z), and assume that f is normalised (a1 = 1), is...

A Spitalfields Day at the Newton Institute was organised on the subject of the recent theorem that any elliptic curve over any totally real field is potentially modular. This article is a survey of the strategy of the proof, together with some history. Introduction: Our main goal in this article is to talk about recent theorems of Taylor and his co...

We state conjectures on the relationships between automorphic representations and Galois representations, and give evidence for them.

In 1987 Serre conjectured that any mod l ("ell", not "1") two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where l is unramified....

We use the p-adic local Langlands correspondence for GL_2(Q_p) to explicitly compute the reduction modulo p of crystalline representations of small slope, and give applications to modular forms. Comment: 10 pages, appeared in IMRN 2009, no. 12. This version does not incorporate any minor changes (e.g. typographical changes) made in proof

This collection of survey and research articles brings together topics at the forefront of the theory of L-functions and Galois representations. Highlighting important progress in areas such as the local Langlands programme, automorphic forms and Selmer groups, this timely volume treats some of the most exciting recent developments in the field. In...

We prove that, near the boundary of weight space, the 2-adic eigencurve of tame level 1 can be written as an infinite disjoint union of annuli, and on each annulus the slopes of the corresponding overconvergent eigenforms tend to zero.

For p=2 and tame level N=1 we prove that the map from the (Coleman-Mazur) Eigencurve to weight space satisfies the valuative criterion of properness. More informally, we show that the Eigencurve has no "holes"; given a punctured disc of finite slope overconvergent eigenforms over weight space, the center can be "filled in" with a finite slope overc...

We formulate a conjecture which predicts, in many cases, the precise p-adic valuations of the eigenvalues of the Hecke operator Tp acting on spaces of classical modular forms. The conjecture has very concrete consequences in the classical theory, but can also be thought of as saying that there is a lot of unexplained symmetry in many of the Coleman...

Coleman and Mazur have constructed “eigencurves”, geometric objects parametrising certain overconvergent p-adic modular forms. We formulate definitions of overconvergent p-adic automorphic forms for two more classes of reductive groups — firstly for GLI over a number field, and secondly for D x , D a definite quaternion algebra over the rationals....

Gouvea and Mazur made a precise conjecture about slopes of modular forms. Weaker versions of this conjecture were established by Coleman and Wan. In this note, we exhibit explicit examples contradicting the full conjecture as it currently stands.

The slope of a p-adic overconvergent eigenform of weight k is the p-adic valuation of its U_p eigenvalue. We find the slope of all 2-adic finite slope overconvergent eigenforms of tame level 1 and weight 0. As a consequence we prove that any finite slope 2-adic overconvergent eigenform of tame level 1 and weight 0 has coefficients in Q_2. These res...

Let f be an overconvergent p-adic eigenform of level Np r , r≥1, with non-zero U p -eigenvalue. We show how f may be analytically continued to a subset of X 1 (Np r ) an containing, for example, all the supersingular locus. Using these results we extend the main theorem of our earlier work with R. Taylor [Ann. Math. (2) 149, No. 3, 905–919 (1999; Z...

We give eight new examples of icosahedral Galois represen-tations that satisfy Artin's conjecture on holomorphicity of their L-function. We give in detail one example of an icosahe-dral representation of conductor 1376 = 2 5 · 43 that satisfies Artin's conjecture. We briefly explain the computations be-hind seven additional examples of conductors 2...

-

If ρ : Gal(ℚ^ac/ℚ))→GL₂(ℂ) is a continuous odd irreducible representation wit nonsolvable image, then under certain local hypotheses we prove that ρ is the representation associated to a weight 1 modular form and hence that the L-function of ρ has an analytic continuation to the entire complex plane.

The theory of “level-lowering” for mod l modular forms is now essentially complete when l is odd, thanks to work of Ribet and others. In the paper R. Taylor [Pac. J. Math., Spec. Issue, 337-347 (1998; Zbl 0942.11031)] explains how one might be able to attack new cases of Artin’s conjecture if (amongst other things) Wiles’ results on lifting of modu...

In this paper we prove the following theorem. Let L/\Q_p be a finite extension with ring of integers O_L and maximal ideal lambda. Theorem 1. Suppose that p >= 5. Suppose also that \rho:G_\Q -> GL_2(O_L) is a continuous representation satisfying the following conditions. 1. \rho ramifies at only finitely many primes. 2. \rho mod \lambda is modular...

LetKbe the splitting field of the characteristic polynomial of the Hecke operatorT2acting on thed-dimensional space of cusp forms of weightkand level 1. We show, for various values ofk, that the Galois group Gal(K/Q) is the full symmetric group on d symbols.

We report on a systematic computation of weight one cuspidal eigen-forms for the group Γ1(N) in characteristic zero and in characteristic p > 2. Perhaps the most surprising result was the existence of a mod 199 weight 1 cusp form of level 82 which does not lift to characteristic zero.