Kevin Buzzard

Analytic continuation of overconvergent eigenforms

Let f be an overconvergent p-adic eigenform of level Np r , r 1, with non-zero Up-eigenvalue. We show how f may be analytically continued to a subset of X1(Np r ) an containing, for example, all the supersingular locus. Using these results we extend the main theorem of [BT] to many ramified cases.

Automorphic Forms and Galois Representations: The conjectural connections between automorphic representations and Galois representations

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

Companion forms and weight one forms.

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 and absolutely irreducible. 3. \rho is unramified at p and \rho(Frob_p) has eigenvalues \alpha and \beta with distinct reductions modulo \lambda. Then there exists a classical weight one eigenform f = \sum_{n=1}^\infty a_m(f) q^m and an embedding of \Q(a_m(f)) into L such that for almost all primes q, a_q(f)=tr(\rho(\Frob_q)). In particular \rho has finite image and for any embedding i of L in \C, the Artin L-function L(i o \rho, s) is entire.

On icosahedral Artin representations

If ρ : Gal(Qac/Q) → GL2(C) is a continuous odd irreducible representation with 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. Introduction E. Artin [A] conjectured that theL-seriesL(r, s) of any continuous representation r : Gal(Qac/Q) −→ GLn(C) is entire except possibly for a pole at s = 1 whenr contains the trivial representation. The case when n = 1 is simply a restatement of the Kronecker-Weber theorem and standard results on the analytic continuation of Dirichlet L-series. Artin proved his conjecture when r is induced from a 1-dimensional representation of an open subgroup of Gal (Qac/Q). Moreover, R. Brauer [ Br] was able to show in general that L(r, s) is meromorphic on the whole complex plane. Since then, the only real progress has been for n = 2, although very recently D. Ramakrishnan [Ra] has dealt with somen = 4 cases. Whenn = 2, such representations can be classified according to the image of the projectivised representation proj : Gal(Qac/Q) −→ PGL2(C). This image is either cyclic, dihedral, the alternating group A4 (the tetrahedral case), the symmetric group S4 (the octahedral case), or the alternating group A5 (the icosahedral case). When the image of projr is cyclic, thenr is reducible and Artin’s conjecture follows from the n = 1 case. When the image of proj is dihedral, thenr is induced from a character of an open subgroup of index 2, and so Artin himself proved the conjecture in this case, although the result is implicit in earlier work of E. Hecke [ H ]. R. Langlands [Langl] proved Artin’s conjecture for tetrahedral and some octahedral representations. J. Tunnell [Tu] extended this to all octahedral representations. These results are based on Langlands’s theory of cyclic base change for automorphic representations of GL 2, DUKE MATHEMATICAL JOURNAL Vol. 109, No. 2, c ©2001 Received 29 December 1999. Revision received 13 October 2000. 2000Mathematics Subject Classification . Primary 11F11, 11F80; Secondary 11F33, 11G18, 14G22. Taylor’s work partially supported by National Science Foundation grant number DMS-9702885 and by the Miller Institute at the University of California at Berkeley.

The 2-adic eigencurve at the boundary of weight space

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 “evenly-spaced” annuli, and on each annulus the slopes of the corresponding overconvergent eigenforms tend to zero.

Slopes of overconvergent 2-adic modular forms

We explicitly compute all the slopes of the Hecke operator U2 acting on overconvergent 2-adic level 1 cusp forms of weight 0: the nth slope is 1 + 2v((3n)!/n!), where v denotes the 2-adic valuation. We formulate an explicit conjecture about what these slopes should be for weight k forms.

On p-adic families of automorphic forms

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. We give several reasons why we believe the objects we construct to be the correct analogue of an overconvergent p-adic modular form in this setting.

L-functions and Galois representations

Preface List of participants 1. Stark-Heegner points and special values of L-series Massimo Bertolini, Henri Darmon and Samit Dasgupta 2. Presentations of universal deformation rings Gebhard Bockle Eigenvarieties Kevin Buzzard 3. Nontriviality of Rankin-Selberg L-functions and CM points Christophe Cornut and Vinayak Vatsal 4. A correspondence between representations of local Galois groups and Lie-type groups Fred Diamond 5. Non-vanishing modulo p of Hecke L-values and application Haruzo Hida 6. Serres modularity conjecture: a survey of the level one case Chandrashekhar Khare 7. Two p-adic L-functions and rational points on elliptic curves with supersingular reduction Masato Kurihara and Robert Pollack 8. From the Birch and Swinnerton-Dyer Conjecture to non-commutative Iwasawa theory via the Equivariant Tamagawa Number Conjecture - a survey Otmar Venjakob 9. The Andre-Oort conjecture - a survey Andrei Yafaev 10. Locally analytic representation theory of p-adic reductive groups: a summary of some recent developments Matthew Emerton 11. Modularity for some geometric Galois representations - with an appendix by Ofer Gabber Mark Kisin 12. The Euler system method for CM points on Shimura curves Jan Nekovar 13. Representations irreductibles de GL(2,F ) modulo p Marie-France Vigneras.

Slopes of Modular Forms

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 (Asterisque 298:1–15, 2005), discuss strategies for making further progress, and examine other related questions.

Explicit reduction modulo p of certain 2-dimensional crystalline representations, II

We complete the calculations begun in [BG09], using the p-adic local Langlands correspondence for GL2(Q_p) to give a complete description of the reduction modulo p of the 2-dimensional crystalline representations of G_{Q_p} of slope less than 1, when p > 2.