Playing simple loony dots and boxes endgames optimally

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 practice in well-played games on a 5x5 board and were in fact developed by the authors in order to improve their over-the-board play.Comment: 20 pages; minor revisions made after referee's report. To be published in "Integers

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.Comment: 15 pages, published version, abstract added in migratio

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.Comment: 10 pages. Correcting a minor typ

Stably uniform affinoids are sheafy

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 is a sheaf; note in particular that we assume no finiteness hypotheses on our rings here. One can use our result to give a new proof that the spectrum of a perfectoid algebra is an adic space.Comment: Version 2 of the manuscript -- the arguments are now presented for general f-adic rings with a topologically nilpotent unit (the original proofs still go through in this generality

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 [Buz05], discuss strategies for making further progress, and examine other related questions.Comment: Final published version (to appear in Proceedings of the 2014 Simons symposium on the trace formula

The 2-adic Eigencurve is Proper

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 overconvergent eigenform

Explicit reduction modulo pp of certain crystalline representations

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 proo