1,546 research outputs found
Local constancy of dimensions of Hecke eigenspaces of Automorphic forms
We use a method of Buzzard to study p-adic families of different types of
modular forms - classical, over imaginary quadratic fields and totally real
fields. In the case of totally real fields of even degree, we get local
constancy of dimensions of spaces of fixed slope and varying weight. For
imaginary quadratic fields we obtain bounds independent of the weight on the
dimensions of such spaces.Comment: Revised version which appeared in Journal of Number Theor
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
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
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
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 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
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