23,001 research outputs found
Modular embeddings of Teichmueller curves
Fuchsian groups with a modular embedding have the richest arithmetic
properties among non-arithmetic Fuchsian groups. But they are very rare, all
known examples being related either to triangle groups or to Teichmueller
curves.
In Part I of this paper we study the arithmetic properties of the modular
embedding and develop from scratch a theory of twisted modular forms for
Fuchsian groups with a modular embedding, proving dimension formulas,
coefficient growth estimates and differential equations.
In Part II we provide a modular proof for an Apery-like integrality statement
for solutions of Picard-Fuchs equations. We illustrate the theory on a worked
example, giving explicit Fourier expansions of twisted modular forms and the
equation of a Teichmueller curve in a Hilbert modular surface.
In Part III we show that genus two Teichmueller curves are cut out in Hilbert
modular surfaces by a product of theta derivatives. We rederive most of the
known properties of those Teichmueller curves from this viewpoint, without
using the theory of flat surfaces. As a consequence we give the modular
embeddings for all genus two Teichmueller curves and prove that the Fourier
developments of their twisted modular forms are algebraic up to one
transcendental scaling constant. Moreover, we prove that Bainbridge's
compactification of Hilbert modular surfaces is toroidal. The strategy to
compactify can be expressed using continued fractions and resembles
Hirzebruch's in form, but every detail is different.Comment: revision including the referee's comments, to appear in Compositio
Mat
Zeta functions of regular arithmetic schemes at s=0
Lichtenbaum conjectured the existence of a Weil-\'etale cohomology in order
to describe the vanishing order and the special value of the Zeta function of
an arithmetic scheme at in terms of Euler-Poincar\'e
characteristics. Assuming the (conjectured) finite generation of some \'etale
motivic cohomology groups we construct such a cohomology theory for regular
schemes proper over . In particular, we obtain
(unconditionally) the right Weil-\'etale cohomology for geometrically cellular
schemes over number rings. We state a conjecture expressing the vanishing order
and the special value up to sign of the Zeta function at
in terms of a perfect complex of abelian groups
. Then we relate this conjecture to
Soul\'e's conjecture and to the Tamagawa number conjecture of Bloch-Kato, and
deduce its validity in simple cases.Comment: 53 pages. To appear in Duke Math.
Three-term arithmetic progressions and sumsets
Suppose that G is an abelian group and A is a finite subset of G containing
no three-term arithmetic progressions. We show that |A+A| >> |A|(log
|A|)^{1/3-\epsilon} for all \epsilon>0.Comment: 20 pp. Corrected typos. Updated references. Corrected proof of
Theorem 5.1. Minor revisions
Multiplicities of Periodic Orbit Lengths for Non-Arithmetic Models
Multiplicities of periodic orbit lengths for non-arithmetic Hecke triangle
groups are discussed. It is demonstrated both numerically and analytically that
at least for certain groups the mean multiplicity of periodic orbits with
exactly the same length increases exponentially with the length. The main
ingredient used is the construction of joint distribution of periodic orbits
when group matrices are transformed by field isomorphisms. The method can be
generalized to other groups for which traces of group matrices are integers of
an algebraic field of finite degree
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