5,029 research outputs found
Mumford dendrograms and discrete p-adic symmetries
In this article, we present an effective encoding of dendrograms by embedding
them into the Bruhat-Tits trees associated to -adic number fields. As an
application, we show how strings over a finite alphabet can be encoded in
cyclotomic extensions of and discuss -adic DNA encoding. The
application leads to fast -adic agglomerative hierarchic algorithms similar
to the ones recently used e.g. by A. Khrennikov and others. From the viewpoint
of -adic geometry, to encode a dendrogram in a -adic field means
to fix a set of -rational punctures on the -adic projective line
. To is associated in a natural way a
subtree inside the Bruhat-Tits tree which recovers , a method first used by
F. Kato in 1999 in the classification of discrete subgroups of
.
Next, we show how the -adic moduli space of
with punctures can be applied to the study of time series of
dendrograms and those symmetries arising from hyperbolic actions on
. In this way, we can associate to certain classes of dynamical
systems a Mumford curve, i.e. a -adic algebraic curve with totally
degenerate reduction modulo .
Finally, we indicate some of our results in the study of general discrete
actions on , and their relation to -adic Hurwitz spaces.Comment: 14 pages, 6 figure
Modular polynomials for genus 2
Modular polynomials are an important tool in many algorithms involving
elliptic curves. In this article we investigate their generalization to the
genus 2 case following pioneering work by Gaudry and Dupont. We prove various
properties of these genus 2 modular polynomials and give an improved way to
explicitly compute them
Lang-Trotter and Sato-Tate Distributions in Single and Double Parametric Families of Elliptic Curves
We obtain new results concerning Lang-Trotter conjecture on Frobenius traces
and Frobenius fields over single and double parametric families of elliptic
curves. We also obtain similar results with respect to the Sato-Tate
conjecture. In particular, we improve a result of A.C. Cojocaru and the second
author (2008) towards the Lang-Trotter conjecture on average for polynomially
parameterized families of elliptic curves when the parameter runs through a set
of rational numbers of bounded height. Some of the families we consider are
much thinner than the ones previously studied
Relations among modular points on elliptic curves
Given a correspondence between a modular curve and an elliptic curve A we
study the group of relations among the CM points of A. In particular we prove
that the intersection of any finite rank subgroup of A with the set of CM
points of A is finite. We also prove a local version of this global result with
an effective bound valid also for certain infinite rank subgroups. We deduce
the local result from a ``reciprocity'' theorem for CL (canonical lift) points
on A. Furthermore we prove similar global and local results for intersections
between subgroups of A and isogeny classes in A. Finally we prove Shimura curve
analogues and, in some cases, higher-dimensional versions of these results.Comment: 48 page
Hypergeometric Properties of Genus 3 Generalized Legendre Curves
Inspired by a result of Manin, we study the relationship between certain
period integrals and the trace of Frobenius of genus 3 generalized Legendre
curves. We show that both of these properties can be computed in terms of
"matching" classical and finite field hypergeometric functions, a phenomenon
that has also been observed in elliptic curves and many higher dimensional
varieties.Comment: 13 page
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