45 research outputs found
Generators of the group of modular units for Gamma1(N) over QQ
We give two explicit sets of generators of the group of invertible regular
functions over QQ on the modular curve Y1(N).
The first set of generators is the most surprising. It is essentially the set
of defining equations of Y1(k) for k<=N/2 when all these modular curves are
simultaneously embedded into the affine plane, and this proves a conjecture of
Maarten Derickx and Mark van Hoeij. This set of generators is an elliptic
divisibility sequence in the sense that it satisfies the same recurrence
relation as the elliptic division polynomials.
The second set of generators is explicit in terms of classical analytic
functions known as Siegel functions. This is both a generalization and a
converse of a result of Yifan Yang.
Our proof consists of two parts. First, we relate our two sets of generators.
Second, we use q-expansions and Gauss' lemma for power series to prove that our
functions generate the full group of modular functions. This second part shows
how a proof of Kubert and Lang for Y(N) can be much simplified and strengthened
when applied to Y1(N).
The link between the two sets of generators also provides a set of generators
of the ring of regular functions of Y1(N), giving a more uniform version of a
result of Ja Kyung Koo and Dong Sung Yoon.Comment: 18 page
On polarised class groups of orders in quartic CM-fields
We give an explicit necessary condition for pairs of orders in a quartic
CM-field to have the same polarised class group. This generalises a simpler
result for imaginary quadratic fields. We give an application of our results to
computing endomorphism rings of abelian surfaces over finite fields, and we use
our results to extend a completeness result of Murabayashi and Umegaki to a
list of abelian surfaces over the rationals with complex multiplication by
arbitrary orders.Comment: 19 pages, v2 strengthened results slightly and changed theorem
numbering, v3 further strengthened results and added more details, v4 eased
the presentation but changed notations and numbering, v5 updated references,
v6 removes mistaken "transitivity" statemen
Examples of CM curves of genus two defined over the reflex field
In "Proving that a genus 2 curve has complex multiplication", van Wamelen
lists 19 curves of genus two over with complex multiplication
(CM). For each of the 19 curves, the CM-field turns out to be cyclic Galois
over . The generic case of non-Galois quartic CM-fields did not
feature in this list, as the field of definition in that case always contains a
real quadratic field, known as the real quadratic subfield of the reflex field.
We extend van Wamelen's list to include curves of genus two defined over this
real quadratic field. Our list therefore contains the smallest "generic"
examples of CM curves of genus two.
We explain our methods for obtaining this list, including a new
height-reduction algorithm for arbitrary hyperelliptic curves over totally real
number fields. Unlike Van Wamelen, we also give a proof of our list, which is
made possible by our implementation of denominator bounds of Lauter and Viray
for Igusa class polynomials.Comment: 31 pages; Updated some reference
Computing Igusa class polynomials
We bound the running time of an algorithm that computes the genus-two class
polynomials of a primitive quartic CM-field K. This is in fact the first
running time bound and even the first proof of correctness of any algorithm
that computes these polynomials.
Essential to bounding the running time is our bound on the height of the
polynomials, which is a combination of denominator bounds of Goren and Lauter
and our own absolute value bounds. The absolute value bounds are obtained by
combining Dupont's estimates of theta constants with an analysis of the shape
of CM period lattices.
The algorithm is basically the complex analytic method of Spallek and van
Wamelen, and we show that it finishes in time Otilde(Delta^(7/2)), where Delta
is the discriminant of K. We give a complete running time analysis of all parts
of the algorithm, and a proof of correctness including a rounding error
analysis. We also provide various improvements along the way.Comment: 31 pages (Various improvements to the exposition suggested by the
referee. For the most detailed exposition, see Chapter II of the author's
thesis http://hdl.handle.net/1887/15572
Generalized class polynomials
The Hilbert class polynomial has as roots the j-invariants of elliptic curves
whose endomorphism ring is a given imaginary quadratic order. It can be used to
compute elliptic curves over finite fields with a prescribed number of points.
Since its coefficients are typically rather large, there has been continued
interest in finding alternative modular functions whose corresponding class
polynomials are smaller. Best known are Weber's functions, that reduce the size
by a factor of 72 for a positive density subset of imaginary quadratic
discriminants. On the other hand, Br\"oker and Stevenhagen showed that no
modular function will ever do better than a factor of 100.83. We introduce a
generalization of class polynomials, with reduction factors that are not
limited by the Br\"oker-Stevenhagen bound. We provide examples matching Weber's
reduction factor. For an infinite family of discriminants, their reduction
factors surpass those of all previously known modular functions by a factor at
least 2.Comment: 28 pp. 5 fi