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We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based, not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.
The jaggedness of an order ideal
$I$
in a poset
$P$
is the number of maximal elements in
$I$
plus the number of minimal elements of
$P$
not in
$I$
. A probability distribution on the set of order ideals of
$P$
is toggle-symmetric if for every
$p\in P$
, the probability that
$p$
is maximal in
$I$
equals the probability that
$p$
is minimal not in
$I$
. In this paper, we prove a formula for the expected jaggedness of an order ideal of
$P$
under any toggle-symmetric probability distribution when
$P$
is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan–López–Pflueger–Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp–Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.
Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over
$\mathbb{Q}$
with good reduction away from 3, up to
$\mathbb{Q}$
-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined and an application to a question of Ihara is discussed.
We describe the construction of a database of genus-
$2$
curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated
$L$
-function. This data has been incorporated into the
$L$
-Functions and Modular Forms Database (LMFDB).
Given a sextic CM field
$K$
, we give an explicit method for finding all genus-
$3$
hyperelliptic curves defined over
$\mathbb{C}$
whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng [J. Ramanujan Math. Soc. 16 (2001) no. 4, 339–372], we give an algorithm which works in complete generality, for any CM sextic field
$K$
, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field
$\mathbb{F}_{p}$
with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo
$p$
.
In this paper, we investigate examples of good and optimal Drinfeld modular towers of function fields. Surprisingly, the optimality of these towers has not been investigated in full detail in the literature. We also give an algorithmic approach for obtaining explicit defining equations for some of these towers and, in particular, give a new explicit example of an optimal tower over a quadratic finite field.
We show how to efficiently evaluate functions on Jacobian varieties and their quotients. We deduce an algorithm to compute
$(l,l)$
isogenies between Jacobians of genus two curves in quasi-linear time in the degree
$l^{2}$
.
We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite-index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.
We study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.
We define an infinite class of fractals, called horizontally and vertically blocked labyrinth fractals, which are dendrites and special Sierpiński carpets. Between any two points in the fractal there is a unique arc α; the length of α is infinite and the set of points where no tangent to α exists is dense in α.
We discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell–Weil sieve algorithm and discuss its efficiency.