20,405 research outputs found
Genus 2 Curves with Complex Multiplication
Genus 2 curves are useful in cryptography for both discrete-log based and pairing-based systems, but a method is required to compute genus 2 curves with Jacobian with a given number of points. Currently, all known methods involve constructing genus 2 curves with complex multiplication via computing their 3 Igusa class polynomials. These polynomials have rational coefficients and require extensive computation and precision to compute. Both the computation and the complexity analysis of these algorithms can be improved by a more precise understanding of the denominators of the coefficients of the polynomials. The main goal of this paper is to give a bound on the denominators of Igusa class polynomials of genus 2 curves with CM by a primitive quartic CM field .
We give an overview of Igusa\u27s results on the moduli space of genus two curves and the method to construct genus 2 curves via their Igusa invariants. We also give a complete characterization of the reduction type of a CM abelian surface, for biquadratic, cyclic, and non-Galois quartic CM fields, and for any type of prime decomposition of the prime, including ramified primes
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
Isogeny graphs with maximal real multiplication
An isogeny graph is a graph whose vertices are principally polarized abelian
varieties and whose edges are isogenies between these varieties. In his thesis,
Kohel described the structure of isogeny graphs for elliptic curves and showed
that one may compute the endomorphism ring of an elliptic curve defined over a
finite field by using a depth first search algorithm in the graph. In dimension
2, the structure of isogeny graphs is less understood and existing algorithms
for computing endomorphism rings are very expensive. Our setting considers
genus 2 jacobians with complex multiplication, with the assumptions that the
real multiplication subring is maximal and has class number one. We fully
describe the isogeny graphs in that case. Over finite fields, we derive a depth
first search algorithm for computing endomorphism rings locally at prime
numbers, if the real multiplication is maximal. To the best of our knowledge,
this is the first DFS-based algorithm in genus 2
Higher dimensional 3-adic CM construction
We find equations for the higher dimensional analogue of the modular curve
X_0(3) using Mumford's algebraic formalism of algebraic theta functions. As a
consequence, we derive a method for the construction of genus 2 hyperelliptic
curves over small degree number fields whose Jacobian has complex
multiplication and good ordinary reduction at the prime 3. We prove the
existence of a quasi-quadratic time algorithm for computing a canonical lift in
characteristic 3 based on these equations, with a detailed description of our
method in genus 1 and 2.Comment: 23 pages; major revie
Primes dividing invariants of CM Picard curves
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
Explicit computations of Serre's obstruction for genus 3 curves and application to optimal curves
Let k be a field of characteristic different from 2. There can be an
obstruction for an indecomposable principally polarized abelian threefold (A,a)
over k to be a Jacobian over k. It can be computed in terms of the rationality
of the square root of the value of a certain Siegel modular form. We show how
to do this explicitly for principally polarized abelian threefolds which are
the third power of an elliptic curve with complex multiplication. We use our
numeric results to prove or refute the existence of some optimal curves of
genus 3.Comment: 24 pages ; added : an explicit model, remarks on the hyperelliptic
and decomposable reduction, reference
Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem
Fix an ordinary abelian variety defined over a finite field. The ideal class
group of its endomorphism ring acts freely on the set of isogenous varieties
with same endomorphism ring, by complex multiplication. Any subgroup of the
class group, and generating set thereof, induces an isogeny graph on the orbit
of the variety for this subgroup. We compute (under the Generalized Riemann
Hypothesis) some bounds on the norms of prime ideals generating it, such that
the associated graph has good expansion properties.
We use these graphs, together with a recent algorithm of Dudeanu, Jetchev and
Robert for computing explicit isogenies in genus 2, to prove random
self-reducibility of the discrete logarithm problem within the subclasses of
principally polarizable ordinary abelian surfaces with fixed endomorphism ring.
In addition, we remove the heuristics in the complexity analysis of an
algorithm of Galbraith for explicitly computing isogenies between two elliptic
curves in the same isogeny class, and extend it to a more general setting
including genus 2.Comment: 18 page
Machine-learning the Sato-Tate conjecture
We apply some of the latest techniques from machine-learning to the arithmetic of hyperelliptic curves. More precisely we show that, with impressive accuracy and confidence (between 99 and 100 percent precision), and in very short time (matter of seconds on an ordinary laptop), a Bayesian classifier can distinguish between Sato–Tate groups given a small number of Euler factors for the L-function. Our observations are in keeping with the Sato-Tate conjecture for curves of low genus. For elliptic curves, this amounts to distinguishing generic curves (with Sato–Tate group SU(2)) from those with complex multiplication. In genus 2, a principal component analysis is observed to separate the generic Sato–Tate group USp(4) from the non-generic groups. Furthermore in this case, for which there are many more non-generic possibilities than in the case of elliptic curves, we demonstrate an accurate characterisation of several Sato–Tate groups with the same identity component. Throughout, our observations are verified using known results from the literature and the data available in the LMFDB. The results in this paper suggest that a machine can be trained to learn the Sato–Tate distributions and may be able to classify curves efficiently
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