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
Abstract. We present an algorithm that, on input of a CM-field K, an integer k ≥ 1, and a prime r ≡ 1 mod k, constructs a q-Weil number π ∈ OK corresponding to an ordinary, simple abelian variety A over the field F of q elements that has an F-rational point of order r and embedding degree k with respect to r. We then discuss how CM-methods over K can be used to explicitly construct A.
Van Wamelen [Math. Comp. 68 (1999) no. 225, 307-320] lists 19 curves of genus two over Q with complex multiplication (CM). However, for each curve, the CM-field turns out to be cyclic Galois over Q, and the generic case of a non-Galois quartic CM-field did not feature in this list. The reason is that the field of definition in that case always contains 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.
In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.2010 Mathematics subject classification: primary 11B39; secondary 11G05. Keywords and phrases: lucas sequence, elliptic divisibility sequence, primitive divisor, function field over number field.
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