2,677 research outputs found
The minimum discriminant of number fields of degree 8 and signature (2,3)
In this paper we describe how to use the algorithmic methods provided by
Hunter and Pohst in order to give a complete classification of number fields of
degree 8 and signature (2,3) with absolute discriminant less than a certain
bound. The choice of this bound comes from the local corrections given by prime
ideals to the lower estimates for discriminants obtained with the
Odlyzko-Poitou-Serre method.Comment: 8 pages: shortened with respect to the previous version, some known
theoretical details have been transferred on the website together with tables
and dat
K3 surfaces with non-symplectic automorphisms of 2-power order
This paper concerns complex algebraic K3 surfaces with an automorphism which
acts trivially on the Neron-Severi group. Complementing a result by Vorontsov
and Kondo, we determine those K3 surfaces where the order of the automorphism
is a 2-power and equals the rank of the transcendental lattice. We also study
the arithmetic of these K3 surfaces and comment on mirror symmetryComment: 19 pages, 1 figure; v3: exposition improved thanks to referee's
comment
On small discriminants of number fields of degree 8 and 9
We classify all the number fields with signature (4,2), (6,1), (1,4) and
(3,3) having discriminant lower than a specific upper bound. This completes the
search for minimum discriminants for fields of degree 8 and continues it in the
degree 9 case. We recall the theoretical tools and the algorithmic steps upon
which our procedure is based, then we focus on the novelties due to a new
implementation of this process on the computer algebra system PARI/GP; finally,
we make some remarks about the final results, among which the existence of a
number field with signature and small discriminant which was not
previously known.Comment: 10 pages. This is a correction of the previous version: typos have
been fixed and references for many details are give
Shimura curve computations via K3 surfaces of Neron-Severi rank at least 19
It is known that K3 surfaces S whose Picard number rho (= rank of the
Neron-Severi group of S) is at least 19 are parametrized by modular curves X,
and these modular curves X include various Shimura modular curves associated
with congruence subgroups of quaternion algebras over Q. In a family of such K3
surfaces, a surface has rho=20 if and only if it corresponds to a CM point on
X. We use this to compute equations for Shimura curves, natural maps between
them, and CM coordinates well beyond what could be done by working with the
curves directly as we did in ``Shimura Curve Computations'' (1998) =
Comment: 16 pages (1 figure drawn with the LaTeX picture environment); To
appear in the proceedings of ANTS-VIII, Banff, May 200
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