487 research outputs found
Tabulation of cubic function fields via polynomial binary cubic forms
We present a method for tabulating all cubic function fields over
whose discriminant has either odd degree or even degree
and the leading coefficient of is a non-square in , up
to a given bound on the degree of . Our method is based on a
generalization of Belabas' method for tabulating cubic number fields. The main
theoretical ingredient is a generalization of a theorem of Davenport and
Heilbronn to cubic function fields, along with a reduction theory for binary
cubic forms that provides an efficient way to compute equivalence classes of
binary cubic forms. The algorithm requires field operations as . The algorithm, examples and numerical data for
are included.Comment: 30 pages, minor typos corrected, extra table entries added, revamped
complexity analysis of the algorithm. To appear in Mathematics of Computatio
Black Box Galois Representations
We develop methods to study -dimensional -adic Galois representations
of the absolute Galois group of a number field , unramified outside a
known finite set of primes of , which are presented as Black Box
representations, where we only have access to the characteristic polynomials of
Frobenius automorphisms at a finite set of primes. Using suitable finite test
sets of primes, depending only on and , we show how to determine the
determinant , whether or not is residually reducible, and
further information about the size of the isogeny graph of whose
vertices are homothety classes of stable lattices. The methods are illustrated
with examples for , and for imaginary quadratic, being
the representation attached to a Bianchi modular form.
These results form part of the first author's thesis.Comment: 40 pages, 3 figures. Numerous minor revisions following two referees'
report
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