487 research outputs found
On number fields with equivalent integral trace forms
Let be a number field. The \textit{integral trace form} is the integral
quadratic form given by In this
article we study the existence of non-conjugated number fields with equivalent
integral trace forms. As a corollary of one of the main results of this paper,
we show that any two non-totally real number fields with the same signature and
same prime discriminant have equivalent integral trace forms. Additionally,
based on previous results obtained by the author and the evidence presented
here, we conjecture that any two totally real quartic fields of fundamental
discriminant have equivalent trace zero forms if and only if they are
conjugated
On the distribution of class groups of number fields
We propose a modification of the predictions of the Cohen--Lenstra heuristic
for class groups of number fields in the case where roots of unity are present
in the base field. As evidence for this modified formula we provide a large set
of computational data which show close agreement.Comment: 14 pages. To appear in Experimental Mat
Ideal class groups of cyclotomic number fields II
We first study some families of maximal real subfields of cyclotomic fields
with even class number, and then explore the implications of large plus class
numbers of cyclotomic fields. We also discuss capitulation of the minus part
and the behaviour of p-class groups in cyclic ramified p-extensions
Modularity of the Consani-Scholten quintic
We prove that the Consani-Scholten quintic, a Calabi-Yau threefold over QQ,
is Hilbert modular. For this, we refine several techniques known from the
context of modular forms. Most notably, we extend the Faltings-Serre-Livne
method to induced four-dimensional Galois representations over QQ. We also need
a Sturm bound for Hilbert modular forms; this is developed in an appendix by
Jose Burgos Gil and the second author.Comment: 35 pages, one figure; with an appendix by Jose Burgos Gil and Ariel
Pacetti; v3: corrections and improvements thanks to the refere
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