37 research outputs found
On Level One Cuspidal Bianchi Modular Forms
In this paper, we present the outcome of vast computer calculations, locating
several of the very rare instances of level one cuspidal Bianchi modular forms
that are not lifts of elliptic modular forms.Comment: final versio
On Fermat's equation over some quadratic imaginary number fields
Assuming a deep but standard conjecture in the Langlands programme, we prove
Fermat's Last Theorem over . Under the same assumption, we also
prove that, for all prime exponents , Fermat's equation
does not have non-trivial solutions over
and .Comment: The present is a revised version, including suggestions from
referees, that was accepted for publication in Research in Number Theory; 16
page
On the cohomology of linear groups over imaginary quadratic fields
Let Gamma be the group GL_N (OO_D), where OO_D is the ring of integers in the
imaginary quadratic field with discriminant D<0. In this paper we investigate
the cohomology of Gamma for N=3,4 and for a selection of discriminants: D >=
-24 when N=3, and D=-3,-4 when N=4. In particular we compute the integral
cohomology of Gamma up to p-power torsion for small primes p. Our main tool is
the polyhedral reduction theory for Gamma developed by Ash and Koecher. Our
results extend work of Staffeldt, who treated the case n=3, D=-4. In a sequel
to this paper, we will apply some of these results to the computations with the
K-groups K_4 (OO_{D}), when D=-3,-4
On Serre's modularity conjecture and Fermat's equation over quadratic imaginary fields of class number one
In the present article, we extend previous results of the author and we show that when K is any quadratic imaginary field of class number one, Fermat's equation does not have integral coprime solutions such that and is prime. The results are conjectural upon the veracity of a natural generalisation of Serre's modularity conjecture
Twisted limit formula for torsion and cyclic base change
Let be the group of complex points of a real semi-simple Lie group whose
fundamental rank is equal to 1, e.g. G= \SL_2 (\C) \times \SL_2 (\C) or
\SL_3 (\C). Then the fundamental rank of is and according to the
conjecture made in \cite{BV}, lattices in should have 'little' --- in the
very weak sense of 'subexponential in the co-volume' --- torsion homology.
Using base change, we exhibit sequences of lattices where the torsion homology
grows exponentially with the \emph{square root} of the volume.
This is deduced from a general theorem that compares twisted and untwisted
-torsions in the general base-change situation. This also makes uses of a
precise equivariant 'Cheeger-M\"uller Theorem' proved by the second author
\cite{Lip1}.Comment: 23 page