128 research outputs found
Dihedral Galois representations and Katz modular forms
We show that any two-dimensional odd dihedral representation \rho over a
finite field of characteristic p>0 of the absolute Galois group of the rational
numbers can be obtained from a Katz modular form of level N, character \epsilon
and weight k, where N is the conductor, \epsilon is the prime-to-p part of the
determinant and k is the so-called minimal weight of \rho. In particular, k=1
if and only if \rho is unramified at p. Direct arguments are used in the
exceptional cases, where general results on weight and level lowering are not
available.Comment: 11 pages, LaTe
On projective linear groups over finite fields as Galois groups over the rational numbers
Ideas and techniques from Khare's and Wintenberger's article on the proof of
Serre's conjecture for odd conductors are used to establish that for a fixed
prime l infinitely many of the groups PSL_2(F_{l^r}) (for r running) occur as
Galois groups over the rationals such that the corresponding number fields are
unramified outside a set consisting of l, the infinite place and only one other
prime.Comment: 7 pages, LaTeX; tiny change
On the faithfulness of parabolic cohomology as a Hecke module over a finite field
In this article we prove conditions under which a certain parabolic group
cohomology space over a finite field F is a faithful module for the Hecke
algebra of Katz modular forms over an algebraic closure of F. These results can
e.g. be used to compute Katz modular forms of weight one with methods of linear
algebra over F.
This is essentially Chapter 3 of my thesis.Comment: 26 pages; small corrections and change
A Short Note on the Bruinier-Kohnen Sign Equidistribution Conjecture and Hal\'asz' Theorem
In this note, we improve earlier results towards the Bruinier-Kohnen sign
equidistribution conjecture for half-integral weight modular eigenforms in
terms of natural density by using a consequence of Hal\'asz' Theorem. Moreover,
applying a result of Serre we remove all unproved assumptions.Comment: 4 pages, main result made unconditional, minor changes due to
referee's report
On modular Galois representations modulo prime powers
We study modular Galois representations mod . We show that there are
three progressively weaker notions of modularity for a Galois representation
mod : we have named these `strongly', `weakly', and `dc-weakly' modular.
Here, `dc' stands for `divided congruence' in the sense of Katz and Hida. These
notions of modularity are relative to a fixed level .
Using results of Hida we display a `stripping-of-powers of away from the
level' type of result: A mod strongly modular representation of some
level is always dc-weakly modular of level (here, is a natural
number not divisible by ).
We also study eigenforms mod corresponding to the above three notions.
Assuming residual irreducibility, we utilize a theorem of Carayol to show that
one can attach a Galois representation mod to any `dc-weak' eigenform,
and hence to any eigenform mod in any of the three senses.
We show that the three notions of modularity coincide when (as well as
in other, particular cases), but not in general
Hilbertian fields and Galois representations
We prove a new Hilbertianity criterion for fields in towers whose steps are
Galois with Galois group either abelian or a product of finite simple groups.
We then apply this criterion to fields arising from Galois representations. In
particular we settle a conjecture of Jarden on abelian varieties.Comment: 18 pages, accepted for publication in Journal f\"ur die reine und
angewandte Mathemati
- …