1,347 research outputs found
Computing the number of certain Galois representations mod
Using the link between mod Galois representations of \qu and mod
modular forms established by Serre's Conjecture, we compute, for every prime
, a lower bound for the number of isomorphism classes of continuous
Galois representation of \qu on a two--dimensional vector space over \fbar
which are irreducible, odd, and unramified outside .Comment: 28 pages, 3 table
Universal deformation rings and generalized quaternion defect groups
We determine the universal deformation ring R(G,V) of certain mod 2
representations V of a finite group G which belong to a 2-modular block of G
whose defect groups are isomorphic to a generalized quaternion group D. We show
that for these V, a question raised by the author and Chinburg concerning the
relation of R(G,V) to D has an affirmative answer. We also show that R(G,V) is
a complete intersection even though R(G/N,V) need not be for certain normal
subgroups N of G which act trivially on V.Comment: 20 pages, 6 figures. The paper has been updated as follows: The
results remain true for more general 2-modular blocks with generalized
quaternion defect groups (see the introduction and Hypothesis 3.1). Sections
4 and 5 have been swapped
Universal deformation rings and tame blocks
Let k be an algebraically closed field of positive characteristic, and let W
be the ring of infinite Witt vectors over k. Suppose G is a finite group and B
is a block of kG of infinite tame representation type. We find all finitely
generated kG-modules V that belong to B and whose endomorphism ring is
isomorphic to k and determine the universal deformation ring R(G,V) for each of
these modules.Comment: 14 page
On the evaluation of modular polynomials
We present two algorithms that, given a prime ell and an elliptic curve E/Fq,
directly compute the polynomial Phi_ell(j(E),Y) in Fq[Y] whose roots are the
j-invariants of the elliptic curves that are ell-isogenous to E. We do not
assume that the modular polynomial Phi_ell(X,Y) is given. The algorithms may be
adapted to handle other types of modular polynomials, and we consider
applications to point counting and the computation of endomorphism rings. We
demonstrate the practical efficiency of the algorithms by setting a new
point-counting record, modulo a prime q with more than 5,000 decimal digits,
and by evaluating a modular polynomial of level ell = 100,019.Comment: 19 pages, corrected a typo in equation (8) and added equation (9
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