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Towards Donovan's conjecture for abelian defect groups
We define a new invariant for a -block, the strong Frobenius number, which
we use to address the problem of reducing Donovan's conjecture to normal
subgroups of index p. As an application we use the strong Frobenius number to
complete the proof of Donovan's conjecture for 2-blocks with abelian defect
groups of rank at most 4 and for 2-blocks with abelian defect groups of order
at most 64
Twists of X(7) and primitive solutions to x^2+y^3=z^7
We find the primitive integer solutions to x^2+y^3=z^7. A nonabelian descent
argument involving the simple group of order 168 reduces the problem to the
determination of the set of rational points on a finite set of twists of the
Klein quartic curve X. To restrict the set of relevant twists, we exploit the
isomorphism between X and the modular curve X(7), and use modularity of
elliptic curves and level lowering. This leaves 10 genus-3 curves, whose
rational points are found by a combination of methods.Comment: 47 page
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