Subscribe: The Quarterly Journal of Mathematics - Advance Access
Preview: The Quarterly Journal of Mathematics - Advance Access

The Quarterly Journal of Mathematics Advance Access

Published: Tue, 24 Oct 2017 00:00:00 GMT

Last Build Date: Tue, 24 Oct 2017 04:47:21 GMT


Comjectures uniformes sur les variÉtÉs abÉliennes


We consider several conjectures on abelian varieties over number fields whose common feature is a bound that depends only on the dimension of the variety and the degree of the number field. For example, Coleman’s conjecture predicts that only a finite number of rings can occur as endomorphism rings once these two parameters are fixed. We show that this conjecture implies the existence of a small polarization as well as a uniform isogeny conjecture (without Faltings height) which in turn implies the uniform torsion conjecture. We then discuss several variants of the Lang–Silverman conjecture on heights and implications between them. In particular, we show how a rather weak version is, under Coleman’s conjecture, equivalent to much more precise versions. We build on Bertrand’s work to give a bound explicit in terms of the polarization.

AR-Components of domestic finite group schemes: McKay-Quivers and Ramification


For a domestic finite group scheme, we give a direct description of the Euclidean components in its Auslander–Reiten quiver via the McKay-quiver of a finite linearly reductive subgroup scheme of SL(2). Moreover, for a normal subgroup scheme N of a finite group scheme G, we show that there is a connection between the ramification indices of the restriction morphism ℙ(VN)→ℙ(VG) between their projectivized cohomological support varieties and the ranks of the tubes in their Auslander–Reiten quivers.