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We define a new class of sets —stable sets —of primes in number fields. For example, Chebotarev sets , with Galois and , are very often stable. These sets have positive (but arbitrarily small) Dirichlet density and they generalize sets with density one in the sense that arithmetic theorems such as certain Hasse principles, the Grunwald–Wang theorem, and Riemann’s existence theorem hold for them. Geometrically, this allows us to give examples of infinite sets with arbitrarily small positive density such that is a (simultaneously for all ).
For a group of order , where is a prime with , we consider the regular subgroups that are normalized by , the left regular representation of . These subgroups are in one-to-one correspondence with the Hopf–Galois structures on separable field extensions with . Elsewhere we showed that if then all such lie within the normalizer of the Sylow -subgroup of . Here we show that one only need assume that all groups of a given order have a unique Sylow -subgroup, and that not be a divisor of the order of the automorphism groups of any group of order . We thus extend the applicability of the program for computing these regular subgroups and concordantly the corresponding Hopf–Galois structures on separable extensions of degree .
We prove a variety of results on tensor product factorizations of finite dimensional Hopf algebras (more generally Hopf algebras satisfying chain conditions in suitable braided categories). The results are analogs of well-known results on direct product factorizations of finite groups (or groups with chain conditions) such as Fitting’s lemma and the uniqueness of the Krull–Remak–Schmidt factorization. We analyze the notion of normal (and conormal) Hopf algebra endomorphisms, and the structure of endomorphisms and automorphisms of tensor products. The results are then applied to compute the automorphism group of the Drinfeld double of a finite group in the case where the group contains an abelian factor. (If it doesn’t, the group can be calculated by results of the first author.)
We prove several basic extension theorems for reductive group schemes via extending Lie algebras and via taking schematic closures. We also prove that, for each scheme , the category in groupoids of adjoint group schemes over whose Lie algebra -modules have perfect Killing forms is isomorphic, via the differential functor, to the category in groupoids of Lie algebra -modules which have perfect Killing forms and which, as -modules, are coherent and locally free.
We classify Hopf actions of Taft algebras on path algebras of quivers, in the setting where the quiver is loopless, finite, and Schurian. As a corollary, we see that every quiver admitting a faithful -action (by directed graph automorphisms) also admits inner faithful actions of a Taft algebra. Several examples for actions of the Sweedler algebra and for actions of are presented in detail. We then extend the results on Taft algebra actions on path algebras to actions of the Frobenius–Lusztig kernel , and to actions of the Drinfeld double of .
Fix a prime . Let be the Galois representation coming from a non-CM irreducible component of Hida’s -ordinary Hecke algebra. Assume the residual representation is absolutely irreducible. Under a minor technical condition we identify a subring of containing such that the image of is large with respect to . That is, contains for some nonzero -ideal . This paper builds on recent work of Hida who showed that the image of such a Galois representation is large with respect to . Our result is an -adic analogue of the description of the image of the Galois representation attached to a non-CM classical modular form obtained by Ribet and Momose in the 1980s.
Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve of equation , we prove that, given linearly independent points on with coordinates in , there are at most finitely many complex numbers such that the points satisfy two independent relations on . This is a special case of conjectures about unlikely intersections on families of abelian varieties.