Registered users receive a variety of benefits including the ability to customize email alerts, create favorite journals list, and save searches.
Please note that a Project Euclid web account does not automatically grant access to full-text content. An institutional or society member subscription is required to view non-Open Access content.
Contact email@example.com with any questions.
We consider t-structures that naturally arise on elliptic fibrations. By filtering the category of coherent sheaves on an elliptic fibration using the torsion pairs corresponding to these t-structures, we prove results describing equivalences of t-structures under Fourier–Mukai transforms.
Consider an abelian variety of dimension defined over . Assume that , where is an integer, and let be linearly independent points. (So, in particular, have infinite order, and if , then the set is empty.) For a rational prime of good reduction for , let be the reduction of at , let for be the reduction of (modulo ), and let be the subgroup of generated by . Assume that and . (Note that this particular assumption forces the inequality , but we can take care of the case , under the right assumptions, also.) Then in this article, in particular, we show that the number of primes for which has at most cyclic components is infinite. This result is the right generalization of the classical Artin’s primitive root conjecture in the context of general abelian varieties; that is, this result is an unconditional proof of Artin’s conjecture for abelian varieties. Artin’s primitive root conjecture (1927) states that, for any integer or a perfect square, there are infinitely many primes for which is a primitive root . (This conjecture is not known for any specific .)
For a Jacobian elliptic surface over a finite field and a prime different from the characteristic of , the points of period on the smooth fibers of yield, for each , a smooth projective curve over by taking Zariski closure in and normalization. We consider the restriction map in -adic étale cohomology . By using an earlier result of ours we prove that, except for at most a finite number of such primes , this map is faithful on the submodule of those classes vanishing on the geometric fibers and on the zero section of , and that it gives an isomorphism between this submodule and the subgroup of of primitive elements in the sense of Serre.
The relation between the Auslander (resp., Bass) class and the class of modules with finite Gorenstein projective (resp., injective) dimension is well known when these mentioned classes are built with a dualizing module over Noetherian -perfect rings. Basically, the results are necessary conditions to ensure that both classes coincide. In this article we try to extend and sometimes improve some of these results by weakening the condition of being dualizing. Among other results, we prove that a Wakamatsu tilting module with some extra conditions is precisely a module such that the Bass class coincides with the class of modules of finite Gorenstein injective dimension.
Let and be odd primes. For a positive integer , let be the ray class field of modulo . We present certain class fields of such that , and we provide a necessary and sufficient condition for . We also construct, in the sense of Hilbert, primitive generators of the field over by using Shimura’s reciprocity law and special values of theta constants.
In this article, the fractional Hardy-type operator of variable order is shown to be bounded from the variable-exponent Herz–Morrey spaces into the weighted space , where is log-Hölder continuous both at the origin and at infinity, with some , and when is not necessarily constant at infinity.
Let be an odd prime, and let . Consider the loop space for with . Then we first determine the condition for the power map on to be an -map. We next assume that is a simply connected -finite -space and that is a primitive st root of unity mod . Our results show that if the reduced power operations act trivially on the indecomposable module and the power map on is an -map with , then is -acyclic.
When a singular projective variety admits a projective crepant resolution and a smoothing , we say that and are related by extremal transition. In this article, we study a relationship between the quantum cohomology of and in some examples. For -dimensional conifold transition, a result of Li and Ruan implies that the quantum cohomology of a smoothing is isomorphic to a certain subquotient of the quantum cohomology of a resolution with the quantum variables of exceptional curves specialized to one. We observe that similar phenomena happen for toric degenerations of , , and by explicit computations.