Pre-Pólya group in even dihedral extensions of ℚ

Investigating on Pólya groups [P. J. Cahen and J. L. Chabert Integer-Valued Polynomials, Mathematical Surveys and Monographs, Vol. 48 (American Mathematical Society, Providence, 1997)] in non-Galois number fields, Chabert [J. L. Chabert and E. Halberstadt, From Pólya fields to Pólya groups (II): Non-Galois number fields, J. Number Theory (2020), https://doi.org/10.1016/j.jnt.2020.06.008 ] introduced the notion of pre-Pólya group [Formula: see text], which is a generalization of the pre-Pólya condition, duo to Zantema [H. Zantema, Integer valued polynomials over a number field, Manuscripta Math. 40 (1982) 155–203]. In this paper, using class field theory, we describe the pre-Pólya group of a [Formula: see text]-field [Formula: see text], for [Formula: see text] an even integer, where [Formula: see text] denotes the dihedral group of order [Formula: see text]. Moreover, for special case [Formula: see text], we improve the Zantema’s upper bound on the maximum ramification in Pólya [Formula: see text]-fields.

One Level Density for Cubic Galois Number Fields

Katz and Sarnak predicted that the one level density of the zeros of a family of L-functions would fall into one of five categories. In this paper, we show that the one level density for L-functions attached to cubic Galois number fields falls into the category associated with unitary matrices.

Modular Abelian Varieties Over Number Fields

The main result of this paper is a characterization of the abelian varieties B/K defined over Galois number fields with the property that the L-function L(B/K; s) is a product of L-functions of non-CM newforms over ℚ for congruence subgroups of the form Γ1(N). The characterization involves the structure of End(B), isogenies between the Galois conjugates of B, and a Galois cohomology class attached to B/K.We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied, we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting.