Let F-q[T] be the polynomial ring over a finite field F-q. We study the endomorphism rings of Drinfeld F-q[T]-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings generated by the Frobenius endomorphism and deduce from this a refinement of a reciprocity law for division fields of Drinfeld modules proved in our earlier paper. We then use these results to give an efficient algorithm for computing the endomorphism rings and discuss some interesting examples produced by our algorithm. (c) 2019 Elsevier Inc. All rights reserved.

Building on Bosca's method, we extend to tame ray class groups the results on capitulation of ideals of a number field by composition with abelian extensions of a subfield first studied by Gras. More precisely, for every extension of number fields K/k, where at least one infinite place splits completely, and every squarefree divisor m of K, we prove that there exist infinitely many abelian extensions F/k such that the ray class group mod m of K capitulates in KF. As a consequence we generalize to tame ray class groups the results of Kurihara on triviality of class groups for maximal abelian pro-extensions of totally real number fields. (C) 2021 Elsevier Inc. Tous droits reserves.

Let F be a finite field, mu be a fixed additive character and s be an integer coprime to |F-x|. For any a is an element of F, the corresponding Weil sum is defined to be W-F,W-s(a) = Sigma(x is an element of F) mu(x(s) - ax). The Weil spectrum counts distinct values of the Weil sum as a runs through the invertible elements in the finite field. The value of these sums and the size of the Weil spectrum are of particular interest, as they link problems in coding and information theory to other areas of math such as number theory and arithmetic geometry. In the setting of Niho exponents, we examine the Weil sum, its bounds and its spectrum. As a consequence, we give a new proof to the Vanishing Conjecture of Helleseth (1971) on the presence of zero in the Weil spectrum in the case of Niho exponents. We also state a conjecture for when the Weil spectrum contains at least five elements, and prove it for a certain class of Weil sums. (c) 2021 Elsevier Inc. All rights reserved.

We generalize some results of Greither and Popescu to a geometric Galois cover X -> Y which appears naturally for example in extensions generated by p(n)-torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz-Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula involving Stickelberger elements and providing a link (similar to the one in [1]) with Goss zeta-function. (C) 2019 Elsevier Inc. All rights reserved.

In this paper, we prove that the adelic Galois representation rho(phi) : Gal(F-q(T)(sep)/F-q(T)) -> lim <- a Aut(phi[alpha]) congruent to GL(3)((A) over cap) associated to the Drinfeld module phi over F-q(T) of rank 3,phi defined by phi(T) = T + tau(2) + Tq-1 tau(3), is surjective. (c) 2020 Elsevier Inc. All rights reserved.

In this paper we determine the perfect powers that are sums of three fifth powers in an arithmetic progression. More precisely, we completely solve the Diophantine equation (x - d)5 + x5 + (x + d)5 = zn, n > 2, where d, x, z is an element of Z and d = 2a5b with a, b > 0. (c) 2021 Elsevier Inc. All rights reserved.