A Note on Open Homomorphisms Between Global Solvably Closed Galois Groups

In the present paper, we study continuous open homomorphisms between the Galois groups of solvably closed Galois field extensions of number fields. In particular, we discuss Uchida's conjecture that asserts that an arbitrary continuous open homomorphism between the Galois groups of solvably closed Galois field extensions of number fields arises from a homomorphism between the given Galois field extensions. In the present paper, we prove that this conjecture is equivalent to the assertion that if the Galois group of a Galois field extension of a number field is isomorphic to an open subgroup of the maximal prosolvable quotient of the absolute Galois group of the field of rational numbers, then, for all prime numbers l and all but finitely many prime numbers p, the given Galois extension field contains l roots of the polynomial t[l]−p. Moreover, we prove that this conjecture is also equivalent to the assertion that if the Galois group of a Galois field extension of an absolutely Galois number field is isomorphic to an open subgroup of the maximal prosolvable quotient of the absolute Galois group of the field of rational numbers, then the given Galois extension field is absolutely Galois

ON THE FUNDAMENTAL GROUPS OF LOG CONFIGURATION SCHEMES

In the present paper, we study the cuspidalization problem for the fundamental group of a curve by means of the log geometry of the log configuration scheme, which is a natural compactification of the usual configuration space of the curve. The goal of this paper is to show that the fundamental group of the configuration space is generated by the images from morphisms from a group extension of the fundamental groups of the configuration spaces of lower dimension, and that the fundamental group of the configuration space can be partially reconstructed from a collection of data concerning the fundamental groups of the configuration spaces of lower dimension.</p

Canonical Liftings of Level Two of Tetrapods in Characteristic Three

In the present paper, we give concrete descriptions of the canonical liftings of level two of tetrapods in characteristic three

A Note on Torsion Points on Ample Divisors on Abelian Varieties

In the present paper, we consider torsion points on ample divisors on abelian varieties. We prove that, for each integer n &le; 2, an effective divisor of level n on an abelian variety does not contain the subgroup of n-torsion points. Moreover, we also discuss an application of this result to the study of the p-rank of cyclic coverings of curves in positive characteristic

Mono-anabelian Reconstruction of Solvably Closed Galois Extensions of Number Fields

A theorem of Uchida asserts that every continuous isomorphism between the Galois groups of solvably closed Galois extensions of number fields arises from a unique isomorphism between the solvably closed Galois extensions. In particular, the isomorphism class of a solvably closed Galois extension of a number field is completely determined by the isomorphism class of the associated Galois group. On the other hand, neither the statement of this theorem nor the proof of this theorem yields an "explicit reconstruction" of the given solvably closed Galois extension. In the present paper, we establish a functorial "grouptheoretic" algorithm for reconstructing, from the Galois group of a solvably closed Galois extension of a number field, the given solvably closed Galois extension equipped with the natural Galois action

Tripod-degrees

Let p, l be distinct prime numbers. A tripod-degree over p at l is defined to be an l-adic unit obtained by forming the image, by the l-adic cyclotomic character, of some continuous automorphism of the geometrically pro-l fundamental group of a split tripod over a finite field of characteristic p. The notion of a tripod-degree plays an important role in the study of the geometrically pro-l anabelian geometry of hyperbolic curves over finite fields, e.g., in the theory of cuspidalizations of the geometrically pro-l fundamental groups of hyperbolic curves over finite fields. In the present paper, we study the tripod-degrees. In particular, we prove that, under a certain condition, the group of tripod-degrees over p at l coincides with the closed subgroup of the group of l-adic units topologically generated by p. As an application of this result, we also conclude that, under a certain condition, the natural homomorphism from the group of automorphisms of the split tripod to the group of outer continuous automorphisms of the geometrically pro-l fundamental group of the split tripod that lie over the identity automorphism of the absolute Galois group of the basefield is surjective

A Note on Fields Generated by Jacobi Sums

In the present paper, we study fields generated by Jacobi sums. In particular, we completely determine the field obtained by adjoining, to the field of rational numbers, all of the Jacobi sums “of two variables” with respect to a fixed maximal ideal of the ring of integers of a fixed prime-power cyclotomic field