On p-adic L-functions and normal bases of rings of integers.
0. Introduction. Since ℤ is a principal ideal domain, every finitely generated torsion-free ℤ-module has a finite ℤ-basis; in particular, any fractional ideal in a number field has an "integral basis". However, if K is an arbitrary number field the ring of integers, A, of K is a Dedekind domain but not necessarily a principal ideal domain. If L/K is a finite extension of number fields, then the fractional ideals of L are finitely generated and torsion-free (or, equivalently, finitely generated and...
Let be a finite abelian extension of , with the ring of algebraic integers of . We investigate the Galois structure of the unique fractional -ideal which (if it exists) is unimodular with respect to the trace form of .
Let be a CM number field, an odd prime totally split in , and let be the -adic analytic space parameterizing the isomorphism classes of -dimensional semisimple -adic representations of satisfying a selfduality condition “of type ”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in has dimension at least . As important steps, and in any rank, we prove that any first order...
Let be a prime number. A finite Galois extension of a number field with group has a normal -integral basis (-NIB for short) when is free of rank one over the group ring . Here, is the ring of -integers of . Let be a power of and a cyclic extension of degree . When , we give a necessary and sufficient condition for to have a -NIB (Theorem 3). When and , we show that has a -NIB if and only if has a -NIB (Theorem 1). When divides , we show that this descent property...
Let K, L be algebraic number fields with K ⊆ L, and , their respective rings of integers. We consider the trace map and the -ideal . By I(L/K) we denote the group indexof in (i.e., the norm of over ℚ). It seems to be difficult to determine I(L/K) in the general case. If K and L are absolutely abelian number fields, however, we obtain a fairly explicit description of the number I(L/K). This is a consequence of our description of the Galois module structure of (Theorem 1). The case...
This paper provides a complete catalog of the break numbers that occur in the ramification filtration of fully and thus wildly ramified quaternion extensions of dyadic number fields which contain (along with some partial results for the more general case). This catalog depends upon the refined ramification filtration, which as defined in [2] is associated with the biquadratic subfield. Moreover we find that quaternion counter-examples to the conclusion of the Hasse-Arf Theorem are extremely rare...
A -adic version of Stark’s Conjecture at is attributed to J.-P. Serre and stated (faultily) in Tate’s book on the Conjecture. Building instead on our previous paper (and work of Rubin) on the complex abelian case, we give a new approach to such a conjecture for real ray-class extensions of totally real number fields. We study the coherence of our -adic conjecture and then formulate some integral refinements, both alone and in combination with its complex analogue. A ‘Weak Combined Refined’ version...