On the realizable classes of the square root of the inverse different in the unitary class group
Copyright policy )On the realizable classes of the square root of the inverse different in the unitary class group
Let [Formula: see text] be a number field with ring of integers [Formula: see text] and let [Formula: see text] be a finite abelian group of odd order. Given a [Formula: see text]-Galois [Formula: see text]-algebra [Formula: see text], write [Formula: see text] for its trace map and [Formula: see text] for its square root of the inverse different, where [Formula: see text] exists by Hilbert’s formula since [Formula: see text] has odd order. The pair [Formula: see text] is locally [Formula: see text]-isometric to [Formula: see text] whenever [Formula: see text] is weakly ramified, in which case it defines a class in the unitary class group [Formula: see text] of [Formula: see text]. Here [Formula: see text] denotes the canonical symmetric bilinear form on [Formula: see text] defined by [Formula: see text] for all [Formula: see text]. We will study the set of all such classes and show that a subset of them forms a subgroup of [Formula: see text].
- Tsinghua University China (People's Republic of)
Class groups and Picard groups of orders, Mathematics - Number Theory, realizable classes, weakly ramified, unitary class group, square root of the inverse different, FOS: Mathematics, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Galois module, Number Theory (math.NT)
Class groups and Picard groups of orders, Mathematics - Number Theory, realizable classes, weakly ramified, unitary class group, square root of the inverse different, FOS: Mathematics, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Galois module, Number Theory (math.NT)
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