Strongly modular models of ℚ-curves

Let [Formula: see text] be a [Formula: see text]-curve without complex multiplication. We address the problem of deciding whether [Formula: see text] is geometrically isomorphic to a strongly modular [Formula: see text]-curve. We show that the question has a positive answer if and only if [Formula: see text] has a model that is completely defined over an abelian number field. Next, if [Formula: see text] is completely defined over a quadratic or biquadratic number field [Formula: see text], we classify all strongly modular twists of [Formula: see text] over [Formula: see text] in terms of the arithmetic of [Formula: see text]. Moreover, we show how to determine which of these twists come, up to isogeny, from a subfield of [Formula: see text].

Prime splitting in abelian number fields and linear combinations of Dirichlet characters

Let 𝕏 be a finite group of primitive Dirichlet characters. Let ξ = ∑χ∈𝕏 aχ χ be a nonzero element of the group ring ℤ[𝕏]. We investigate the smallest prime q that is coprime to the conductor of each χ ∈ 𝕏 and that satisfies ∑χ∈𝕏 aχ χ(q) ≠ 0. Our main result is a nontrivial upper bound on q valid for certain special forms ξ. From this, we deduce upper bounds on the smallest unramified prime with a given splitting type in an abelian number field. For example, let K/ℚ be an abelian number field of degree n and conductor f. Let g be a proper divisor of n. If there is any unramified rational prime q that splits into g distinct prime ideals in ØK, then the least such q satisfies [Formula: see text].

Cyclotomic units and class groups in p-extensions of real abelian fields

For a real abelian number field F and for a prime p we study the relation between the p-parts of the class groups and of the quotients of global units modulo cyclotomic units along the cyclotomic p-extension of F. Assuming Greenberg's conjecture about the vanishing of the λ-invariant of the extension, a map between these groups has been constructed by several authors, and shown to be an isomorphism if p does not split in F. We focus in the split case, showing that there are, in general, non-trivial kernels and cokernels.

THE DISCRIMINANT OF ABELIAN NUMBER FIELDS

A formula for computing the discriminant of any Abelian number field K is given. It is presented as a function of the conductor m of K and of the degrees of the fields K ∩ ℚ(ζpα) over ℚ, where p runs through the set of primes that divide m, and pα is the greatest power of p that divides m.