Pulling back torsion line bundles to ideal classes

Abstract

Abstract. We prove results concerning the specialization of torsion line bundles on a variety V defined over Q to ideal classes of number fields. This gives a new general technique for constructing and counting number fields with large class group.

The class group pairing and p -descent on elliptic curves

“…We now state a result inspired by the work of Gillibert and Levin [16]. For the convenience of the reader, we give a complete proof, following the method of Gillibert and Levin [16]. Unlike with our previous conventions, p = 2 is allowed in this section.…”

“…. By a variant of Hilbert's irreducibility theorem (see [16,Theorem 2.1]), it is possible to ensure that F is linearly disjoint from L. Then K is linearly disjoint from Q[ζ p ], which is a subfield of L. Combining this with the fact that the primes in S are totally ramified in K, we have O × K,S /p = Z × S /p. Therefore, the Kummer exact sequence over U reads…”

Elliptic surfaces over ℙ1 and large class groups of number fields

“…In the present paper, we investigate this conjecture in the specific case when d = p 2 − 1. Our strategy is closely related to the techniques developed in [GL12] and [BG18]. The main new ingredient is the following result of [GL18]: given a non-isotrivial elliptic surface over k(t) with large rank, for almost all primes p one is able to produce a curve C which admits a morphism C → P 1 k of degree p 2 − 1, and whose Picard group has large p-rank (see Theorem 2.1).…”

A Geometric Approach to Large Class Groups: A Survey

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The purpose of this note is twofold. First, we survey results from [24], [18] and [3] on the construction of large class groups of number fields by specialization of finite covers of curves. Then we give examples of applications of these techniques.

The surveyThe geometric techniques we shall report on are in fact explanations of geometric nature of a strategy which has been used from the beginning of the subject.

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“…The purpose of this note is twofold. First, we survey results from [24], [18] and [3] on the construction of large class groups of number fields by specialization of finite covers of curves. Then we give examples of applications of these techniques.…”