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Let v be a henselian valuation of arbitrary rank of a field K with value group G(K) and residue field R(K) and be the unique extension of v to a fixed algebraic closure of K with value group . It is known that a complete distinguished chain for an element θ belonging to with respect to (K, v) gives rise to several invariants associated to θ, including a chain of subgroups of , a tower of fields, together with a sequence of elements belonging to which are the same for all K-conjugates of θ. These invariants satisfy some fundamental relations. In this paper, we deal with the converse: Given a chain of subgroups of containing G(K), a tower of extension fields of R(K), and a finite sequence of elements of satisfying certain properties, it is shown that there exists a complete distinguished chain for an element associated to these invariants. We use the notion of lifting of polynomials to construct it. [ABSTRACT FROM AUTHOR]

In this paper, we shall study type-definable groups in a simple theory with respect to one or several stable reducts. While the original motivation came from the analysis of definable groups in structures obtained by Hrushovski's amalgamation method, the notions introduced are in fact more general, and in particular can be applied to certain expansions of algebraically closed fields by operators. We prove the following (Theorem 3.1): Theorem. Let T be simple and T0 be a stable reduct of T. If G is a type-definable group in T, then there are a *-interpretable group H in T0 and a definable homomorphism φ V G0 ! H such that for independent generic elements g, g0 of G we can name a set D independent of g, g0 with acl.g/; acl.g0/j0 acl.φ.gg0//\acl0.acl.g//;acl.g0//acl.gg0/; where j0 and acl0 denote independence and algebraic closure respectively in the reduct T0 over D. Thus φ captures all possible 0-dependence relations resulting from the group operation. In order to ensure that the morphism is non-trivial, we need to add further hypotheses on the geometric complexity of T over T0: relative one-basedness and relative CM-triviality. The kernel of φ is finite in the first case (Theorem 4.9) and virtually central in the latter (Theorem 5.7). Examples for relatively one-based theories are differentially closed fields in characteristic 0 and fields with a generic automorphism. We therefore recover, up to isogeny, the characterisation of definable groups in these structures from [20, 24]. Finally, we show that all coloured fields and fusions constructed by Hrushovski amalgamation are relatively CM-trivial (Theorem 6.4). [ABSTRACT FROM AUTHOR]

Let E be a modular elliptic curve over a totally real cubic field. We have a cubic-triple product motive over Q constructed from E through multiplicative induction; it is of rank 8. We show that, under certain assumptions on E, the nonvanishing of the central critical value of the L-function attached to the motive implies that the dimension of the associated Bloch-Kato Selmer group is 0. [ABSTRACT FROM AUTHOR]

In this paper, we study an algebraically closed field Ω expanded by two unary predicates denoting an algebraically closed proper subfield k and a multiplicative subgroup Γ. This will be a proper expansion of algebraically closed field with a group satisfying the Mann property, and also pairs of algebraically closed fields. We first characterize the independence in the triple (Ω,k,Γ). This enables us to characterize the interpretable groups when Γ is divisible. Every interpretable group H in (Ω,k,Γ) is, up to isogeny, an extension of a direct sum of k-rational points of an algebraic group defined over k and an interpretable abelian group in Γ by an interpretable group N, which is the quotient of an algebraic group by a subgroup N1, which in turn is isogenous to a cartesian product of k-rational points of an algebraic group defined over k and an interpretable abelian group in Γ. [ABSTRACT FROM AUTHOR]