Universal Covers of Commutative Finite Morley rank Groups

Abstract

We give an algebraic description of the structure of the analytic universal cover of a complex abelian variety which suffices to determine the structure up to isomorphism. More generally, we classify the models of theories of "universal covers" of rigid divisible commutative finite Morley rank groups.

Model theory of special subvarieties and Schanuel-type conjectures

“…The following is an extension of the previous result to abelian varieties incorporated in [6] by M. Bays, B. Hart and A. Pillay Theorem A AbV . Let A be an Abelian variety over a number field k 0 ⊂ C and such that every endomorphism θ ∈ O (complex multiplication) is defined over k 0 .…”

“…In this setting Theorem A holds for an arbitrary commutative algebraic group over algebraically closed field of arbitrary characteristic. In fact, [6] building up on [7] by Bays, Gavrilovich and Hils shows how to generalise the statement to an arbitrary commutative finite Morley rank group and proves it in this formulation.…”

“…On the other hand Bays, Hart and Pillay in [6] prove a broad generalisation of A-style theorems at the cost of fixing more parameters in the "natural axiomatisation". Their axioms include the complete diagram of the prime model.…”

“…In this regard there is a substantial difference between results of type A (Theorems A) and the main result about the "one-sorted" pseudo-exponentiation stated in 2.3. In the first situation, as shown in [6], one can use essentially the same techniques to classify models of the first order theory of the two-sorted structure in question.…”

Pseudo-exponential maps, variants, and quasiminimality

“…The result in this case follows from the first two cases. This can be seen modeltheoretically in the context of [BHP14] as a matter of transitivity of atomicity, but we give here a direct argument.…”