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For my master's thesis, I am studying Hrushovski's model-theoretic proof of the Manin-Mumford Conjecture. Among the references I have used are the following:

  1. Lecture notes 'Model Theory of Difference Fields' by Chatzidakis
  2. 'A Survey on the Model Theory of Difference Fields' by Chatzidakis
  3. 'Théorie des modèles et conjecture de Manin-Mumford' by Bouscaren https://eudml.org/doc/110272
  4. Hrushovski's article on http://www.sciencedirect.com/science/article/pii/S0168007201000963

My first goal is to understand the following proposition (crl. 4.1.13 in (4)): Let $A$ be an Abelian variety, defined over $Fix(\sigma)$. Let $p(T)$ be a polynomial with integer coefficients. Then $\ker(p( \sigma))$ is LMS $\iff$ $p$ has no cyclotomic factors.

Here, a definable group $B$ is LMS (stable, stably embedded, locally modular) if every definable subset of $B^n$ (with parameters possibly outside $B$) is a finite Boolean combination of cosets of definable subgroups of $B^n$.

References 2. and 3. both give the big steps in the proof without providing all details. As far as I understand, the strategy of the proof is as follows:

  1. Reduce to the case that $A$ is a simple Abelian variety.
  2. Let $H = \ker(p(\sigma))$. Towards contradiction, suppose that $H$ is not LMS. If $H = \ker(f_1\cdot \dots \cdot f_k)$ with $f_i$ irreducible in $\mathbb{Q}\otimes \operatorname{E}(A)$, where $E(A)$ the definable endomorphisms of $A$, then one of the $\ker(f_i)$ is not LMS. So reduce to the case that $H=\ker(f_i)$.
  3. It follows that $H$ is $c$-minimal and has finite $\sigma$-degree.
  4. By the classification of weakly normal groups of Hrushovski and Pillay, it follows that $H$ is not modular and stable.
  5. Then by the Dichotomy theorem in characteristic 0 (Chatzidakis and Hrushovski), $H$ is not orthogonal to $\sigma(x)=x$.
  6. It follows that $\ker(f_i)\subseteq \ker(\sigma^N -1)$ for some $N$. Now by choosing $N$ large enough, $f_i$ becomes invertible, which contradicts the assumption that it was irreducible.
  7. Therefore, $H$ is LMS.

However, I do not succeed in matching these steps to the actual proof in the article by Hrushovski. I have the following specific questions:

(i). In reference no. 2., the dichotomy theorem is about stable formulas in the sense of stability theory. In reference no 3., an algebraic condition for a stable set is given. Why are these definitions equivalent in this context?

(ii). What is the definition of a stable type? How does it follow that a superficially stable type is stable and stably embedded?

(iv). Apparently, it follows from the classification of $c$-minimal subgroups of abelian varieties (Hrushovski prop. 4.1.2) and the theorem by Hrushovski and Pillay that because $\ker(f_i)$ is $c$-minimal, that it is not stable and modular. However, I do not see how that follows, because the notion of weakly normal groups is not used in Hrushovski's article.

Could somebody perhaps help me find an answer to these questions or refer me to literature?

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