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• W2330698946 abstract "In [B1] the first author proved a theorem about differential algebraic groups (in the sense of Cassidy-Kolchin [C], [K2]) that implied the geometric LangMordell conjecture [L]; cf. Theorem 1 below. The proof in [B1] used an analytic argument, based on “Big Picard”. Hrushovski [H] was able to replace this analytic argument by a remarkable model theoretic argument, based in its turn on the difficult theory developed in [HZ]. Now Hrushovski’s model theoretic methods further lead to striking new results on differential algebraic groups [HS]; cf. Theorem 3 below. So a natural challenge presents itself: to find a proof for these new results that is free from model theory. In this note we shall give in particular a quick analytic proof of Theorem 3, based, again, on “Big Picard”. The main result of this note is the “Gap” Theorem 2 below, which should be viewed as a significant complement to Theorem 1; Theorems 1 and 2 will then easily imply Theorem 3. Some comments are in order concerning the role of the authors in this paper. The second author asked whether analytic methods as in [B1] could yield strong minimality and local modularity of the “Manin kernel” of a simple abelian variety which does not descend to the constants (Theorem 3 below). The first author then proved strong minimality (assertion 1 of Theorem 3) by proving a weaker version of Theorem 2 (namely for simple abelian varieties). The second author suggested some general ideas, analogous to those in [P1], for obtaining local modularity. In trying to carry out details, the first author came up with the proof of the present Theorem 2, from which everything follows. Recall some basic terminology of differential algebra [K1], [C], [B2]. (The definitions below will suffice to understand the statements of the Theorems below, without assuming any previous knowledge of differential algebra; for the proofs, however, familiarity with [B1], [B2] is required.) Let F be a δ−field (i.e. a field of characteristic zero equipped with a derivation δ.) One defines the ring of δ−polynomials F{y1, ..., yN} as the ring of usual polynomials with F−coefficients in the variables δyj , 1 ≤ j ≤ N, i ≥ 0. There is an obvious notion of order for δ−polynomials. F is said to be δ−closed if for any A, B ∈ F{y}, B = 0, such that the order of A is strictly bigger than the order of B there" @default.
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• W2330698946 date "1997-01-01" @default.
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• W2330698946 title "A gap theorem for abelian varieties over differential fields" @default.
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• W2330698946 doi "https://doi.org/10.4310/mrl.1997.v4.n2.a4" @default.
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