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The existence of an abelian variety over \(\overline{\mathbb{Q}}\) isogenous to no Jacobian. (English) Zbl 1250.14032

In [ibid. 176, No. 1, 589–635 (2012; Zbl 1263.14032)], Ch.-L. Chai and F. Oort proved the following theorem: let \(A_g\) be the moduli space of principally polarized abelian varieties of dimension \(g\) over \(\mathbb{\overline Q}\), and \(X \subset A_g\) a proper closed subvariety. If the André-Oort conjecture holds, then there is a point \([A] \in A_g(\mathbb{\overline Q})\) such that \(A\) is not isogenous to \(B\), for any \([B] \in X\) [C. Chai and F. Oort, Ann. Math. 176, 589–635 (2012; Zbl 1263.14032)]. (The title takes its name from the special case when \(g \geq 4\) and \(X\) is the Torelli locus.)
In this article, the same statement is proven, without use of the André-Oort conjecture. The strategy is to modify Klingler-Yafaev’s conditional proof of André-Oort assuming the GRH [B. Klingler and A. Yafaev, “The André-Oort conjecture”, preprint (2008)]. They need the GRH to produce ‘many’ ‘small’ split primes for certain CM fields. What Tsimerman does is prove in this particular case the existence of sufficiently many CM fields with enough small split primes ‘by hand’, using powerful equidistribution results, after which he can carry out Klingler-Yafaev’s proof unconditionally.

MSC:

14K10 Algebraic moduli of abelian varieties, classification
11G15 Complex multiplication and moduli of abelian varieties
14K02 Isogeny
14K22 Complex multiplication and abelian varieties

Citations:

Zbl 1263.14032
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References:

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