A bound for the torsion on subvarieties of abelian varieties (Q6567118)

This is a counterpart in the context of abelian varieties to the theorem proved in [J. Reine Angew. Math. 755, 103--126 (2019; Zbl 1427.14051)] by the second author in the context of tori. That is, for a subvariety \(V\) of an abelian variety \(A\), the authors give a bound on the number of subvarieties of \(V\) that are abelian subvarieties of \(A\) translated by a torsion point and maximal among such subvarieties.\N\NThe bound is simply \((\deg V)^{\dim A}\), but that hides a constant that depends on \(A\) and on a polarisation. Such bounds can be seen as analogues of the Manin-Mumford conjecture (Raynaud's theorem) and the methods used here include many ideas that have been used for other extensions of that. In other respects, the broad outline of the proof is similar to the one for the torus case. The hidden constant is given in terms of a constant that appears in a well-known theorem of Serre: according to a conjecture of Lang, if \(L\) is a finite extension of the field of definition of \(A\) then the adelic representation induced by the action of \(\operatorname{Gal}_{L/K}\) on the \(\ell\)-adic Tate module for all \(\ell\) should contain an open subgroup of the group of homotheties, but all that is known is Serre's result that it contains some power of the homothety group. This power is a mysterious constant and occurs in the description of the constant here.\N\NSerre's result is also a main tool for the proof, because it is used to reduce the size of the problem by replacing \(V\) by a typically much smaller subvariety that still contains the torsion cosets. Namely, one intersects \(V\) with a Galois conjugate that acts by homothety on torsion. If \(V\) is a curve, that is enough (and indeed gives a stronger result), because the torsion can be bounded by an application of Bézout's theorem, but applying that repeatedly in the general case is too weak, and the interpolation approach used in the torus case is required.