Finiteness conjecture for optimal subvarieties in mixed Shimura varieties

Let MM be a mixed Shimura variety, let ZMZ\subset M be a subvariety, and let δ(A)=dimAdimA\delta(A)=\dim\langle A\rangle-\dim A denote the defect of a subvariety AA, where A\langle A\rangle is the smallest special subvariety containing it. A subvariety AZA\subset Z is optimal if it is maximal for its defect: whenever ABZA\subset B\subset Z and δ(B)δ(A)\delta(B)\leq\delta(A), one has B=AB=A. Finiteness conjecture for optimal subvarieties. For every subvariety ZMZ\subset M, there are only finitely many optimal subvarieties. This is presented as formally equivalent to the strongest form of the Zilber–Pink conjecture for mixed Shimura varieties, but its status is not specified in the source.

Sources & referencesView supporting material

Primary source

Jonathan Pila and Jacob Tsimerman, “Independence of CM points in Elliptic Curves”, arXiv:1907.02737 (2019).

Additional references

2 papers in this index state this conjecture (2019). The statement above is taken from the most recent of them; the others are arXiv:1905.00827.

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