Categoricity of the two sorted j-function

We show that a natural, two sorted $\cL_{ω_1,ω}$ theory involving the modular $j$-function is categorical in all uncountable cardinaities. It is also shown that a slight weakening of the adelic Mumford-Tate conjecture for products of elliptic curves is necessary and (along with a couple of other results from arithmetic geometry) sufficient for categoricity.

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