Mumford-Tate conjecture cases with small $l$-adic monodromy groups

Question

My question concerns the Mumford-Tate conjecture for abelian varieties over number fields.

Most proven cases (that I am familiar with) show that the l-adic monodromy group is as large as it can possibly get because of the conditions imposed and has the same rank as the Mumford-Tate group. A notable exception is the paper of Zhao on four-folds with endomorphism ring $\mathbb{Z}$ which starts out with abelian varieties such that the derived subgroup $G_l^{\mathrm{der}}$ of the $l$-adic monodromy group is a product of three copies of $SL_2$ and shows that the same is true for the Mumford-Tate group (as opposed to the latter being $GSp_{8}$).

Are there any other known results of this type?

Answer 1

This is not a clear answer, but let me attempt to clarify the question a little, and also explain why the problem, properly interpreted, is so difficult:

Deligne's theorem that Hodge classes are absolute Hodge shows that the identity component of the $\ell$-adic monodromy group is a subgroup of the Mumford-Tate group. Hence in some sense, every case of the Mumford-Tate conjecture will involve showing that the $\ell$-adic monodromy group is "as large as it can possibly get because of conditions imposed" on the Mumford-Tate group.

Focusing on the size of the $\ell$-adic monodromy group is a bit of a red herring, I think.

First, because the Mumford-Tate group is an upper bound, one can prove the conjecture under the assumption that the Mumford-Tate group is quite small, as Pink did in Theorem 5.15 of https://people.math.ethz.ch/~pinkri/ftp/AMT-v3.pdf.

Second, one can prove the conjecture also in the case that the $\ell$-adic monodromy group is as small as possible (i.e. abelian) because then by Falting's theorem the abelian variety has many endomorphisms and thus must be CM.

So to state things more precisely, the situation that is hard to rule out is when the $\ell$-adic monodromy group is less than the maximum possible size that commutes with the endomorphism algebra of the abelian variety, but the Mumford-Tate group is larger.

I am not aware of any other results of this type.

Basically, the reason it is so difficult is that we have very few general tools to relate the $\ell$-adic monodromy group and the Mumford-Tate group together, other than the ones already mentioned ( absolute Hodge cycles, Falting's theorem / Hodge conjecture in codimension $1$) and so progress has mostly revolved around proving in as many cases as possible that there is a unique $\ell$-adic monodromy group with a given endomorphism algebra.