Restriction to the diagonal of Hilbert eigenforms

Question

Do you know of any reference that discusses whether the restriction to the diagonal of a Hilbert eigenform is an (elliptic) eigenform?

Answer 1

It is extremely unusual for the restriction of a Hilbert modular form to the diagonal to be an elliptic modular eigenform. It happens occasionally in some small cases (by coincidence, essentially), but there is nothing systematic which forces it to occur.

For instance, when $[K : \mathbf{Q}] = 2$, it turns out that there is a formula for the Petersson product $\langle res(\mathbf{f}), g\rangle$, where $\mathbf{f}$ is a Hilbert eigenform of weight $(k_1, k_2)$ over $K$, $res(\mathbf{f})$ is its diagonal restriction, and $g$ is an elliptic eigenform of weight $k_1 + k_2$. By a theorem of Paul Garrett (edit: and Harris--Kudla), the product $\langle res(\mathbf{f}), g \rangle$ is (up to various fudge factors) equal to the square root of the central value of a degree 8 L-function -- the Rankin convolution of $g$ with the Asai L-function of $\mathbf{f}$. It would be extremely unlikely for this L-function to vanish for all but one of the possible eigenforms $g$ of the relevant level, so one wouldn't expect $res(\mathbf{f})$ to be an eigenform.