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I'm interested in understanding how one may associate modular symbols to the L-functions and $p$-adic L-functions associated to the Rankin Selberg convolution of two modular forms/ elliptic curves and the symmetric square of a modular form/elliptic curves. Looking through the literature I see that the $\operatorname{GL}_{n-1}\times \operatorname{GL}_n$ case has been treated in general, for instance "Modular symbols for reductive groups and p-adic Rankin-Selberg convolutions over number fields"-Januszewski (arXiv/publisher link) and "Relative modular symbols and $p$-adic Rankin-Selberg convolutions"-Schmidt (behind paywall).

Is there a reference for the theory of modular symbols in the $\operatorname{GL}_{2}\times \operatorname{GL}_2$ and $\operatorname{Sym}^2$ cases?

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I do not think there is a reference for this theory, because as far as I know no such theory exists. I have spent a substantial portion of my career studying the arithmetic of the special values of the $GL_2 \times GL_2$ Rankin--Selberg L-function, and I am not aware of a theory of modular symbols in this setting.

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  • $\begingroup$ I understand that there are technicalities involved, but would it be possible to state what it makes the $\operatorname{GL}_2\times \operatorname{GL}_3$ tractable and the $\operatorname{GL}_2\times \operatorname{GL}_2$ case out of reach? $\endgroup$
    – user130124
    Commented Oct 14, 2019 at 20:36
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    $\begingroup$ The group $G = GL_3 \times GL_2$ has a reductive subgroup $H$, namely the copy of $GL_2$ embedded via $g \mapsto (g \oplus 1, g)$, such that $H$ has an open orbit on the Borel flag variety of $G$ with trivial stabiliser. No such subgroup exists in $GL_2 \times GL_2$. $\endgroup$
    – David Loeffler
    Commented Oct 14, 2019 at 20:57

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