Let $f$, $g$ and $h$ be three general automorphic forms on $\operatorname{GL}(2)$.
Do we know that $L(s, f\times g\times h)$ comes from an automorphic form on $\operatorname{GL}(8)$?
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2$\begingroup$ The short answer is no. $\endgroup$– LuciaCommented May 12, 2014 at 22:24
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3$\begingroup$ The slightly longer answer is that proving such a result using traditional methods would be to check nice analytic properties for L-functions of twists, e.g. L(s, f x g x h x j) where j is an automorphic form on GL(7) (or a bit lower if you're lucky). That's out of range for now, as far as I can tell. $\endgroup$– MartyCommented May 13, 2014 at 6:53
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$\begingroup$ By Cogdell--Piatetski-Shapiro, j automorphic on GL(n) for n up to 6 is sufficient. So, the long answer is no as well. $\endgroup$– RaminCommented May 16, 2014 at 3:11
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$\begingroup$ But suppose $f$, $g$, and $h$ are holomorphic modular forms. Is there a way to prove (potential) automorphy of $f \times g \times h$? $\endgroup$– RaminCommented May 16, 2014 at 3:21
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