# Rankin-Selberg convolutions of motivic L-series

Background: Let $M_{f_i}, i=1,2$ be two modular motives associated to cusp forms $f_i \in S_{w_i}(\Gamma_0(N_i))$ of weight $w_i$ and level $N_i$ respectively. The Rankin-Selberg convolution associates an L-series $L(f_1\otimes f_2,s)$ to this pair of modular forms. In the framework of automorphic motives a natural question is whether the L-series $L(M,s)$ of a motive $M$ can be represented in terms of modular submotives $M_{f_i}$ as $L(M,s) \stackrel{?}{=} L(f_1\otimes f_2,s)$.

Question: Is there a (practical) test that can be applied to a given (motivic) L-series as to whether it admits a Rankin-Selberg product representation?

• Clearly, there are some necessary conditions that follow from the structure of the Euler factors, but what do you mean by a "practical test"? What is given and what operations can be performed? – Victor Protsak Aug 20 '10 at 2:36
• Dear Laie, I am putting this here to direct your attention to my comment below David Hansen's answer, in case you don't otherwise notice it. – Emerton Aug 22 '10 at 4:54

If $L(s,M)$ is an irreducible degree 4 motivic L-function, and $L(s,\mathrm{sym}^2(M))$ has a pole, then either $M=f_1 \otimes f_2$ for a pair of distinct classical modular forms $f_1, f_2$, or $M=\mathrm{Asai}(f)$ for $f$ cuspidal on $GL_2(K)$, with $K/\mathbb{Q}$ quadratic. You can rule out the Asai case if $L(s,\mathrm{sym}^2(M)\otimes \chi)$ is entire for any nontrivial quadratic character (in particular, entire for $\chi$ the character of $K$). If by "practical" you mean "local", then I think the answer is no.
• Well, self-dual cuspidal representations of $GL_4$ are either orthogonal, in which case they come from $GO_4$, or symplectic, in which case they come from $GSp_4$. These cases are determined by whether the symmetric or exterior square L-function has a pole, respectively. Then the possible split forms of $GO_4$ are $P(\mathrm{Res}_{K/\mathbb{Q}}GL_2)$ for $K/\mathbb{Q}$ a quadratic etale algebra. I don't know a good reference for my actual statements, but you can extract them from what I just said + Asai's paper (do a "by hand" computation of the Euler factors of $\mathrm{sym^2 Asai}(f)$). – David Hansen Aug 20 '10 at 17:36
• Dear Laie, Do you mean compute formally (say as an Euler product), in which case you can look at the Euler factors; or do you mean compute as a fuction of $s$? (So that you just have a whole bunch of numbers, obtained by evaluating your function at various values of $s$.) – Emerton Aug 22 '10 at 4:52
Dear Laie, If the $f_i$ correspond to the motives $M_{f_i}$, then $L(f_1\otimes f_2,s)$ is the $L$-function of the product $M_{f_1}\times M_{f_2}$. (If you like, this is an interpretation of the Kunneth formula in cohomology.)