Representation-theoretic operations on modular forms

Question

Let $A$ and $B$ be Hecke eigenforms of some weight $k$ and level $N$. We know that there are irreducible representations $\rho_a$, $\rho_b$ of the absolute Galois group of $\mathbb{Q}$ whose trace of Frobenius of $p$ is given by $a_p$ or $b_p$, the Fourier coefficients of $A$ or $B$.

$AB$ is a modular form of weight $2k$. It isn't clearly a Hecke eigenform, but its Fourier coefficients are still algebraic numbers, and it can be expressed as a sum of Hecke eigenforms and possibly an Eisenstein series of weight $2k$.

We also have $\rho_a \otimes \rho_b$ a representation. It isn't irreducible, but can be expressed as a sum of irreducibles. The obvious hope is that this would correspond to the product, but that cannot be true because the characters don't match up. But convolution if appropriately defined would produce a form related to $\rho_a \otimes \rho_b$.

So my two questions: what is multiplication of modular forms representation-theoretically, and what is tensoring of representations on the modular forms side?

Answer 1

Modular forms correspond to a $2$-dimensional Galois representations. Tensoring two of those produces a $4$-dimensional Galois representation, which, unless it decomposes into two $2$-dimensional representations, will be related to higher-dimensional automorphic forms, not modular forms.

Conversely:

Galois representations correspond to eigenforms, and multiplying two eigenforms does not usually (does not always? does not ever?) produce an eigenform. For instance $g_2$ and $g_3$ are eigenforms, but $g_2^3$ and $g_3^2$ are not - instead, $g_2^3-27g_3^2$ is.

So each operation takes you out of the domain of consideration, and hence can't directly correspond to anything on the other side!

Much of the difficulty in proving the correspondence of modular forms and Galois representations, or more general Langlands correspondences between automorphic forms and Galois representations, is that constructions which are very natural on one side often are much less natural or have no known analogue on the other.

Answer 2

With the notation of the question, the tensor product $\rho_a \otimes\rho_b$ is a degree $4$ Galois representation so corresponds to an automorphic representation for ${\rm GL}(4)$. This representation cannot be constructed in an obvious way from the modular forms $A$ and $B$, but its $L$-function can be obtained as the (now classical) Rankin-Selberg convolution of $A$ and $B$.