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In his book "Automorphic Forms on GL(2), II", Springer Lecture Notes vol. 278, Jacquet defines the Rankin--Selberg L-function of $\pi_1 \times \pi_2$, where $\pi_i$ are automorphic representations of $\operatorname{GL}_2$ over a global field, and proves its analytic continuation and functional equation.

Is there any reference which explains the ideas of Jacquet's theory in a somewhat more concrete and explicit way? In particular, I'm interested in the case where the field is $\mathbf{Q}$ and the $\pi_i$ correspond to holomorphic modular eigenforms -- is there any account of the theory which makes explicit the dictionary between Jacquet's representation-theoretic approach and more "classical" objects?

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  • $\begingroup$ I think Jacquet's theory is pretty much a generalisation of the original Rankin-Selberg method, and specialising to modular forms is exactly the Rankin-Selberg method. This is written explicitly in Jacquet-Gelbart's Relation Between Automorphic Representations of GL(2) and GL(3), in chapter 5, though for the case of symmetric square. Bump's Automorphic Forms and Representations has a section on classical Rankin-Selberg for modular forms, and later in the book a section on the modern viewpoint, explaining a lot, but not all, of Jacquet's theory. $\endgroup$
    – Dror Speiser
    Commented Nov 10, 2015 at 20:12
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    $\begingroup$ @DrorSpeiser I am aware that the basic strategy -- expressing the L-function as an integral, over $GL(2)$, of a product of a pair of cusp forms with a family of Eisenstein series -- is the same as in Rankin and Selberg's works. But it is not true that Jacquet's approach, specialised to modular forms, gives only results that were previously known. The chief difficulty in this theory is to get the "right" local factors at the bad primes (where $\pi_1$ or $\pi_2$ is ramified); Jacquet was the first to even define these factors in general. $\endgroup$
    – David Loeffler
    Commented Nov 10, 2015 at 20:57
  • $\begingroup$ Also I was wondering if (at least in some simple cases) some kind of Rankin-Selberg convolution of L-functions over number fields (ie. things like $\sum_{I \in \mathcal{O}_K} a_I(F) a_I(G) N(I)^{-s}$) could let us construct some non-trivial Artin-L functions from Hecke L-functions. $\endgroup$
    – reuns
    Commented Nov 24, 2017 at 23:30

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I'm not an expert on integral representations, but I don't think such a reference should exist. The method of automorphic representations are crucial to Jacquet's approach which, analogous to Tate's thesis, is to define the local Rankin-Selberg L-factor as a gcd of zeta integrals $Z(s,\phi_v \times \phi_v')$ where $\phi_v, \phi_v'$ run over vectors in the associated local representations. (There are also parameters for additive characters and Eisenstein series, which I supress.) It seems to me there should be no classical translation of this approach, as there is no classical version of varying $\phi_v, \phi_v'$ locally in a $GL_2(\mathbb Q_v)$-representation.

So now you may ask: how can you relate the $L$-function to a classical Rankin-Selberg integral?

In the unramified case, Jacquet proves that $Z(s,\phi_v \times \phi_v') = L(s, \phi_v \times \phi_v')$ where $\phi_v, \phi_v'$ are new vectors. This means the classical approach to Rankin-Selberg should give the right factors at the unramified primes. However, Jacquet does not determine "test vectors" $\phi_{v}, \phi_{v}'$ where the zeta integral spits out the $L$-factor on the nose in general. Indeed, determining test vectors is a nontrivial problem and they are known not to always exist in more general settings (i.e., the gcd may not be attained for any choice of $\phi_{v}, \phi_v'$). In this setting however, test vectors probably do exist, and as I recall under some conditions they were determined in the (unpublished but accessible) thesis of Kim Mi-Kyung, a fairly recent student of Cogdell.)

The problem is that without knowing how $Z(s,\phi_v \times \phi_v')$ relates to $L(s, \phi_v' \times \phi_v')$ when $\phi_v, \phi_v'$ are newvectors for ramified representations, you can't relate the complete $L$-function to a classical Rankin-Selberg integral exactly. (One also needs the analogous archimedean theory; in this case Jacquet does determine test vectors.) Possibly one can do more with the classical Rankin-Selberg integrals now in light of Kim's thesis (I don't remember exactly what she did), but I'm not aware of any such work so far.

If you still want to understand the idea of Jacquet's approach, besides Bump's book and his surveys (discussing more general Rankin-Selberg cases), Cogdell has several nice notes on the Rankin-Selberg method which include the more general setting of $GL(n) \times GL(m)$ (e.g., his Fields lecture notes).

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  • $\begingroup$ So you're basically saying that my question doesn't deserve to be answered because it's morally unsound? (Don't worry, I'm only joking! :-) ). But I think that some "middle way" between Jacquet's approach and the adelic approach does exist, and since posting my question I sat down and worked out the details. $\endgroup$
    – David Loeffler
    Commented Dec 12, 2015 at 10:11
  • $\begingroup$ ... The point is that test vectors at infinity are explicitly known, and one can write down the finite parts of the automorphic representations $\pi$ and $\pi'$ as explicit spaces of holomorphic functions on the upper half-plane (or more accurately on (upper half-plane) * ($\widehat{\mathbf{Z}}^*)$, to keep track of the component groups). One can also write down a collection of (non-holomorphic) Eisenstein series depending on a choice of a Schwartz function on $\mathbf{A}_f^2$; $\endgroup$
    – David Loeffler
    Commented Dec 12, 2015 at 10:13
  • $\begingroup$ .. and unless I'm very much mistaken, the Petersson product of these Eisenstein series with vectors from (our model of) $\pi_f$ and $\pi'_f$ are exactly Jacquet's adelic integrals, so the GCD over all choices of the two cusp forms and the Schwartz function is (by definition) the $L$-function. $\endgroup$
    – David Loeffler
    Commented Dec 12, 2015 at 10:14
  • $\begingroup$ @DavidLoeffler Ah, I didn't realize you could model $\pi_f$ classically. What does this classical model look like? And what does (upper half-plane)*($\widehat{\mathbb Z}^*$) mean? $\endgroup$
    – Kimball
    Commented Dec 12, 2015 at 15:08
  • $\begingroup$ @DavidLoeffler Oh, do you mean to essentially look at the $SL_2(\mathbb Q)$ translates of a modular form? I guess this won't quite give you all of $\pi_f$, but a dense subset, which should be enough to get you Jacquet's global result, but it's not clear to me if you can translate his local analysis into this setting. $\endgroup$
    – Kimball
    Commented Dec 12, 2015 at 17:04

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