I was just solving an Exercise, where we looked at an analytic function $\phi : \mathbf{H} \to \mathbf{C}$, which is automorphic and $\phi (z) = \mathcal{O}(y^{-C})$ for all $C > 0$ as $z \to i \infty$ (we write $z = x + iy$). I think that this implies that $\phi$ is constant, since it is a modular form of weight zero.

The solution to the Exercise however is three pages long and is about proving that $$\Lambda_\phi (s) := 2 \pi^{-s} \Gamma (s) \zeta(s) \mathcal{M}(\phi)(s - 1)$$ satisfies the functional equation $$\Lambda_\phi (s) = \Lambda_\phi(1 - s)$$ where $\mathcal{M}(\phi)(s)$ is the Mellin-transform $\int_0^\infty \phi(x + iy) y^{s - 1} dy$.

This is referred to as "the simplest case of the Rankin-Selberg-method". The statement gets rather easy if $\phi$ is constant. So my question is: What was the actual point of the Exercise? I was hoping for someone to recognize a formulation of the Rankin-Selberg-method (which I have only seen stated differently in the sources I considered) or of a general tool that is used in the theory.


Added: Thinking about the problem again and looking at the specific example of the non-holomorphic Eisenstein series $$E(z, s) = \sum_{(m, n) \in \mathbf{Z}^2, \, (m, n ) \neq (0, 0)} \frac{y^s}{\vert m z + n \vert^{2s}}$$ I suppose that the only weakening on $\phi$ that is necessary is to make $\phi$ a smooth function in $x$ and $y$, instead of a holomorphic function. But my question is still: "What is the context of this exercise?"

  • $\begingroup$ One thing that's constant in the study of automorphic and modular forms is the drive to classify the forms. It's hard to have a complete classification if one isn't sure what the simplest pieces are of whatever you're trying to classify, and the different ways of taking one or more objects and using them to find more. The Rankin-Selberg method is a way to take an old modular (resp. automorphic) form, or (I believe) a pair of them (with the Rankin-Selberg Convolution) and get a new one. Are you reading Iwaniec's book? $\endgroup$ – Benjamin Gadoua Jul 28 '16 at 18:31
  • $\begingroup$ I'm reading a bit in Iwaniec's book every here and there, but the exercise from above was a class-exercise. $\endgroup$ – Steven Jul 28 '16 at 18:34
  • $\begingroup$ Do you mind if I ask who's teaching a class on automorphic forms? $\endgroup$ – Benjamin Gadoua Jul 28 '16 at 18:35
  • $\begingroup$ @BenjaminGadoua are we considering any fixed $x$ here, or $\lim_{x \to 0^+}$ ? $\endgroup$ – reuns Jul 28 '16 at 18:37
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    $\begingroup$ if we know that $\phi(-1/z) = z^k\phi(z)$ with $M_x(\phi)(s) = \int_0^\infty \phi(x+iy) y^{s-1}dy = \int_0^\infty \phi(x+i/y) y^{-s-1}dy $ we have at least formally $M_0(\phi)(s) = \lim_{x \to 0^+} M_x(\phi)(s) = \lim_{x \to 0^+}\int_0^\infty \phi(x+i/y) y^{-s-1}dy = \lim_{x \to 0^+}\int_0^\infty \phi(\frac{-1}{x+iy}) y^{-s-1}dy =\lim_{x \to 0^+}\int_0^\infty (x+iy)^k \phi(x+iy) y^{-s-1}dy = M_0(\phi)(k-s) e^{i \pi k s/2}$ $\endgroup$ – reuns Jul 28 '16 at 18:42

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