Hypergeometric representation of Eisenstein series

It is well known (Fricke ?) that $ E_4^{1/4}$ and $ E_6^{1/6}$ can be represented as Gauss hypergeometric functions of $ 1728/j$ and $ 1728/(1728-j)$ respectively. The same result is true in levels $ 2$ , $ 3$ , and $ 4$ for instance, in which case $ E_4$ , $ E_6$ , and $ j$ must be replaced by analogous modular functions. The standard way of proving this is to show that both sides are solutions of the same linear differential equation of order $ 2$ with suitable additional conditions to ensure uniqueness. But this seems to involve in each case some ad hoc and not completely trivial computations. Is there a simple, more elegant manner to prove these results in a unified way without too much computation ?

Is there an analogy to twin primes in the rational integers among the Gaussian and Eisenstein integers?

Twin primes, like (29, 31) and (137, 139) are interesting to study. I have been exploring the parallels of the Gaussian and Eisenstein integers with the rational integers. For instance, they have primes and composites in common. But are there “twin primes”? There may not be a direct analogy, since the relationships > and < are ambiguous in the Gaussian and Eisenstein integers – unless you are talking about the norm.

What is the meaning of the “constant term of Eisenstein series” in terms of Fourier analysis

Let $ G$ be a connected, reductive group over $ \mathbb Q$ , with parabolic subgroup $ P = MN$ . Let $ \pi$ be a cuspidal automorphic representation of $ M(\mathbb A)$ . For a smooth, right $ K$ -finite function $ \phi$ in the induced space $ \operatorname{Ind}_{P(\mathbb A)}^{G(\mathbb A)} \pi$ (realized in a suitable way as a function $ \phi: G(\mathbb Q) \backslash G(\mathbb A )\rightarrow \mathbb C$ ), we can associate the Eisenstein series

$ $ E(g,\phi) = \sum\limits_{\delta \in P(\mathbb Q) \backslash G(\mathbb Q)} \phi(\delta g)$ $ Assuming $ \pi$ is chosen so that this series converges absolutely, one can define the constant term of the Eisenstein series along a parabolic subgroup $ P’$ with unipotent radical $ N’$ :

$ $ E_{P’}(g,\phi) = \int\limits_{N'(\mathbb Q) \backslash N'(\mathbb A)}E(n’g,\phi)dn’ \tag{0}$ $

I see the constant term defined in this way without reference to Fourier analysis. Is it possible to always realize this object as the constant term of an honest Fourier expansion on some product of copies of $ \mathbb A/\mathbb Q$ ?

This can be done when $ G = \operatorname{GL}_2$ and $ P = P’$ the usual Borel. The unipotent radical identifies with the additive group $ \mathbb G_a$ , and for fixed $ g \in G(\mathbb A)$ the function $ \mathbb A/\mathbb Q \rightarrow \mathbb C, n \mapsto \phi(ng)$ has an absolutely convergent Fourier expansion

$ $ E(ng,\phi) = \sum\limits_{\alpha \in \mathbb Q} \int\limits_{\mathbb A/\mathbb Q} E(n’ng,\phi) \psi(-\alpha n’)dn’ \tag{1}$ $ where $ \psi$ is a fixed nontrivial additive character of $ \mathbb A/\mathbb Q$ . The constant term is

$ $ \int\limits_{\mathbb A/\mathbb Q} E(n’ng,\phi) dn’$ $ Setting $ n = 1$ in (1) gives us a series expansion for $ E(g,\phi)$ and (0) is the constant term of this series.

Question about the Fourier expansion of adelic Eisenstein series for $\operatorname{GL}_2$

My reference is Daniel Bump’s book, Automorphic Forms and Representations, Chapter 3.7. Let $ k$ be a number field, $ G = \operatorname{GL}_2$ , $ B$ and $ T$ the usual Borel subgroup and maximal torus of $ G$ . For $ \chi$ an unramified character of $ T(\mathbb A)/T(k)$ , let $ V$ be the space of “smooth” functions $ f: G(\mathbb A) \rightarrow \mathbb C$ satisfying $ f(bg) = \chi(b) \delta_B(b)^{\frac{1}{2}}f(g)$ which are right $ K$ -finite. For $ f \in V$ and $ g \in G(\mathbb A)$ , define the Eisenstein series

$ $ E(g,f) = \sum\limits_{\gamma \in B(k) \backslash G(k)} f(\gamma g)$ $

For suitable $ \chi$ , the series converges absolutely for all $ g \in G(\mathbb A)$ . Now $ E(g,f)$ has a “Fourier expansion,” which as explained by Bump is gotten as follows: the function $ $ \Phi: \mathbb A/k \rightarrow \mathbb C$ $ $ $ \Phi(x) = E( \begin{pmatrix} 1 & x \ & 1 \end{pmatrix}g,f)$ $ is continuous, hence is in $ L^2(\mathbb A/k)$ , and therefore has a “Fourier expansion” over the characters of $ \mathbb A/k$ . If $ \psi$ is a fixed character of $ \mathbb A/k$ , then $ \psi_{\alpha}: x \mapsto \psi(\alpha a)$ comprise the rest of them, for $ \alpha \in k$ . Then

$ $ \Phi(x) = \sum\limits_{a \in k} c_{\alpha}(g,f) \psi_{\alpha}(x) \tag{1}$ $

$ $ c_{\alpha}(g,f) = \int\limits_{\mathbb A/k} E( \begin{pmatrix} 1 & y \ & 1 \end{pmatrix} g,f) \psi(-\alpha y) dy$ $

According to Bump, we may simply set $ x = 0$ , giving us the Fourier expansion for the Eisenstein series

$ $ E(g,f) = \sum\limits_{\alpha \in k} c_{\alpha}(g,f)$ $

My question: Why is this last step valid? The right hand side of equation (1) converges to $ \Phi(x)$ in the $ L^2$ -norm. As far as I know, this is not an equation of pointwise convergence. In general, the Fourier series of a continuous function need not converge pointwise to that function everywhere (in the classical case $ \mathbb R/\mathbb Z$ , the Fourier series of a continuous function converges pointwise to that function almost everywhere).