# History of points of view on Eisenstein series

What is the history of Eisenstein series? Did the mathematician Eisenstein actually encounter them?

There are, as far as I know, two major perspectives on what Eisenstein series are. The first is in the study of modular forms. By summing over lattice points, one obtains a holomorphic modular form

$$E_{2k}(z) = \sum\limits_{(m,n) \neq (0,0)} \frac{1}{(m+nz)^{2k}}$$

for $$\operatorname{SL}_2(\mathbb Z)$$ which together with the cusp forms exhausts the space of modular forms of a given weight.

The other perspective is in considering a unitary representation $$(\pi, V)$$ of a Levi subgroup $$M$$ of a semisimple real Lie group $$G$$. The representation considered should be left $$\Gamma \cap M$$- invariant for a lattice $$\Gamma$$ of $$G$$. If $$P$$ is a parabolic subgroup of $$G$$ containing $$M$$ as a Levi subgroup, one forms the induced representation $$\mathcal V = \operatorname{Ind}_P^G \pi$$ of $$G$$ and, wishing to embed $$\mathcal V$$ into $$L^2(\Gamma \backslash G)$$, one associates to each $$\varphi_{\pi} \in \mathcal V$$ the Eisenstein series

$$E(\varphi_{\pi},g) = \sum\limits_{\gamma \in \Gamma \cap P \backslash \Gamma \cap G} \varphi_{\pi}(\gamma g). \tag{1}$$

There is a bit more to it, as one needs to take continuous sums (integrals) of such Eisenstein series $$E(\varphi_{\pi_s},g)$$ for various unramified twists $$\pi_s$$ of $$\pi$$, and analytically continue $$E(\varphi_{\pi_s},g)$$ so that the formula (1) no longer makes sense.

These perspectives can be connected by associating modular forms with automorphic forms on $$\operatorname{SL}_2(\mathbb Z) \backslash \operatorname{SL}_2(\mathbb R)$$.

Did these two points of view on Eisenstein series develop independently? Who were the first people to connect them?

• some aspects are discussed in mathoverflow.net/a/220535/11260 Jul 5 at 21:01
• Surely the lattice point version came up first, and the second version out of attempts to generalize it Jul 6 at 0:36
• The first occurence was probably as the Laurent coefficients of (Weierstrass) elliptic functions at $0$, without thinking to the action of $SL_2(\Bbb{Z})$ on the upper half plane. The modular functions parametrize complex tori so it is natural to study complex tori first. Jul 6 at 9:52 where $$w=\alpha m+\beta n$$ and the sum is over the two integers $$m,n$$.