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?



History of points of view on Eisenstein series
**Q:** *What is the history of Eisenstein series? Did the mathematician Eisenstein actually encounter them?*


**A:** Yes, he did, see [Elliptic Functions According to
Eisenstein and Kronecker](https://ems.press/content/serial-article-files/8714).


The reference is  [Beiträge zu Theorie der elliptischen Functionen. VI. Genaue Untersuchungen der unendlichen Doppelproducte, aus welchen die elliptischen Functionen als Quotienten zusammengesetzt sind, und der mit ihnen zusammenhangenden Doppelreihen](https://resolver.sub.uni-goettingen.de/purl?PPN243919689_0035) (1847).


Eisenstein writes the double sum (on page 223) as  

![](https://i.stack.imgur.com/MDGa8.png)  

where $w=\alpha m+\beta n$ and the sum is over the two integers $m,n$.


