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?

  • 1
    $\begingroup$ some aspects are discussed in mathoverflow.net/a/220535/11260 $\endgroup$ Jul 5, 2022 at 21:01
  • $\begingroup$ Surely the lattice point version came up first, and the second version out of attempts to generalize it $\endgroup$
    – Will Sawin
    Jul 6, 2022 at 0:36
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    $\begingroup$ 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. $\endgroup$
    – reuns
    Jul 6, 2022 at 9:52

1 Answer 1


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.

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 (1847).

Eisenstein writes the double sum (on page 223) as

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


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