# Eisenstein's last theorem

Hopefully the following is appropriate for MathOverflow; it's possible the question (of a somewhat historical nature) is unanswerable, but I think there's some hope it can be answered, as I'll explain at the end.

I recently had occasion to look through Eisenstein's final paper, written in 1852, the year he died of tuberculosis at the age of 29. In it, Eisenstein states the following

Theorem. (Eisenstein, 1852 [1]) Let $$f(z)=\sum a_iz^i$$ be the Taylor expansion of an algebraic function, i.e. $$f$$ satisfies a polynomial with coefficients in the field of rational functions $$\mathbb{Q}(z)$$. Then there exists an integer $$A$$ such that $$A^{i+1}a_i$$ is an integer for all $$i$$.

This implies, for example, that there are only finitely many primes dividing the denominators of the $$a_i$$. Eisenstein goes on to give a one-sentence sketch of the proof, and then says:

Die wichtigsten Anwendungen der so erhaltenen Sätze habe ich auf Fälle gemacht, in denen die algebraischen Funktionen als Integrale von Differential-Gleichungen definirt werden, und dies Differential-Gleichungen für einfache Reihen-Entwicklung geeignet sind, während die vielleicht sehr complicirte Darstellung in endlicher Form ganz unbekannt bleibt und für diesen Zweck auch wirkliich ganz aus dem Spiele gelassen werden kann. Das Einzelne der hieranf bezüglichen Untersuchungen mag für eine künftige Mittheilung vorbehalten bleiben.

My poor English translation:

The most important applications of the theorems thus obtained I have made to cases in which the algebraic function is defined as the integral of a differential equation suitable for simple series expansion, while a perhaps-very-complicated representation in finite form remains completely unknown and can actually be completely ignored. The details of the investigations relating to this may be reserved for a future communication. (DL: Emphasis mine.)

This being Eisenstein's final paper, the details relating to this investigation do not in fact appear in any of his future papers. My question is:

What were these "most important applications" of Eisenstein's theorem to solutions to differential equations that he is referring to in this final paper of his?

My hope is that the answer can be found in some of Eisenstein's letters. It seems like he wrote a lot of letters to his contemporaries (Gauss, Richelot, Stern, etc.), some of which were in 1852, contemporaneous with this last paper of his. I've tried to scan the letters I could find (in particular, those in Eisenstein's "Collected Works") without any luck, though this might not mean much as my German is quite bad, and it's unclear if there are other extant letters of his.

The following is just speculation--in 1904, Landau used Eisenstein's theorem to reproduce Schwarz's 1873 list of hypergeometric functions which were also algebraic. I wonder if Eisenstein could have had something along these lines in mind (in which case it would have, of course, predated Schwarz's work)?

[1] Eisenstein, Gotthold. "Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Funktionen." Bericht der Königl. Preuss. Akademie der Wissenschaften zu Berlin (1852): 441-443.

• By 1852, Eisenstein was already very ill. so that even writing letters was an extreme burden. If there were any letters except those in his Werke, then Biermann probably would have found them. A few years back I discovered a letter of Eisenstein's parents to Gauss (see gauss.adw-goe.de/handle/gauss/2588? and arxiv.org/pdf/2101.04519) online (and one by Legendre in an online auction, of all places - see images-archive.math.cnrs.fr/…), so who knows . . . Commented Aug 2 at 13:51
• @FranzLemmermeyer: Thanks for this interesting info (and for improving my transcription/translation)! Commented Aug 2 at 14:14