# The historical development of automorphic geometry

Background: Today the notion of automorphic geometry is often framed in the context of the Langlands program, in particular what is sometimes called the Langlands reciprocity conjecture. This is often phrased as the expectation that all motives are automorphic. The historical development of this notion is fairly extensive, and can certainly be traced back more than a century to the work of Felix Klein and his student Adolf Hurwitz. One historical thread that therefore could be identified is the "elliptic lineage", with important contributions from Klein, Hurwitz, Hecke, Artin, Eichler, Taniyama, Shimura, Weil, Langlands, Deligne, Serre, Wiles, Taylor, and probably others.

I'm mainly interested in whether the notion of automorphic geometry (motives) has roots that go back farther than Klein and Hurwitz, but I'd also like to use this historical framework to ask what other "automorphic lineages" might be identified, possibly with overlaps with the elliptic lineage above.

Questions:

1) Is there work on automorphic geometry that predates that of Klein and Hurwitz?

2) Are there other lineages in the automorphic genealogy?

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What work of Klein are you thinking of? –  Moosbrugger Jan 12 '12 at 21:51
There is earlier work that is summarized in the books by Klein and Fricke. –  Laie Jan 12 '12 at 23:04
Sure. Euler was the first to emphasize $\mathbb{Z}(1)$, the kernel of exponential, as a mathematical object of interest (albeit with different notation). He also related the Gamma function to periods. This predates Klein by quite a bit. –  S. Carnahan Jan 13 '12 at 2:59
By the way, in what sense is Langlands in the elliptic lineage? Elliptic curves do not strike me as a particularly important motivation for him, compared to, say, the study of $L$-functions. –  Olivier Jan 13 '12 at 12:32
I agree with that assessment, in particular because he didn't know about geometry until Weil pointed him to his papers. My reason for mentioning Langlands is more his emphasis of the Galois part of the story, the key tool for everything that followed. One could probably formulate a more minimalistic elliptic lineage but that wasn't the goal. –  Laie Jan 13 '12 at 23:05

## 3 Answers

The assertion of Gauss' quadratic reciprocity (and Eisenstein's cubic and quartic, etc), construed qualitatively as asserting that the quadratic symbol $p\rightarrow (D/p)_2$ has a "conductor", is an assertion that a Galois object (motive?) is an analytical/automorphic object (Dirichlet character), albeit for $GL(1)$.

Depending on one's taste, one might imagine that Kronecker's youth-dream can be construed as asking about Galois properties of periods. This is certainly compatible with work on special values of L-functions in the last 40 years, and p-adic interpolation, very active in recent years. Again, to my mind, anyway, the fact that an analytic/automorphic object or some of its attributes might be "algebraic" (or vice-versa) is a point of interest.

There is also "potential automorphy" and applications to Sato-Tate (Clozel-Harris-ShephardBarron-Taylor, et alia).

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Interesting take, but a bit hard to make into proper history, I should think.

Start with the idea that, post-Gauss, 19th century mathematics was mainly not about number theory. This could be hard to understand now: but consider how long it took to get from the Abel-Jacobi theorem to the Mordell-Weil theorem being conjectured (I think around 1910). In particular the theory of special functions, precursor of representation theory in a way, was mainly not about number theory (and mainly was about mathematical physics and scientific computation). In the absence of any sort of general diophantine theories, how many people would have appreciated any sense in talking at the breadth of the current raft of conjectures? The name of the play might be "Waiting for Hilbert".

So I think you're looking for exceptions to general trends. I doubt Klein was really interested in number theory, for example. We tend to know the names (Dirichlet, Kummer and so on). What Eisenstein actually thought about appears enigmatic. I'm not aware of anything in Kummer for your theme, but he wrote a lot. Kronecker now - Langlands did quite a good job of suggesting that the Jugendtraum was some sort of wrong turning. The revisionist theory of what Galois was about might be one place to look. A bit like clutching at straws, really.

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"Langlands did quite a good job of suggesting that the Jugendtraum was some sort of wrong turning." Where did he do that? –  anon Jan 13 '12 at 0:01
Perhaps Langlands didn't really intimate such a thing. Some of L's prose was circuitous, for one thing. –  paul garrett Jan 13 '12 at 0:25
Some contemporary problems with origins in the Jugendtraum is easy to find online. Langlands says that Hilbert's 12th problem has been neglected, whether justly or not, and then goes on to talk about other stuff. –  Charles Matthews Jan 14 '12 at 11:21

A common answer to question 1 is to mention the entries of Gauss's diary from 1814, including famously (but not restricted to) the last one, in which he studies some properties of biquadratic reciprocity and their links with the division of the lemniscate. Gauss understood these properties to derive from properties of Punktgitter (lattices of points) in the complex planes and his formulation is surprisingly close to the statement that (some) elliptic curves with CM are automorphic objects.

Though I am not sure that this is the best way to think about the work of Gauss from a historical point of view (he himself seems to have been unsatisfied with this work and to have considered it of relatively low value compared to the latter approach of Einsenstein) and though these findings had no impact on mathematics at the time for the simple reason that they were kept private for over 80 years, it certainly counts as work in "automorphic geometry" done in the early XIX° century.

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