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Are there any historical articles about the origins of zeta functions of curves over global fields (undoubtedly starting with $\mathbb{Q}$)? In particular who (and when did this happen) first have the idea of creating such a thing? Was it by analogy with the zeta functions of number fields?

asked Mar 12 2011 at 15:39

Victor Miller

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The earliest reference to zeta functions over global fields that I am aware of is Serre's short article "Zeta and L-functions", in Arithmetical Algebraic Geometry, New York: Harper and Row, 1965; this paper is also in Vol.2 of Serre's Oeuvres (pp. 249-260). – Tim Dokchitser Mar 12 2011 at 16:25

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It's called the Hasse-Weil zeta function --- that should be a clue. – mephisto Mar 12 2011 at 18:31

KConrad · Accepted Answer · 2011-03-13 09:19:51Z UTC

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On the last page of chapter 18 of Ireland and Rosen's book, they say that Weil defined zeta-functions of smooth varieties over number fields in his 1954 paper "Abstract versus Classical Algebraic Geometry". It's in Weil's Collected Works, vol. II, pp. 550--558.

Edit: In an article by Lang about the history of the Taniyama--Shimura conjecture, available at http://www.ams.org/notices/199511/forum.pdf, he attributes the conception of the zeta-function of a variety over a number field as a product of local factors to Hasse in the 1930s. (It was too early then to get the correct definition of local factors at the bad places and Hasse's conjecture was a meromorphic continuation and functional equation.) Weil is credited with bringing the conjecture to a wider audience at the 1950 ICM. I don't know if Hasse in the 1930s was really thinking about varieties of dimension > 1 or just curves. After all, number theory in the 1930s was still pretty one-dimensional when it came to the overlap with algebraic geometry. Since Victor's question is about the case of curves, then the answer to the question appears to be Hasse.

edited Mar 13 2011 at 9:19

KConrad

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So, since it's called the Hasse-Weil zeta function did Hasse previously define it for elliptic curves? – Victor Miller Mar 12 2011 at 20:43

On p. 303 of Ireland and Rosen (section 2 of Chapter 18) they say that the analytic continuation of zeta-functions of elliptic curves over Q to the whole complex plane was conjectured by Hasse, although they don't cite a paper. They also say that the first proof of special cases of the conjecture was in Weil's paper "Jacobi sums as Grossencharakete", which is in Transactions AMS 73 (1952), 487--495. So maybe in that paper you'll find a citation by Weil to the appropriate earlier paper of Hasse (the same one in which Hasse proved RH for ell. curves over finite fields?). – KConrad Mar 12 2011 at 20:53

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The story goes (I think I read this in the commentary of Weil's collected works) that Hasse suggested as a thesis problem to a student to prove that the zeta function of an elliptic curve satisfied a functional equation. He asked Weil his opinion of the suitability of problem and Weil thought that it might a bit too hard for a student. – Felipe Voloch Mar 12 2011 at 21:29

@KConrad: I just looked at Weil's paper. He has no references! It looks like (for elliptic curves), he only treats the case of complex multiplication. What I'm after is who (possibly Artin?) had the idea of making a global $L$-function by taking the product of the local factors? On a related note, when the modularity conjecture was first stated (say by Taniyama and/or Shimura) was it in terms of uniformization of elliptic curves by modular functions or the fact that the global $L$-series was the Mellin transform of a particular eigen-form on $\Gamma_0(N)$ of weight 2? – Victor Miller Mar 13 2011 at 0:27

@ KConrad. Wait, I thought Deuring proved something of the form "L-functions of CM elliptic curves are grossencharakter L-functions", before Weil. – David Hansen Mar 13 2011 at 4:25

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quid · Answer 2 · 2011-03-12 22:30:10Z UTC

This does not adress the actual question, but might be useful, if somebody wants to trace back the literature as aluded to in another answer.

Peter Roquette has writteen a series of articles entitled

The Riemann hypothesis in characteristic p, its origin and development several parts, total length close to 200 pages. Bibliographic details and pdf are avalaible here: http://www.rzuser.uni-heidelberg.de/~ci3/manu.html (see papers 26,27,36).

In particular, in later parts various of Hasse's paper related to this are discussed.

Historical Articles about zeta functions of curves

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