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?

On the last page of chapter 18 of Ireland and Rosen's book, they say that Weil defined zetafunctions 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. 550558. Edit: In an article by Lang about the history of the TaniyamaShimura conjecture, available at http://www.ams.org/notices/199511/forum.pdf, he attributes the conception of the zetafunction 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 onedimensional 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. 


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.uniheidelberg.de/~ci3/manu.html (see papers 26,27,36). In particular, in later parts various of Hasse's paper related to this are discussed. 

