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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. (Obviously it was too early then to get the correct definitions 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.

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.

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. (Obviously it was too early then to get the correct definitions 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.

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. (Obviously it was too early then to get the correct definitions 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.

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.

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. (Obviously it was too early then to get the correct definitions 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.