Let $X$ be an arithmetic scheme, that is, a scheme of finite type over the integers. We denote the set of closed points of $X$ by $|X|$. For every $x\in|X|$, write $N(x)$ for the cardinality of the residue field $\kappa(x)$.

The arithmetic zeta function of $X$ is defined as $$\zeta_X(s)=\prod_{x\in|X|}\frac{1}{1-N(x)^{-s}}.$$

This definition (up to a change in variable) can be found in

A. Grothendieck, Formule de Lefschetz et rationalité des fonctions $L$, Séminaire Bourbaki 279 (1964), 41-55.

Grothendieck attributes this definition to Weil, but as far as I know, Weil only defined the Hasse-Weil zeta function: if $X$ is a smooth projective variety over $\mathbb{F}_q$ and $N_r=|X(\mathbb{F}_{q^r})|$, then $$Z_X(t)=\exp\left(\sum_{r=1}^\infty N_r(X)\frac{t^r}{r}\right).$$

Of course, it is easy to show these two functions satisfy $$\zeta_X(s)=Z_X(q^{-s}),$$ but Weil did not address the notion of the zeta function of a scheme, at least not in the original paper:

A. Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497-508.

Where was the zeta function of an arithmetic scheme first defined? If anyone knows the actual paper in which this first appears, that would be optimal. (Of course, it's very possible that this definition was well-known but unpublished for some time; that would be an acceptable answer too.)

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    $\begingroup$ The earliest reference I know of is Serre's "Zeta and L-functions", Arithm. Alg. Geom., Proc. Conf. Purdue Univ. 1963, 82-92 (1965), <books.google.co.uk/…>. There he defines zeta-functions of general schemes and proves some basic properties. $\endgroup$ – Tim Dokchitser Mar 16 '13 at 21:04
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    $\begingroup$ You are looking for the Hasse-Weil zeta function of a variety over a number field. With a change of language, one can also call this the zeta function of the arithmetic scheme over the ring of integers defined by the variety. I don't know when the Hasse-Weil zeta function was first considered, but already by about 1950 Deuring knew how to express the Hasse-Weil zeta function of a CM elliptic curve in terms of Hecke characters. There's probably no first paper --- if you look hard enough you can find such a zeta function in the work of Gauss. $\endgroup$ – anon Mar 17 '13 at 2:03

I accidentally found the answer to this question in the book "The Abel Prize. 2003-2007 The First Five Years", page 74:

"A first lecture on zeta and L-functions in the setting of the theory of schemes (of finite type over $\mathrm{Spec}(\mathbb{Z})$) was given by Serre in [S112, 64(1965)]."

The paper is, of course,

  • Jean-Pierre Serre, Zeta and L functions (1965) Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) pp. 82-92 Harper & Row, New York [also in Collected Papers, vol. II, n° 64)]

This confirms Tim Dokchitser comment.

On a side note, the concept of scheme didn't even exist at the time of Weil's paper!


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