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The only way I know to write the Hasse-Weil zeta function of an elliptic curve is as a product over the local zeta factors which are rational functions. To me, this appears like an Euler product.

Is there a natural summation version of the Hasse-Weil zeta function where the terms have some geometrical significance, say in analogy with the Dedekind zeta function?

I apologize for the naive question.

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    $\begingroup$ I mean, the coefficient of $n^{-s}$ parameterizes zero-cycles of norm $n$, where zero-cycles are formal sums of closed points of the arithmetic scheme, and the norm of a sum of closed points is the products of the orders of the closed points' residue fields. $\endgroup$
    – Will Sawin
    Commented Feb 8, 2021 at 3:44
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    $\begingroup$ @Kapil I don't think it can be quite that because as soon as $R$ has dimension $>1$ there are many ideals supported at a single point of a given degree. $\endgroup$
    – Will Sawin
    Commented Feb 8, 2021 at 14:05
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    $\begingroup$ Yes, it's correct. No, the generic fiber does not directly enter in this definition. Why is it not defined this way? One reason is that the summation perspective is not helpful in this setting for getting analytic information about the zeta function (as it is in the Dedekind case), and in fact the main known way to get analytic information is to factor the zeta function into automorphic $L$-functions, which is done most easily in the Euler product representation. $\endgroup$
    – Will Sawin
    Commented Feb 8, 2021 at 14:10
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    $\begingroup$ If you'd like some formal nonsense, if $\mathfrak{p}$ is a prime lying over the generic fiber, then $R/\mathfrak{p}$ is infinite, so $N(\mathfrak{p})$ should be infinity, and thus $N(\mathfrak{p})^{-s}$ should be zero. I would take this with several grains of salt, though. $\endgroup$
    – David E Speyer
    Commented Feb 8, 2021 at 14:12
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    $\begingroup$ @DavidESpeyer When we expand the Artin L-function of a (complex? $\ell$-adic works too when $n^{-s}$ is taken formally) representation we get $L(s,\rho)=\det(F(s))$ where $F(s)=\sum_n n^{-s} \prod_j \rho(Frob_{p_j})^{v_{p_j}(n)}$, here $\prod_j$ is a non-commutative product of matrices, assuming we fixed one for all an order on the primes. Do you know if there can / cannot be some ordering of primes making the obtained Dirichlet series with matrix coefficients $F(s)$ "meaningful" ? We can also add some conjugating factors $Q_j \rho(Frob_{p_j})Q_j^{-1}$ increasing the degree of freedom. $\endgroup$
    – reuns
    Commented Feb 8, 2021 at 16:14

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