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Let $X/K$ be a smooth projective variety over a number field $K$. I have seen two definitions of the L-function $L^i(s)$ attached to it's cohomology groups $H^i(X)$.

Both definitions agree that at a bad prime $\mathfrak{p}$ take the euler factor should be $\frac{1}{P(N\mathfrak{p}^s)}$, where $P(t) = det(1- Fr*t)$. $Fr$ is the inverse of the frobenius map (which takes $x \mapsto x^{N\mathfrak{p}}$) acting on $H^i_{et}(X \times \bar{K}_\mathfrak{p}, \mathbb{Q}_\ell)^{I_\mathfrak{p}}$ (by acting on the factor $\bar{K}_\mathfrak{p}$). Here, $I_\mathfrak{p}$ is the inertia group and as usual $(\ell, N\mathfrak{p}) = 1$.

$\textbf{Question:}$ The bad factors of the L-function are given by geometric frobenius. I've been told that the same definition gives both the good and bad euler factors. However, for good primes, Serre (Facteurs locaux des fonctions Zeta ..., section 1.2 ) appears to use (but isn't very explicit) the ordinary frobenius map (that is, he doesn't take the inverse in the description of the euler factors above).

Do we use frobenius or its inverse in good factors of the L-function?

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    $\begingroup$ I think it is standard to use arithmetic Frobenius on $H^i$ or equivalently take the contragredient representation of geometric Frobenius when thinking in terms of Galois representations on $H^i$ (even when this isn't explicitly mentioned). That way the definitions coincide for elliptic curves using the natural isomorphism $H^1_{et}(\overline{E}, \mathbf{Q}_\ell)^\vee \simeq T_\ell(E)\otimes \mathbf{Q}_\ell$. $\endgroup$
    – Matt
    Commented Oct 27, 2013 at 21:27
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    $\begingroup$ This is pretty wierd. (1) Of course, fixed pts of arithmetic or geometric frobenius of vars/finite fields obviously agree, only their characteristic polynomials disagree. (2) The eigenvalues of geometric frobenius are algebraic integers, (hence not arithmetic frobenius). (3) Charpolys appearing in Weil conjectures (hence local zeta funcs) are of geometric frob. and I thought that the Hasse-Weil zeta function of vars/num fields are products of the local zeta functions (evaluated at various pts). If this last result is true, do we have a contradiction? $\endgroup$
    – LMN
    Commented Oct 28, 2013 at 2:06
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    $\begingroup$ I'd have to re-look at places I've seen this, but I think when the local zeta functions are defined with geometric Frobenius they don't use the "characteristic polynomial" but rather $det(1-p^{-s}\rho(Frob_{\mathfrak{p}}))=P_{\mathfrak{p}}(p^{-s})$ or something? I was really worried about this as well at one point, and never really came to a definitive conclusion, but decided that there were enough minus signs and duals in papers by careful people that it seemed to work out as the same... $\endgroup$
    – Matt
    Commented Oct 28, 2013 at 4:22
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    $\begingroup$ For example, the place I tend to see this is with modularity of varieties, in particular Calabi-Yau threefolds, and Noriko Yui has notes here: arxiv.org/abs/1212.4308 that talks about this (maybe I just messed up the terms "arithmetic and geometric" in my previous post now that I look at this?) $\endgroup$
    – Matt
    Commented Oct 28, 2013 at 4:29
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    $\begingroup$ Great! Here's a precise reference: Serre's Abelian L-adic representations book defines the L-function via arithmetic frobenius (p.I-16) and specifies the euler factor as the characteristic polynomial evaluated at the appropriate number. Thanks! $\endgroup$
    – LMN
    Commented Oct 28, 2013 at 18:05

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