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As a beginner who just get in touch with Euler characteristics in this field, could I get some intuition for the arithmetic meaning of Euler characteristics of bounded complexes, for example Selmer complex? I just feel vaguely that it has some relation with $p$-adic $L$-function or $p$-part of BSD.

Besides, is the characteristic equal to 0 having special meaning in these cases?

Recommendation of reference is also helpful!

Grateful for your guidance!

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  • $\begingroup$ A guess is that there should be a reasonable analogy between the characteristic $0$ and characteristic $p$ cases. The fact of the matter is that in characteristic $0$, we don't have a cohomological interpretation of Artin $L$ functions (different than char $p$). In char $p$, we have good cohomology theories like etale and crystalline, as well as a Lefschez fixed point formula and a description of the absolute Galois group. I imagine we would need to understand much more about the latter before we establish these analogies in char $0$. $\endgroup$
    – kindasorta
    Commented Aug 20 at 6:50

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If $\rho$ is a Galois representation of geometric origin and if you consider a cohomology complex $C$ computing Galois cohomology satisfying (supplementary cleverly) defined arithmetic conditions, then the Euler-Poincaré characteristic of $C$ is conjectured to be the order of vanishing of the $L$-function of the dual Galois representation $\rho^{*}(1)$ at zero, in very close analogy to the fact that the degree of the Zeta function of a scheme over a field of positive characteristic $p$ is the Euler-Poincaré characteristic of the étale cohomology with compact support of the scheme. If $\rho$ is a $p$-adic family of Galois representations, then indeed one may conjecture a relation with the order of vanishing of some $p$-adic $L$-function though things are much more complicated even to state.

Even for a character of the absolute Galois group, establishing that the conjecture is true is a deep result.

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