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I'd like to ask the following easy question, since I can't find a reference.

Let $X/\mathbb{Q}$ a smooth projective variety. How does one express the $L$-function of the twist, $L(H^i(X)(r), s)$ in terms of $L(H^i(X),s)$. Do they differ by only a shift in $s$?

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  • $\begingroup$ Yes. See: Serre, Jean-Pierre, Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures). (available at numdam.org/item?id=SDPP_1969-1970__11_2_A4_0) $\endgroup$
    – jmc
    Sep 20, 2014 at 17:26
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    $\begingroup$ Sorry, that was a bit too fast. The answer is still yes. But the reference is wrong. See my answer below. $\endgroup$
    – jmc
    Sep 20, 2014 at 17:38

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The answer is yes. On page 12 of [1], line 7 from the top, it reads $$ L(M (n), s) = L(M, s + n). $$

The article deals with compatible systems of cohomological realisations (one way to think and work with motives). Every variety gives rise to such a system, and every such system gives you an $L$-function.

Actually, if you carefully trace the definitions, this identity rolls out fairly easy. For a good overview of the definitions, see [2].

[1]: Minhyong Kim. “An introduction to motives I: classical motives and motivic $L$-functions” (available at http://ucl.ac.uk/~ucahmki/ihes3.pdf ).

[2]: Serre, Jean-Pierre. “Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures).” (available at http://archive.numdam.org/article/SDPP_1969-1970__11_2_A4_0.pdf)

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    $\begingroup$ That's correct. Lest someone get a wrong impression from the reference to a recent paper, this is not a theorem: just a tautology from the definition. $\endgroup$
    – Joël
    Sep 20, 2014 at 19:34

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