11
$\begingroup$

In the proceedings "Algebraic Geometry - Arcata 1974" edited by R. Hartshorne there is an article by Nick Katz called "$p$-adic $L$-functions via moduli of elliptic curves". He starts by recalling $p$-adic measures. In particular, he characterizes all $p$-adic measures on $\mathbb{Z}_p$, with values in $\mathbb{Z}_p$, which correspond to rational functions in $\mathbb{Z}_p[[T-1]]$ under the Iwasawa isomorphism. Then, on page 488, he makes the following observation:

The measures $\mu_F$ we considered above correspond exactly to the rational functions in $\mathbb{Z}_p[[T-1]]$ (the "rationality of the zeta function"!).

Assuming he is referring to the rationality of the zeta function in the Weil conjectures, my question is:

What is the relation between $p$-adic measures and the rationality of the zeta function of an algebraic variety over a finite field?

The exclamation mark in his claim disturbs me. Is this something obvious?

I have studied Dwork's proof (from his paper and from Koblitz book), but I'm not familiar with Deligne et al proof of the whole conjecture.

Thank you very much.

Edit: Apparentely, after reading the comments below, Katz is remarking here the "algebraic" construction due to Iwasawa of the $p$-adic zeta function as a quotient of power series evaluated in a power of a topological generator of a $p$-adic multiplicative group.

Improve this question
$\endgroup$
12
  • $\begingroup$ EFinat-S : to answer your question about "User"'s question (now deleted by the poster) on Riemann's zeta function, it was probably downvoted because his/her question did not make any sense... $\endgroup$
    – js21
    Commented Jun 23, 2017 at 16:32
  • 2
    $\begingroup$ Here I would understand "rational" in the sense that the Iwasawa transform of $\mu^{(a)}$ is rational (as a power series). $\endgroup$
    – js21
    Commented Jun 23, 2017 at 16:53
  • 13
    $\begingroup$ @ Matt F. In the article, Katz writes this with the exclamation point. I am quoting it verbatim. $\endgroup$
    – efs
    Commented Jun 23, 2017 at 17:34
  • 1
    $\begingroup$ The exclamation point makes me think it is an act of humor: the algebraic gadget describing the p-adic zeta function "is" a rational function. This reminds of the Weil conjecture, of course, but is quite unrelated. $\endgroup$
    – ACL
    Commented Jun 23, 2017 at 18:43
  • 1
    $\begingroup$ It's important to note that Iwasawa's construction of the $p$-adic zeta function was inspired partially by Weil's construction of function field zeta functions, which are rational. So ACL is not quite right to say that this is quite unrelated to the Weil conjectures - while there is no formal relation, they can be described in similar ways. $\endgroup$
    – Will Sawin
    Commented Jun 29, 2018 at 5:50

0

Reset to default

You must log in to answer this question.