L-functions of Calabi-Yau varieties

Question

This question might not be suitable for MO since i know nothing about Calabi-yau varieties aside the fact that they are used in string theory to compactify additional dimensions, but still, it makes want to explore a bit this subject. It seems, judging by a quick Google search, that one can associates a(n) (Hasse-Weil) l-function to a Calabi-Yau variety. I also know that there exist pairs of Calabi-yau varieties linked by a so-called "mirror symmetry" so that such mirror pairs, when used to compactify additional dimensions, describe the same physics. My question is thus: do mirror pairs share the same l-function? Thanks in advance.

Answer 1

The short answer is "in general, no". Here are some reasons:

1. The L-function is not even well-defined unless we fix a model over a number field (to so say, choose equations for your variety with coefficients in a number field). Most notions of mirror symmetry do not take this into account.

2. Ignore the previous point for a second, and for simplicity say that everything is defined over Q. If the two L-functions are going to be equal, then the local factors must agree. Here we have a serious issue, because under mirror symmetry the Betti numbers can change (at least for CY of dimension at least 3) hence the degrees of the characteristic polynomials of the Frobenius can change.

Point 1 is a real problem but is not so bad. On the other hand, point 2 is a real problem (not only a possibility) and it is really bad. The study of the local factors (aka [local] zeta function) is already very interesting and a quick internet search will give you some papers to look at. You may want to look at:

http://arxiv.org/abs/hep-th/0012233

To make the long story short: the zeta functions (i.e. local factor of L-function) of mirror pairs are not expected to be equal in general, but there should be some relation between them.

I hope this helps.

Answer 2

Sorry for answering my own question, but it may be useful to some people. It seems, judging by http://arxiv.org/pdf/1301.2225v2.pdf, that the "right" notion of L-function for a Calabi-Yau variety is not the Hasse-Weil L-function, but the L-function of the "$\Omega$-motive" of such a variety, whatever it might be. As said in the abstract "Mirror symmetry in particular predicts that the L-functions of the $\Omega$-motive of such pairs are identical." The author also says the result he proves (namely the isomorphy of $\Omega$-motives for the considered mirror pairs) may have applications to the Langlands program, and this was actually the motivation for my question, even though it was way too vague in my mind.

Edit November 7th, 2023: the isomorphy of $\Omega$-motives of mirror pairs may be related to automorphisms of any L-rig (see Are there infinitely many L-rigs?) preserving the corresponding L-function having order at most $2$.