Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case.

I am looking for references about the Hasse-Weil zeta for arbitrary variety and number field, particularly analytic continuation and functional equation (this is, not focused on special values or zeroes).

Also, I have Serre's "Facteurs locaux", so probably anything published before 1970 would be redundant.

Edit. For future reference, Ivan Fesenko's research, particularly "Analysis on arithmetic schemes" parts I, II and III, fit perfectly in this context. Also, some older papers like Alexei Parshin, for example, "Chern classes, adeles and L-functions".

  • $\begingroup$ I do not know any other coherent reference than "facteurs locaux".... $\endgroup$ – Damian Rössler Jan 25 '15 at 13:28
  • 1
    $\begingroup$ There is a reason why much of the literature deals with elliptic curves and/or abelian varieties: the analytic continuation and functional equation of the $L$-function is a wide open problem for arbitrary varieties. $\endgroup$ – Olivier Feb 6 '15 at 14:08

For analytic continuation you can often devissage back to zeta functions of number fields. See e.g. section 9.1.7 of Serre's "Lecture on N_X(p)" and the references therein for sample theorems.

  • $\begingroup$ It doesn't cover much of what I am looking for, but it is a nice reference. Thanks! $\endgroup$ – Myshkin Feb 6 '15 at 1:32

This recent preprint may be of interest for you, as the author first considers L-functions and then finds back the algebraic variety they come from.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.