# References for general Hasse-Weil zeta function

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".

• I do not know any other coherent reference than "facteurs locaux".... – Damian Rössler Jan 25 '15 at 13:28
• 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. – Olivier Feb 6 '15 at 14:08

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

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.

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

Kahn has quite a panoramic view of L-functions in arithmetic geometry. He has written a small book in french, which has been translated to english.

Bruno Kahn makes a clear distinction between zeta-functions which count points in fibers of arithmetic varieties, and L-functions which are associated to varieties "indirectly" via their motives and depend only on the generic fiber of arithmetic varieties, on varieties over fields.

• The CUP website link you gives lists this book as being one of their offerings for number theory as well as "recreational mathematics"! – KConrad Mar 22 '20 at 11:04
• I did not notice. The explanation is that the editor probably made a straightforward "translation job". So they aligned their marketing on the french editor's. The french series is called "nano", it's meant to be invitations to topics for aspiring mathematicians. calvage-et-mounet.fr This editor has quite a few major reference works in french, with great editing, very creative. I recommend anyone buy some about their favorite topics, you won't be disappointed. The physical books are also bery pleasing to hold, especially softcovers. – plm Mar 23 '20 at 1:46
• ...but you know, recreational math in France is a cut above that in the rest of the world... – plm Mar 23 '20 at 1:49
• It has to be if L-functions in arithmetic geometry are part of French recreational math. :) – KConrad Mar 23 '20 at 3:04