1. You should learn about the Weil Conjectures (which are theorems)
for smooth varieties over finite fields, e.g.,
the Galois representations, the determination of the zeta-function.
See Weil Conjectures in wikipedia for a start.
2. Basics on modular forms (classical modular forms, quasimodular
forms, Hilbert, Siegel modular forms)
The book by Zagier et al. the 1-2-3 of modular forms.
3. Modularity theorems for some Calabi-Yau varieties over Q or number
fields
The book by Ch. Meyer FIM 22 would be a good start.
What we want to work on during the summer is to understand how to
compute the zeta-functions and more generally L-series for families
of
Calabi-Yau varieties, starting with one or more parameter families
of hypersurfaces. For instance, some work has been done already for
the Dwork families, i.e., one-parameter family of quartic K3 surfaces,
one-parameter family of quintic CY threefolds, and D. Wan's work on
arithmetic mirror symmetry.
Our hope is to try with two-or-three parameter families of Calabi-Yau
threefolds.
Here are some of the articles you should read before your arrival
at the Fields Institute. You should try to make yourself familiar
with the articles with *.
Yui, N.
*Update on the modularity of Calabi-Yau varieties with appendix by
H. Verrill, in FIC 38
Yui, N. et al.
Arithmetic and Geometry of K3 surfaces and Calabi-Yau threefolds,
FIC 64
Yui, N.
Modularity of Calabi-Yau varieties: 2011 and beyond in FIC 67.
Lee, Edward
*Update on modular non-rigid Calabi-Yau threefolds, in FIC 54
Goto, Y, Kloosterman, R., and Yui, N.
*Zeta-functions of certain K3-fibered Calabi-Yau threefolds
Internat. J. Math. 22, no-.1 2011.
Garbagnati, A., and van Geemen, B.
*The Picard-Fuchs equations of a family of Calabi-Yau threefolds without
maximal unipotent monodromy, Int. Math. Res. Notes 16, 3134-3143 (2010).
Garbagnati, A.
*New families of Calabi-Yau threefolds without maximal unipotent monodromy,
Manuscr. Math. 140, 273-294 (2013).
D. Wan
*Mirror symmetry for zeta functions, in Mirror Symmetry V Moment zeta-function
Arithmetic mirror symmetry
P. Candels et al.
*Zeta-function of quintic Calabi-Yau threefolds
S. Kadir
*Arithmetic of mirror symmetry for two-parameter family of Calabi-Yau
manifolds, in Mirror Symmetry V
