  This work has two main purposes. On the one side we investigate in this work
a question of H. Esnault on congruence formula in a construction of H. Esnault
and C. Xu for the number of rational points on the closed fiber of a singular
model of the projective plane over a local field. From the viewpoint of
asymptotic analysis, the question is quite familiar with a question of N.
Koblitz, which in turn has some meaningful applications in cryptography. We
don't try to solve those questions in this work, but rather concentrate on
studying asymptotic behaviours with very elementary techniques of generating
functions. On the other side we extend the discussion on generating functions
to the global situation, which inherit hybrid-properties from the Riemann zeta
function and the $L$-function of elliptic curves. At the end we will look at an
example of modular forms, where we are able to prove an analytic result, which
is similar to a result of P\'olya for the Riemann $\xi$-function.
