Dear All,

This question may appear elementary to all the experts in number theory , but forgive me. I really wanted to know how did the $L$-functions came into existence, especially the Hasse-Weil L-functions . Do they have some specific meaning in their formulation or they are just framed heuristically to build some thing else , as scaffolding .

I do know that Zeta functions and L-functions of the curve act as spies in collecting the secret information about the local part of curves and embed that information inside them, but its really a great trouble in understanding the formulation. I referred to many books and they have started saying " Let $L(s,E)$ be the ...." in an assuming manner .

I just wanted to know , why should one consider $$\zeta_{C/\mathbb{F_q}}(u)=\exp \bigg(\ \sum_{n=1}^{\infty}\frac{ | C(\mathbb{F_{q^n}})|}{n} u^{n} \bigg)$$ where $C$ is a projective curve with non-negative genus over finite field $\ \mathbb{F_q}$. Here are my pointers :

I didn't understand about the reason behind introducing exponential function on the right side .

I understood that there is some measure of points taking a ratio of the cardinality ( on R.H.S ) of the solutions, but why is the ratio needed ? I got this doubt when I looked at some other heuristic consideration $\prod\frac{N_p}{p}$ ( Where $N_p$ is the cardinality of solution set at some prime $p$ ) , why is the need to take the ratio ? Isn't it not sufficient to look at just $N_p$ ? We get the cardinality directly, why should we find the ratio again by dividing it with $p$ ?

Similarly , why is the formulation of local part of $L$-series ( Hasse Weil L-function ) appear as $L_p(T)=1-a_pT+pT^2$ when the curve has good reduction at $p$ ( here $a_p=p+1-N_p$ and has some other formulation like $L_p(T) = 1-T$ and $1+T $ when the curve has split multiplicative and non-split multiplicative reductions at $p$ respectively , and $L_p(T)=1$ when the curve has additive reduction at $p$.

How was the quadratic equation on R.H.S ( i.e $1-a_pT+pT^2$ ) formulated ? Was it a scaffolding to get some heuristic output later , or it has a specific meaning derived from something, or what ? Same with $1-T$ and $1+T$ .

Please do explain me , I am sorry my learned friends, if I have wasted your time, but every book I referred starts with Let, and I thought that its just a setting . If you want me to suggest some book that does the same task of explaining what I asked, you are welcome to suggest me .

Cordially,

Shanmukha Srinivasan.

Arithmetical Algebraic Geometry. Proceedings of a Conference held at Purdue University, December 5-7, 1963. Edited by O. F. G. Schilling, Harper & Row, Publishers, New York 1965. And en.wikipedia.org/wiki/Local_zeta-function which you are supposed to read before posting here, in fact. Sandwiches which are too think can dislocate your jaw. Split multiplicative has the local zeta of the projective line minus two points: think about that for intuition. $\endgroup$ – Charles Matthews Nov 6 '12 at 19:29