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.



Blueprint of L-functions and need for introducing them ( Hasse-Weil L-functions ) 
There is an excellent reason why the exponential term and the division by $n$ are there, although they look a bit mysterious at first. 


Firstly, a correction to your formula: it should be $|C(\mathbb{F}\_{q^n})|$, the number of solutions over the field with *q* elements, not $|C(\mathbb{F}\_{q})|$. (Notice that this means that the $n$th term really depends on $n$ and $C$ in a subtle way, because it "knows" how many points C has over every extension of the original field.) 


With this correction made, a miracle occurs: the quantity $\zeta\_{C / \mathbb{F}\_q}$ -- a priori just some formal power series -- is a **rational function**. 


To get some idea of the magic that's going on here, let's consider some simple examples. Firstly, you can take $C$ to be $\mathbb{P}^1$. That's not a very interesting curve, I know, but it's a curve. Then $C$ has exactly $q^n + 1$ points over $\mathbb{F}\_{q^n}$ -- one for each element of the field, together with the point at $\infty$ -- and we get


$$ \zeta\_{C / \mathbb{F}\_q} = \exp( \sum\_{n \ge 1} \frac{q^n + 1}{n} u^n) = \exp(-\log (1-u) - \log (1 - qu)) = \frac{1}{(1 - u)(1 - qu)}.$$


As promised, this is a rational function! If the funny exponential term and the $1/n$ factor hadn't been there in the definition, we wouldn't have got anything so nice. 


It turns out that this happens for any curve $C$ (this was proved by Andre Weil) and in fact for higher-dimensional varieties too (this was proved by Dwork). 


PS. If $C$ is an elliptic curve, then one can show (e.g. this is in Silverman's book "The Arithmetic of Elliptic Curves", in section V.5) that $\zeta\_{C/\mathbb{F}\_q}$ is given by
$$ \zeta\_{C/\mathbb{F}\_q}(u) = \frac{1 - a u + q u^2}{(1 - u)(1 - qu)}$$
where $a = q + 1 - |C(\mathbb{F}\_q)|$. So this quadratic term appearing here really appears for a reason; it wasn't just plucked out of midair. I hope that answers another part of your question, which is where this quadratic comes from.


