$L$-functions of elliptic curves over $\mathbb{Q}$ How to find out the $P_{v}(E/\mathbb{Q},X)$ theoretically given below in the definition of $L$-functions for elliptic curves over $\mathbb{Q}$ $?$ Please cite some references for the same.
For an elliptic curve $E$ over a number field $\mathbb{Q}$, the $L$-function $L(E/\mathbb{Q},s)$ is given by the Euler product 
$$ L(E/\mathbb{Q},s)=\prod_{v}L_{v}(E/\mathbb{Q},s)=\prod_{v}P_{v}(E/\mathbb{Q},v^{-s})^{-1},  $$ 
where $v$ runs over primes of $\mathbb{Q}$ and the polynomials $ P_v(E/\mathbb{Q},T) $ depending on the reduction type of $E$ over $ \mathbb{Q}_{p} $ are given explicitly by:
$$    P_{v}(E/\mathbb{Q},X) =
                 \begin{cases}
              1-a_{v}X+vX^{2}, & \text{good reduction} \\
              1-X,       & \text{split multiplicative reduction} \\
              1+X,       & \text{non-split multiplicative reduction} \\
              1,      & \text{additive reduction} 
                 \end{cases}
 $$
where $ a_{v}= v+1- \mid\widetilde{E}(\mathbb{F}_{v})\mid$ and $\sim$ is reduction of $E$ at $v$.
 A: There's a lot of mathematics going on behind that definition. 
At the primes of good reduction, the Euler factor is essentially the (denominator of the) local zeta function of the reduction of $E$ modulo $p$. See Hasse-Weil zeta function. 
At the other primes, the justification for the definition is less straightforward, and requires étale cohomology, or in this case the poor man's étale cohomology in the form of the Tate module. For each prime $\ell$, one has the $\ell$-adic Tate module $T_\ell(E) = (\varprojlim_n E[\ell^n](\overline{\mathbf Q})) \otimes_{\mathbf Z_\ell} \mathbf Q_\ell$ which is a $2$-dimensional representation of $G_\mathbf{Q} = \text{Gal}(\overline{\mathbf Q}/\mathbf Q)$ over the field $\mathbf Q_\ell$. It turns out that that this collection of representations, indexed by the primes $\ell$, forms a compatible system of $\ell$-adic representations, which is something we can build an $L$-function from, using the formalism of Artin $L$-functions. At the primes of good reduction, the Euler factors coincide with those already constructed by Hasse-Weil, and at the primes of bad reduction it gives the Euler factors which you have written down.
Most of this can be found in Silverman's book(s) on elliptic curves.
