Let $E$ be an elliptic curve.
My book reads, we denote '$L$ series of $E$ regrading $l$ adit representation of $Gal(\overline{\Bbb{Q}}/\Bbb{Q})$' as $L(E/\Bbb{Q}):= \prod_{p}\frac{1}{1-a_pp^{-s}+ε_p p^{1-2s}}=\sum_{n\ge 1}\frac{a_n}{n^s}$. Here, for given $p$, if we admit the former $a_p$ and latter $a_n$ of this equation coincides when $n=p$, we can define '$a_p$ of $E$' by $L$ series.
I have questions about this definition.
- What is definition of $a_p$ and $ε_p$ here ? My book does not write about that, but I guess $a_p$ is trace of Frobenius and $ε_p$ is determinant of Frobenius.
- why the former $a_p$ and latter $a_p$ coincides for all $p$ ?
- The definition seems to be the same as 'Hasse Weil $L$ function' and $l$ does not appear in the definition above. Why we(the book) calls this $L$ as '$L$ series of $E$ regrading $l$ adit representation of $Gal(\overline{\Bbb{Q}}/\Bbb{Q})$' ?