In the preface of "A first course in modular forms", the author considered the quadratic equation $Q: x^2=d, \ \ d \in \mathbb{Z}, d \not=0$, and for each prime $p$ define an integer $a_p(Q)=\#\tilde Q(\mathbb{F}_p)-1$, where $\#\tilde Q(\mathbb{F}_p)$ is the number of solutions $x$ of equation $Q$ working modulo $p$. Then it can be shown that the set $\{a_2(Q),a_3(Q),a_5(Q), \cdots\}$ arises as a system of eigenvalues on a finite dimensional complex vector space associated to the equation $Q$ be defining a suitable linear operator $T_p$ on the vector space $V_N=\{f: (\mathbb{Z}/N\mathbb{Z})^* \rightarrow \mathbb{C}\}$, where $N=4|d|.$ Then we have $T_pf=a_p(Q)f$ for all $p$. And hence $\{a_p(Q)\}$ is a system of eigenvalues.
As an analogous result, consider the elliptic curve $E:y^2=4x^3-g_2x-g_3$ and let $a_p(E)=p-\#\tilde E(\mathbb{F}_p)$, then the famous modulaity theorem asserts that the sequence of solution counts $\{a_p(E)\}$ arises as a system of eigenvalues and the corresponding linear operator $T_p$ is the Hecke operator.
I want to ask if there is simlar result for other curves? I think the case for $C:x^n=d$ can be done by similar method in the book, and by some reciprocity law? But how can we choose the $a_p$ and $T_p$. How about other curve? Is there any book or paper that talked about this? I want to do a mini-research on this topic and obtain some result for other classes of curves, can it be done by an undergraduate? I have learned basic linear and abstract algebra, number theory, representation theory and some very basic theory of modular forms and elliptic curves. what else do i need? Do I need to learn some algebraic geometry? thank you in advance.
