Modular forms and Period Polynomials

1.) What is the importance of special values of L functions in connection to weakly holomorphic modular forms? Why is the study of special values a subject of intense study except the fact it is useful for some important conjectures?

2.) How can it be shown that special values of L - functions (cuspidal) Hecke eigenforms correspond to the coefficients of Period Polynomials in case of holomorphic and weakly holomorphic modular forms?

• I find the question (1) puzzling. You ask "Why is the study of special values a subject of intense study except the fact it is useful for some important conjectures?". Doesn't that question answer itself? What better reason could there be for studying a mathematical theory than "because it is useful for some important conjectures"? – David Loeffler Nov 24 '18 at 9:54
• How is a screw useful in the context of a screwdriver? I think you're looking at the question backwards; we use weakly-holomorphic modular forms to help us understand L-values, not the other way round. – David Loeffler Nov 24 '18 at 11:40

But the idea is that a cuspidal modular form of weight $$k$$ and level $$1$$ is completely determined by its period polynomial, i.e. by the spacial values $$L(f,i)$$ for $$i=1,...,k-1$$. In fact you should distinguish the even part and odd part of the period polynomial, each of which characterizes $$f$$. Zagier (Inventiones Math. 104, 1991) even extended the construction to Eisenstein series (you don't quite get a polynomial, but rather a rational function).
Of course, all this generalizes to higher level. The object you get is usually called modular symbols''. I recommend the paper of Loïc Merel (Universal Fourier expansions of modular forms) for modular symbols in the most general setting, in particular for the study of Hecke operators.
• Don't you want to look at $\int_{\gamma(i\infty)}^{i\infty} f(z) P(z-x) dz, \deg(P) \le k-2$ for each $\gamma \in SL_2(\mathbb{Z})$, not only $\gamma(i\infty ) = 0$ ? – reuns Nov 24 '18 at 6:27
• It is part of the theory of Eichler-Shimura-Manin that it suffices to integrate between $0$ and $\infty$. The reason is basically the surjectivity of the Manin symbol map (Manin's continued fraction trick). – Emmanuel Lecouturier Nov 24 '18 at 7:17
• @reuns The common definition is to look at all $\int_{\gamma 0}^{\gamma \infty}$ with $\gamma \in \Gamma \backslash \mathrm{SL}_2(\mathbb{Z})$. – François Brunault Nov 24 '18 at 8:53