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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?

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  • $\begingroup$ 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"? $\endgroup$ – David Loeffler Nov 24 '18 at 9:54
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    $\begingroup$ 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. $\endgroup$ – David Loeffler Nov 24 '18 at 11:40
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I don't know much about weakly holomorphic modular forms, so what follows is only about holomorphic modular forms. The answer to question 2 is just that this follows from the definition of the period polynomial as an integral and the usual relation between the L function and the Mellin transform.

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).

More than that: you get a Hecke equivariant isomorphism between odd period polynomials and cusp forms (you have to be carefull if you want to include Eisenstein series). The action of Hecke operators on period polynomials is explicit. Thus, period polynomials is a useful tools to compute with modular forms, and even study arithmetic properties like congruences between modular forms.

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.

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    $\begingroup$ 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 $\gamma(i\infty ) = 0$ ? $\endgroup$ – reuns Nov 24 '18 at 6:27
  • $\begingroup$ 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). $\endgroup$ – Emmanuel Lecouturier Nov 24 '18 at 7:17
  • $\begingroup$ @reuns The common definition is to look at all $\int_{\gamma 0}^{\gamma \infty}$ with $\gamma \in \Gamma \backslash \mathrm{SL}_2(\mathbb{Z})$. $\endgroup$ – François Brunault Nov 24 '18 at 8:53

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