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Special values of Dirichlet series, monodromy, and the periods of automorphic forms. (English) Zbl 0536.10023
Mem. Am. Math. Soc. 299, 116 p. (1984).
The author explores a relationship between the classical cusp forms for subgroups of finite index in $$\text{SL}(2,\mathbb Z)$$ and certain ordinary differential equations, and he develops a connection between the equations’ monodromy representation and the special values in the critical strip of the Dirichlet series associated to the cusp forms.
More precisely, let $$g$$ be a cusp form of weight $$k\geq 3$$ with respect to a subgroup $$\Gamma$$ of finite index in $$\text{SL}(2,\mathbb Z)$$. The author associates to $$g$$ an $$n$$th order linear homogeneous differential equation on $${\mathbb P}^ 1_{{\mathbb C}}$$ having three regular singular points $$(0,1,\infty)$$ with the specific basis of solutions in a sector at $$\infty$$ and a closed path based in the sector such that the special values of the Mellin transform $$i^ s M_ g(s)$$, $$s\in \{1,2,...,k-1\}$$ of $$g$$ can be expressed as a rational linear combination of the entries in the monodromy matrix $$(a_{ij})$$ in the specific basis around the given path $i^ s M_ g(s)=\int^{i\infty}_{0}g(z) z^{s-1}\, dz=\sum_{i,j}c_{ij} a_{ij},\quad c_{ij}\in {\mathbb Q}.$ The differential equation associated with $$g$$ arises from an extension of a local system by a symmetric power of the Gauss-Manin connection. The paper contains some interesting examples including of course the hypergeometric equation.
Reviewer: A. Venkov

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F11 Holomorphic modular forms of integral weight 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 34A30 Linear ordinary differential equations and systems 14J99 Surfaces and higher-dimensional varieties
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