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.

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 |