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**On the critical values of certain Dirichlet series and the periods of automorphic forms.**
*(English)*
Zbl 0656.10018

Arithmeticity questions involving special (critical) values of zeta functions (Dirichlet series) associated with automorphic forms of the Hilbert modular type (or like Eisenstein series) or concerning periods of automorphic forms (and, in particular, of abelian integrals on arithmetic curves) have, recently for several years, been deeply investigated in a series of papers by the author. The present important paper is yet another milestone in the same direction, offering not only generalizations of some of his earlier arithmeticity theorems (now, without parity conditions or other exceptions and covering the ‘perpetually present’ forms of half-integral weight, too) but also conjectures expected to have far-reaching consequences and open up ‘new territory’.

Reviewer: S.Raghavan

### MSC:

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11F27 | Theta series; Weil representation; theta correspondences |

11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |

11F33 | Congruences for modular and \(p\)-adic modular forms |

### Keywords:

critical values; Hilbert modular forms; zeta functions; Dirichlet series; automorphic forms; periods of automorphic forms; abelian integrals on arithmetic curves; arithmeticity theorems; half-integral weight### References:

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