Quasi-periodic functions and Drinfeld modular forms.

*(English)*Zbl 0703.11066Let E be an elliptic curve over the complex numbers \({\mathbb{C}}\). As is well-known there is a lovely way to describe the de Rham cohomology of E via use of the “universal additive extension of E”. One then derives the classical formulation of periods, quasi-periods, (with associated Legendre relation) etc...

Now let K be a function field in one variable over the finite field \({\mathbb{F}}_ q\). Let \(\infty\) be a fixed place of K and let A be the Dedekind domain of those functions regular away from \(\infty\). There is a (by now) well-known analogy between the theory of elliptic curves over \({\mathbb{C}}\) and the theory of Drinfeld modules over \(K_{\infty}\) \((=\) the completion of K with respect to \(\infty)\). This analogy motivated Pierre Deligne, a few years ago, to give the “correct” definition of DeRham Cohomology for Drinfeld module E. (That this notion could be defined at all is really quite remarkable - indeed the underlying space of E is just the additive group \(G_ a\). Thus the “cohomology” lives not in the space but in the A-action associated to the E.) Jing Yu suggested an interpretation \(H^*_{DR}(E)\) in terms of a “derivation formalism” which is quite useful for calculations. Greg Anderson then established that Yu’s formulation and Deligne’s were the same. Moreover, Anderson established (using the appropriate definitions) the Legendre relation for quasi-periods.

In the paper being reviewed, the author presents another approach to \(H^*_{DR}(E)\) in the case where A is the polynomial ring and the rank of E is two. This is done by establishing a remarkable connection with the theory of modular forms on the algebraic upper half-plane \(\Omega\). Once that is accomplished the author provides a proof of the Legendre relation.

Finally a definition of the “Gauss-Manin” connection is given. This is done only over \(\Omega\). Thus the important problems remain of a) generalizing to arbitrary A and rank; and b) extending the definition to work over all of Spec(A) (not just generically).

Now let K be a function field in one variable over the finite field \({\mathbb{F}}_ q\). Let \(\infty\) be a fixed place of K and let A be the Dedekind domain of those functions regular away from \(\infty\). There is a (by now) well-known analogy between the theory of elliptic curves over \({\mathbb{C}}\) and the theory of Drinfeld modules over \(K_{\infty}\) \((=\) the completion of K with respect to \(\infty)\). This analogy motivated Pierre Deligne, a few years ago, to give the “correct” definition of DeRham Cohomology for Drinfeld module E. (That this notion could be defined at all is really quite remarkable - indeed the underlying space of E is just the additive group \(G_ a\). Thus the “cohomology” lives not in the space but in the A-action associated to the E.) Jing Yu suggested an interpretation \(H^*_{DR}(E)\) in terms of a “derivation formalism” which is quite useful for calculations. Greg Anderson then established that Yu’s formulation and Deligne’s were the same. Moreover, Anderson established (using the appropriate definitions) the Legendre relation for quasi-periods.

In the paper being reviewed, the author presents another approach to \(H^*_{DR}(E)\) in the case where A is the polynomial ring and the rank of E is two. This is done by establishing a remarkable connection with the theory of modular forms on the algebraic upper half-plane \(\Omega\). Once that is accomplished the author provides a proof of the Legendre relation.

Finally a definition of the “Gauss-Manin” connection is given. This is done only over \(\Omega\). Thus the important problems remain of a) generalizing to arbitrary A and rank; and b) extending the definition to work over all of Spec(A) (not just generically).

Reviewer: D.Goss

##### MSC:

11R58 | Arithmetic theory of algebraic function fields |

11G09 | Drinfel’d modules; higher-dimensional motives, etc. |

14F40 | de Rham cohomology and algebraic geometry |

11F11 | Holomorphic modular forms of integral weight |

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\textit{E.-U. Gekeler}, Compos. Math. 69, No. 3, 277--293 (1989; Zbl 0703.11066)

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##### References:

[1] | G. Anderson : t-motives . Duke Math. J. 53 (1986) 457-502. · Zbl 0679.14001 |

[2] | G. Anderson : Drinfeld modules and tensor products . In preparation. |

[3] | V.G. Drinfeld : Elliptic modules (russian) . Math. Sbornik 94 (1974) 594-627; English translation: Math. USSR-Sbornik 23 (1976) 561-592. · Zbl 0321.14014 |

[4] | E.U. Gekeler : Modulare Einheiten für Funktionenkörper . J. reine angew. Math. 348 (1984) 94-115. · Zbl 0523.14021 |

[5] | E.U. Gekeler : Drinfeld modular curves . Lecture Notes in Mathematics 1231. Springer-Verlag (1986). · Zbl 0607.14020 |

[6] | E.U. Gekeler : On the coefficients of Drinfeld modular forms . Inv. Math. 93 (1988) 667-700. · Zbl 0653.14012 |

[7] | L. Gerritzen and M. van der Put: Schottky groups and Mumford curves . Lecture Notes in Mathematics 817. Springer-Verlag (1980). · Zbl 0442.14009 |

[8] | N. Katz : P-adic properties of modular schemes and modular forms . Lecture Notes in Mathematics 350 69-190. Springer-Verlag (1973). · Zbl 0271.10033 |

[9] | E.U. Gekeler : On the de Rham isomorphism for Drinfeld modules . IAS-Preprint Princeton 1988. |

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