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On singular moduli for rank 2 Drinfeld modules. (English) Zbl 0744.11032

Let \(A\) be the polynomial ring \(\mathbb{F}_ q[T]\) over the finite field with \(q\) elements. It is well known that rank 2 Drinfeld \(A\)-modules are classified by a modular invariant \(j\). By analogy with the case of elliptic curves (i.e., the results evolving from the work of Gross and Zagier), it is natural to consider the prime factorization of singular invariants, or more generally, of their differences. A completely satisfactory description is given in the main result, Thm. 1.4, of the paper. It is interesting for its own sake, but may also be considered as a first step towards a result like Gross-Zagier’s over function fields. The proof relies on a lifting property for Drinfeld modules provided with an endomorphism and the calculation of the local intersection pairing on the moduli scheme. In the last section, the author gives some nice and very enlightening examples.

MSC:

11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R58 Arithmetic theory of algebraic function fields
11T55 Arithmetic theory of polynomial rings over finite fields
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References:

[1] P. Deligne and D. Husemöller , Survey of Drinfeld Modules, Current Trends in Arithmetical Algebraic Geometry (ed. K. Ribet) Contemporary Mathematics 67 (1986) American Math. Soc., Providence, Rhode Island. · Zbl 0627.14026
[2] D.R. Dorman , Special values of the elliptic modular function and factorization formulae , J. reine und ang. Math. 383 (1988), 207-220. · Zbl 0626.10022 · doi:10.1515/crll.1988.383.207
[3] V. Drinfeld , Elliptic Modules (Russian) Math. Sbornik 94 (1974) 594-627, English translation: Math. USSR-Sbornik 23 (1976) 561-592. · Zbl 0321.14014 · doi:10.1070/SM1974v023n04ABEH001731
[4] E.-U. Gekeler , Zur Arithmetic von Drinfeld-Moduln , Math. Ann. 262 (1983) 167-182. · Zbl 0536.14028 · doi:10.1007/BF01455309
[5] B.H. Gross , On canonical and quasi-canonical liftings , Invent. Math. 84 (1986) 321-326. · Zbl 0597.14044 · doi:10.1007/BF01388810
[6] B.H. Gross and D.B. Zagier , On singular moduli , J. reine und ang. Math. 355 (1985), 191-220. · Zbl 0545.10015
[7] David R. Hayes, Explicit class field theory in global function fields , in (ed. G. Rota) Studies in Algebra and Number Theory, Advances in Mathematical Supplementary Studies 16 (1979) 173-217, Academic Press. · Zbl 0476.12010
[8] Toyofumi Takahashi , Good reduction of elliptic modules , J. Math. Soc. Japan 34(2) (1982) 475-487. · Zbl 0476.14010 · doi:10.2969/jmsj/03430475
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