×

zbMATH — the first resource for mathematics

On modular units. (English) Zbl 0663.10025
Let n be a positive integer, \(U_ n\) the group of modular units of level n, and \(U_{n/{\mathbb{Z}}}\) the subgroup of integral modular units, as defined by D. Kubert and S. Lang [Math. Ann. 218, 175-189 (1975; Zbl 0311.14005)]. Kubert and Lang determined the rank of \(U_{n/{\mathbb{Z}}}\) by finding generators for \(U_ n\) and calculating their q-expansions. In this note we use the geometry of the moduli schemes of elliptic curves (following an idea of Beilinson) to give a description of \(U_{n/{\mathbb{Z}}}\otimes {\mathbb{Q}}\) as a module over the Hecke algebra. In particular, this gives a simple way to calculate the rank of the group of integral modular units for any reasonable congruence subgroup of \(SL_ 2({\mathbb{Z}})\).
Reviewer: A.J.Scholl

MSC:
11F99 Discontinuous groups and automorphic forms
14K99 Abelian varieties and schemes
11F06 Structure of modular groups and generalizations; arithmetic groups
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Beilinson, A.A.: Higher regulators and values ofL-functions. J. Soviet Math.30, 2036-2070 (1985) · Zbl 0588.14013 · doi:10.1007/BF02105861
[2] Casselman, W.: An assortment of results on representations ofGL 2 (k). Modular functions of one variable II. Lect. Notes Math.349, 1-54. Berlin Heidelberg New York: Springer 1973
[3] Deligne, P.: Courbes elliptiques: formulaire. Modular functions of one variable IV. Lect. Notes Math.476, 79-88. Berlin Heidelberg New York: Springer 1975
[4] Deligne, P., Rapoport, M.: Les schémas de modules des courbes elliptiques. Modular functions of one variable II. Lect. Notes Math.349, 143-316. Berlin Heidelberg New York: Springer 1973
[5] Jacquet, H., Langlands, R.P.: Automorphic forms onGL(2). Lect. Notes Math.114, Berlin Heidelberg New York: Springer 1970 · Zbl 0236.12010
[6] Katz, N.M., Mazur, B.: Arithmetic moduli of elliptic curves. Ann. Math. Stud.108 (1985) · Zbl 0576.14026
[7] Kubert, D., Lang, S.: Units in the modular function field II. Math. Ann.218, 175-189 (1975) · Zbl 0311.14005 · doi:10.1007/BF01370818
[8] Schappacher, N., Scholl, A.J.: Beilinson’s theorem on modular curves. Beilinson’s conjectures on special values ofL-functions. Rapoport, M., Schappacher, N., Schneider, P. (eds.). New York London: Academic Press 1988 · Zbl 0676.14006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.