On the period matrix of a Riemann surface of large genus (with an appendix by J. H. Conway and N. J. A. Sloane). (English) Zbl 0814.14033

Let \(H\) be a polarization on an abelian variety \(X = \mathbb{C}^ g/\Lambda\), considered as a hermitian form on the vector space \(\mathbb{C}^ g\). The number \(m = m(X,H)\), defined as the minimum of all values \(H(\lambda,\lambda)\) over all \(0 \neq \lambda \in \Lambda\) is an invariant of the polarized abelian variety \((X,H)\). Geometrically \(m\) is the square of the length of the shortest closed geodesic on the corresponding flat real torus. By a well-known compactness theorem of Mahler \(m\) may be thought of as a distance function to the boundary of the corresponding moduli space. Using results of the geometry of numbers the authors prove some explicit inequalities for the invariant \(m\). The surprising fact is that for large genus the whole Schottky locus lies in a very small neighborhood of the boundary of the moduli space of principally polarized abelian varieties. – This is applied to give examples of families of period matrices which are not Jacobians. – There are some applications concerning arithmetic lattices. It is shown that with the exception of a finite number every arithmetic lattice in \(\text{SL}_ 2(\mathbb{R})\) has a subgroup of index at most 2 which is noncongruence.
This paper contains several appendices: Appendix 0: Polarisations and quadratic forms (p. 50-51). – Appendix 1: Some explicit examples (p. 51- 53). – Appendix 2 (by J.H. Conway and N. J. A. Sloane): \(D_ 4\), \(E_ 8\), Leech and certain other lattices are symplectic (p. 53-55).
Reviewer: H.Lange (Erlangen)


14H42 Theta functions and curves; Schottky problem
14K20 Analytic theory of abelian varieties; abelian integrals and differentials
32G20 Period matrices, variation of Hodge structure; degenerations
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[1] [B-W] Barnes, E.S., Wall, G.E.: Some extreme forms defined in terms of abelian groups. J. Aust. Math. Soc.1, 47-63 (1959) · Zbl 0109.03304
[2] [B-M-S] Bass, H., Milnor, J., Serre, J.P.: Solutions of the congruence subgroup problem for SL(n) (n?3) and Sp(2n) (n?2). Publ. Math., Inst. Hautes Étud. Sci.33, 59-137 (1967) · Zbl 0174.05203
[3] [Be] Beardon, A.F.: The Geometry of discrete groups. (Grad. Texts Math., vol. 91) Berlin Heidelberg New York: Springer 1983 · Zbl 0528.30001
[4] [Bol] Bolza, O.: On binary sextics with linear transformations into themselves. Am. J. Math.10, 47-70 (1888) · JFM 19.0488.01
[5] [Bo] Borel, A.: Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Sc. Norm. Super. Pisa, IV. Ser.8, 1-33 (1981) · Zbl 0473.57003
[6] [Bu] Buser, P.: Geometry and spectra of compact Riemann surfaces. Boston Basel Stuttgart: Birkhäuser 1992 · Zbl 0770.53001
[7] [Ch] Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Gunning, R. (ed.) Problems in analysis, pp. 195-199. Princeton: Princeton University Press 1970 · Zbl 0212.44903
[8] [C-S] Conway, J.H. Sloane, N.J.A.: Sphere packings, lattices and groups. (Grundlehren Math. Wiss., vol. 290) Berlin Heidelberg New York, 1988
[9] [E] Epstein, D.B.A.: Curves on 2-manifolds and isotopies. Acta Math.115, 83-107 (1966) · Zbl 0136.44605
[10] [F] Farkas, H.: Schottky-Jung theory. In: Ehrenpreis, L., Gunning, R.C. (eds.), Theta functions, Bowdoin 1987. (Proc. Symp. Pure Math., vol. 49, 459-483) Providence, RI: Am. Math. Soc. 1989 · Zbl 0698.14025
[11] [F-K] Farkas, H., Kra, I.: Riemann surfaces. Berlin Heidelberg New York: Springer 1980 · Zbl 0475.30001
[12] [F-L-P] Fathi, A., Laudenbach, F., Poénaru, V.: Travaux de Thurston sur les surfaces. Seminaire Orsay. Astérisque66-67 (1979)
[13] [Fay] Fay, J.D.: Theta functions on Riemann surfaces. (Lect. Notes Math., vol. 352) Berlin Heidelberg New York: Springer 1973 · Zbl 0281.30013
[14] [G-G-PS] Gel’fand, I.M., Graev, M.I., Pyatetskii-Shapiro, I.I.: Representation theory and automorphic functions. (Generalized functions, vol. 6) Philadelphia London Toronto: Saunders 1969
[15] [Gro] Gromov, M.: Systoles and intersystolic inequalities. (Preprint 1993) · Zbl 0877.53002
[16] [Gr] Gross, B.H.: Group representations and lattices. J. Am. Math. Soc.3, no. 4, 929-960 (1990) · Zbl 0745.11035
[17] [Gru-Le] Gruber, P.M., Lekkerkerker, C.G.: Geometry of numbers, Amsterdam: North Holland 1987 · Zbl 0611.10017
[18] [Gu] Gunning, R.C.: Some curves in abelian varieties. Invent. Math.66, 377-389 (1982) · Zbl 0485.14009
[19] [I] Igusa, J.: On the irreducibility of Schottky’s divisor. J. Fac. Sci. Tokyo28, 531-544 (1982) · Zbl 0501.14026
[20] [K-L] Kabatiansky, G., Levenshtein, V.: Bounds for packings of a sphere in space. Prob. Peradachi Inf.14, no. 1 (1978)
[21] [L] Lang, S.: Introduction to algebraic and abelian functions. (Grad. Texts Math., vol. 89) Berlin Heidelberg New York: Springer 1982 · Zbl 0513.14024
[22] [L-L-L] Lenstra, A.K., Lenstra, H.W. Jr., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann.261, 515-534 (1982) · Zbl 0488.12001
[23] [Ma] Mazur, B.: Arithmetic on curves. Bull. Am. Math. Soc.14, 207-259 (1986) · Zbl 0593.14021
[24] [M] Mumford, D.: Curves and their Jacobians. Ann Arbor. University of Michigan Press 1976 · Zbl 0316.14010
[25] [Ra] Raghunathan, M.S.: On the congruence subgroup problem. Publ. Math., Inst. Hautes Étud. Sci.46, 107-161 (1976) · Zbl 0347.20027
[26] [R1] Randol, B.: Small eigenvalues of the Laplace operator on compact Riemann surfaces. Bull Am. Math. Soc.80, 996-1000 (1974) · Zbl 0317.30017
[27] [R2] Randol, B.: Cylinders in Riemann surfaces. Coment. Math. Helv.54, 1-5 (1979) · Zbl 0401.30036
[28] [Rn] Rankin, R.A.: The modular group and its subgroups. Madras: Ramanujan Inst. 1969
[29] [Sa] Sarnak, P.: Some applications of modular forms. (Cambridge Tracts Math., vol. 99) Cambridge: Cambridge University Press 1990 · Zbl 0721.11015
[30] [S-X] Sarnak, P., Xue, X.: Bounds for multiplicities of automorphic representations. Duke Math. J. 64 (no. 1), 207-227 (1991) · Zbl 0741.22010
[31] [Se] Selberg, A.: On the estimation of Fourier coefficients of modular forms. In: Collected Works, pp. 506-520. Berlin Heidelberg New York: Springer 1988
[32] [Sh] Shiota, T.: Characterization of Jacobian varieties in terms of soliton equations. Invent. Math.83, 333-382 (1986) · Zbl 0621.35097
[33] [Si] Siegel, C.L.: Symplectic geometry. Amer. J. Math.65, 1-86 (1943) · Zbl 0138.31401
[34] [T] Thurston, W.: The geometry and topology of 3-manifolds. (Princeton University Notes · Zbl 0483.57007
[35] [Va] Vaughan, R.C.: The Hardy Littlewood method. (Cambridge Tracts Math., vol. 80) Cambridge: Cambridge University Press 1981 · Zbl 0455.10034
[36] [Vi] Vignéras, M.F.: Quelques remarques sur la conjecture ?1-1/4. In: Bertin, M.-J. (ed.) Sém. de théorie des nombres, Paris 1981-1982. (Prog. Math., vol. 38, pp. 321-343) Boston Basel Stuttgart: Birkhäuser 1983
[37] [Z] Zograf, P.: A spectral proof of Rademacher’s conjecture for congruence subgroups of the modular group. J. Reine Angew. Math.414, 113-116 (1991) · Zbl 0709.11031
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