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Zeta-functions of binary Hermitian forms and special values of Eisenstein series. (English) Zbl 0659.10021

The work under review is a systematic investigation of zeta-functions of binary Hermitian forms over the ring \(\mathfrak o\) of integers of an imaginary quadratic number field \(K={\mathbb{Q}}(\sqrt{D})\) with discriminant \(D<0\). First we develop a theory of representations of numbers by binary Hermitian forms over \(\mathfrak o\) which parallels the classical theory of representations of integers by binary quadratic forms over \(\mathbb{Z}.\)
A key result is a bijection described in Theorem 2.3. This bijection means that the sum of the numbers of representations (modulo \(\mathrm{SL}(2,\mathfrak o)\)-units) of an integer \(k\neq 0\) by a set of representatives of the \(\mathrm{SL}(2,\mathfrak o)\)-classes of binary Hermitian forms over \({\mathfrak o}\) with discriminant \(\Delta\) is equal to the number of cosets \(\lambda +k{\mathfrak o}\) with \(\lambda\in {\mathfrak o}\), \(\lambda \bar{\lambda}+\Delta \equiv 0 \bmod k\).
The number of these cosets is explicitly computed, and the result is applied to the study of zeta-functions of (definite or indefinite) Hermitian forms. For simplicity we restrict in this review to the case of positive definite forms over \(\mathfrak o\).
Let \(f\) be a positive definite binary Hermitian form over \(\mathfrak o\) with \(\mathrm{SL}(2,\mathfrak o)\)-unit group \(\mathcal E_ 1(f)\) and define \[ Z_{\mathfrak o}(f,s):=\frac{1}{| {\mathcal E}_ 1(f)|}\sum_{u,v\in \mathfrak o, <u,v>=\mathfrak o}f(u,v)^{-1-s},\quad Z(\Delta,s):=\sum_{[f]}Z_{\mathfrak o}(f,s), \] where the latter sum extends over a representative system of the \(\mathrm{SL}(2,\mathfrak o)\)-orbits of positive definite binary Hermitian forms over \(\mathfrak o\) with discriminant \(\Delta\). Then the analytic version of the results from above says that \[ Z(\Delta,s)=\theta (\Delta,s) \zeta (s) L(s+1,\chi_ D)^{-1}, \] where \(\theta (\Delta,s)\) is a finite Euler product whose factors are explicitly known.
Second we consider the genus \(\mathcal G(f)\) of the positive definite form \(f\) and its associated zeta-function \(\hat Z(\mathcal G(f),s)\). The main result of H. Braun [Abh. Math. Semin. Hansische Univ. 14, 61–150 (1941; Zbl 0025.01603)] expresses \(\hat Z(\mathcal G(f),s)\) in terms of a Dirichlet series whose coefficients are determined by certain local densities. We explicitly compute those local densities in all cases. The results are most easily formulated in terms of the Hilbert symbol, but the mere length of the theorem is still considerable (see Theorem 6.1). This enables us to evaluate \(\hat Z(\mathcal G(f),s)\) in terms of sums of certain products \(L(s,\chi)L(s+1,\chi)\) for suitable \(\chi\); the very lengthy results are written out in Theorems 7.4–7.6.
Third we reformulate the preceding results in terms of the Eisenstein series for \(\mathrm{PSL}(2,\mathfrak o)\) considered by the authors in [J. Reine Angew. Math. 360, 160–213 (1985; Zbl 0555.10012)]. In particular, we compile tables of the various \(\mathrm{SL}(2,\mathfrak o)\)-classes of positive definite forms over \(\mathfrak o\) for some small discriminants \(\Delta\) and for \(D=-4,-8,-8\). Then we apply our general results to these examples. This yields a wealth of results on special values of Eisenstein series and on explicit representation numbers. In particular, we obtain several explicit representation numbers which were already communicated by Liouville more than 100 years ago, and we obtain some representation numbers which we could not locate in the literature.
In some cases the formulae for the representation numbers involve Fourier coefficients of cusp forms of weight 2, and the Dirichlet series associated with these cusp forms are known to be equal to \(L\)-series of certain elliptic curves. This yields some intriguing formulae for the periods of the elliptic curves in question.

MSC:

11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
11F27 Theta series; Weil representation; theta correspondences
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11M35 Hurwitz and Lerch zeta functions
11F03 Modular and automorphic functions
11P05 Waring’s problem and variants
11E16 General binary quadratic forms
14H45 Special algebraic curves and curves of low genus
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References:

[1] Ananda-Rau, K.: Application of modular equations to some quadratic forms. J. Indian Math. Soc.24, 77-130 (1960) · Zbl 0107.26802
[2] Andrianov, A.N.: On the representation of numbers by certain quadratic forms and the theory of elliptic curves. Am. Math. Soc. Transl. Ser. 2,66, 191-203 (1968) [Trans]. Izv. Akad. Nauk SSSR, Ser. Mat.29, 227-238 (1965)] · Zbl 0179.07302
[3] Bianchi, L.: Geometrische Darstellung der Gruppen linearer Substitutionen mit ganzen complexen Coefficienten nebst Anwendungen auf die Zahlentheorie. Math. Ann.38, 313-333 (1891) (=OpereI, 1, pp. 233-258) · JFM 23.0216.01 · doi:10.1007/BF01199425
[4] Borevich, Z.I., Shafarevich, I.R.: Number theory. New York, London, Academic Press 1966 (Russian Edition: Moscow (1964) · Zbl 0121.04202
[5] Braun, H.: Zur Theorie der hermitischen Formen. Abh. Math. Semin. Univ. Hamb.14, 61-150 (1941) · Zbl 0025.01603 · doi:10.1007/BF02940742
[6] Cassels, J.W.S.: Rational quadratic forms. London, New York: Academic Press 1978 · Zbl 0395.10029
[7] Cohn, H.: Advanced number theory. New York: Dover 1980 (Corrected republication of the work: A second course in number theory. New York: Wiley 1962)
[8] Dickson, L.E.: Linear groups, with an exposition of the Galois field theory. Leipzig: Teubner 1901 (Reprint: New York: Dover 1958) · JFM 32.0128.01
[9] Efrat, I., Sarnak, P.: The determinant of the Eisenstein matrix and Hilbert class fields. Trans. Am. Math. Soc.290, 815-824 (1985) · Zbl 0576.10017 · doi:10.1090/S0002-9947-1985-0792829-1
[10] Elstrodt, J., Grunewald, F., Mennicke, J.: Eisenstein series on three-dimensional hyperbolic space and imaginary quadratic number fields. J. Reine Angew. Math.360, 160-213 (1985) · Zbl 0555.10012
[11] Elstrodt, J., Grunewald, F., Mennicke, J.: Eisenstein series for imaginary quadratic number fields. In: The Selberg trace formula and related topics. D. A. Hejhal, P. Sarnak, A. A. Terras (eds.). Contemp. Math.53, 97-117 (1986) · Zbl 0601.10017
[12] Fomenko, O.M.: Application of Eichler’s reduction formula to the representation of numbers by certain quaternary quadratic forms. Math. Notes9, 41-44 (1971) [Transl. Mat. Zametki9, 71-76 (1971) · Zbl 0223.10008
[13] Gogishvili, G.P.: On representation of numbers by quaternary quadratic forms with coefficients equal to 1 and 11. Bull. Acad. Sci. Georgian SSR49, 9-12 (1968) (Russian) · Zbl 0159.06601
[14] Gongadze, R.: On representation of numbers by certain quadratic forms in four variables. Bull. Acad. Sci. Georgian SSR28, 385-392 (1962) (Russian) · Zbl 0136.33604
[15] Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Fifth edition. Oxford: Oxford University Press 1979 · Zbl 0423.10001
[16] Hecke, E.: Vorlesungen über die Theorie der algebraischen Zahlen. Leipzig: Akademische Verlagsgesellschaft 1932. (Reprint: New York: Chelsea 1970; Engl.: Berlin, Heidelberg, New York: Springer 1983) · Zbl 0208.06101
[17] Hecke, E.: Mathematische Werke. 2. Aufl. Göttingen: Vandenhoeck & Ruprecht 1970 · Zbl 0205.28902
[18] Hermite, Ch.: Sur la théorie des formes quadratiques. J. Reine Angew. Math.47, 313-342, 343-368 (1854) (= EuvresI, 200-233, 234-263) · ERAM 047.1273cj · doi:10.1515/crll.1854.47.313
[19] Humbert, G.: Sur la réduction des formes d’Hermite dans un corps quadratique imaginaire. C.R. Acad. Sci. Paris161, 189-196 (1915) · JFM 45.1255.04
[20] Humbert, G.: Sur les représentations d’un entier par certaines formes quadratiques indéfinies C.R. Acad. Sci. Paris166, 581-587 (1918) · JFM 46.0234.01
[21] Humbert, G.: Sur les représentations propres d’un entier par les formes positives d’Hermite dans un corps quadratique imaginaire. C.R. Acad. Sci. Paris169, 309-315 (1919) · JFM 47.0136.02
[22] Humbert, G.: Sur les représentations d’un entier par les formes positives d’Hermite dans un corps quadratique imaginaire. C.R. Acad. Sci. Paris169, 360-365 (1919) · JFM 47.0137.01
[23] Humbert, G.: Sur la mesure de l’ensemble des classes positives d’Hermite, de discriminant donné, dans un corps quadratique imaginaire. C.R. Acad. Sci. Paris169, 407-414 (1919) · JFM 47.0137.02
[24] Humbert, G.: Sur les formes quadratiques positives d’Hermite dans un corps quadratique imaginaire. C.R. Acad. Sci. Paris170, 349-355 (1920) · JFM 47.0139.01
[25] Jacobowitz, R.: Hermitian forms over local fields. Am. J. Math.84, 441-465 (1962) · Zbl 0118.01901 · doi:10.2307/2372982
[26] Kloosterman, H.D.: On the representation of numbers in the formax 2+by 2+cz 2+dt 2. Proc. Lond. Math. Soc. II. Ser.25, 143-173 (1926) · JFM 52.0170.02 · doi:10.1112/plms/s2-25.1.143
[27] Kogan, L.A.: On the representation of integers by certain positive definite quadratic forms. Publishing House ?FAN? of the Uzbek SSR, Tashkent, 1971 (Russian)
[28] Kogan, L.A., Tashpulatov, B.G., Faîziev, S.R.: Representation of numbers by quadratic forms. Publishing House ?FAN? of the Uzbek SSR, Tashkent, 1980 (Russian)
[29] Landau, E.: Vorlesungen über Zahlentheorie. Band I: Aus der elementaren Zahlentheorie. Leipzig: Hirzel 1927 (Reprinted New York: Chelsea 1950) · JFM 53.0123.17
[30] Ligozat, G.: Courbes modulaires de genre 1. Bull. Soc. Math. Fr. Mémo.43, 80 pp. (1975) · Zbl 0322.14011
[31] Liouville, J.: Sur la formeX 2+2Y 2+2Z 2+4T 2. J. Math. Pures Appl., II. Ser.7, 1-4 (1862)
[32] Liouville, J.: Sur la formex 2+y 2+8z 2+8t 2. J. Math. Pures Appl., II. Ser.7, 109-112 (1862)
[33] Liouville, J.: Sur la formex 2+y 2+16z 2+16t 2. J. Math. Pures Appl., II. Ser.7, 117-120 (1862)
[34] Liouville, J.: Sur la formex 2+2y 2+8z 2+16t 2. J. Math. Pures Appl., II. Ser.7, 153-154 (1862)
[35] Liouville, J.: Sur la formex 2+xy+y 2+z 2+zt+t 2. J. Math. Pures Appl., II. Ser.8, 141-144 (1863)
[36] Liouville, J.: Sur la formex 2+xy+y 2+2z 2+2zt+2t 2. J. Math. Pures Appl., II. Ser.8, 308-310 (1863)
[37] Liouville, J.: Sur la formex 2+xy+y 2+6z 2+6zt+6t 2. J. Math. Pures Appl. II. Ser.9, 181-182 (1864)
[38] Liouville, J.: Sur la forme2x 2+2xy+2y 2+3z 2+3zt+3t 2. J. Math. Pures Appl., II. Ser.9, 183-184 (1864)
[39] Liouville, J.: Sur la formex 2+xy+y 2+3z 2+3zt+3t 2. J. Math. Pures Appl., II. Ser.9, 223-224 (1864)
[40] Liouville, J.: Sur la forme2x 2+2xy+3y 2+2z 2+2zt+3t 2. J. Math. Pures Appl., II. Ser.10, 9-13 (1865)
[41] Liouville, J.: Sur la forme2x 2+2xy+5y 2+2z 2+2zt+5t 2. J. Math. Pures Appl. II. Ser.10, 21-24 (1865)
[42] Liouville, J.: Sur les deux formesx 2+y 2+6z 2+6t 2, 2x 2+2y 2+3z 2+3t 2. J. Math. Pures Appl., II. Ser.10, 359-360 (1865)
[43] Lomadse, G.: Über die Darstellung der Zahlen durch einige quaternäre quadratische Formen. Acta Arith.5, 125-170 (1959) · Zbl 0089.26705
[44] Malyshev, A.V.: On representation of integers by definite quadratic forms. Trudy Mat. Inst. Steklova65, 1-212 (1962) (Russian)
[45] Otremba, G.: Zur Theorie der hermiteschen Formen in imaginär-quadratischen Zahlkörpern. J. Reine Angew. Math.249, 1-19 (1971) · Zbl 0221.12007 · doi:10.1515/crll.1971.249.1
[46] Pepin, P.: Sur quelques formes quadratiques quaternaires. J. Math. Pures Appl. IV. Ser.6, 5-67 (1890) · JFM 22.0219.01
[47] Petersson, H.: Modulfunktionen und quadratische Formen. Berlin, Heidelberg, New York: Springer 1982 · Zbl 0493.10033
[48] Ramanathan, K.G.: On the analytic theory of quadratic forms. Acta Arith.21, 423-436 (1972) · Zbl 0265.10014
[49] Sarnak, P.: The arithmetic and geometry of some hyperbolic three manifolds. Acta Math.151, 253-295 (1983) · Zbl 0527.10022 · doi:10.1007/BF02393209
[50] Siegel, C.L.: Über die analytische Theorie der quadratischen Formen. Ann. Math.36, 527-606 (1935) (=Gesammelte Abhandlungen, Bd.I, S. 326-405. Berlin, Heidelberg, New York: Springer 1966) · JFM 61.0140.01 · doi:10.2307/1968644
[51] Studien zur Theorie der quadratischen Formen. B. L. van der Waerden, H. Gross (Hrsg.). Basel, Stuttgart: Birkhäuser 1968
[52] Tate, J.: The arithmetic of elliptic curves. Invent. Math.23, 179-206 (1974) · Zbl 0296.14018 · doi:10.1007/BF01389745
[53] Vepkhvadze, T.V.: On a formula of Siegel. Acta Arith.40, 125-142 (1982) (Russian) · Zbl 0478.10012
[54] Vepkhvadze, T.V.: On the analytic theory of quadratic forms. Tr. Tbilis. Mat. Inst. Razmadze72, 12-31 (1983) (Russian) · Zbl 0535.10024
[55] Walfisz, A.: Zur additiven Zahlentheorie. VI. Tr. Tbilis. Mat. Inst. Razmadze5, 197-254 (1938) · Zbl 0021.00901
[56] Weil, A.: Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann.168, 149-156 (1967) (=Collected Papers, Vol.III, p. 165-172) · Zbl 0158.08601 · doi:10.1007/BF01361551
[57] Weil, A.: Elliptic functions according to Eisenstein and Kronecker. Berlin, Heidelberg, New York: Springer 1976 · Zbl 0318.33004
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