On the rank of universal quadratic forms over real quadratic fields.(English)Zbl 1396.11061
Consider a real quadratic field \(\mathbb{Q}(\sqrt{D})\) with discriminant \(\Delta\) and let \(m(D)\) be the minimal number of variables required by a totally positive definite diagonal universal quadratic form. Let \(\omega_D= \sqrt{D}\) resp. \({1+\sqrt{D}\over 2}\) (for \(D\equiv 2,3\mod 4\), resp. \(D\equiv 1\mod 4\)) and \(\omega_D= [u_0,\overline{u_1,u_2,\dots, u_s}]\) the periodic continued fraction expression with \(s\) minimal. If \(s\) is odd, let \(M_D= u_1+\dots +u_s\).
The authors obtain lower and upper bounds for \(\omega(D)\), e.g., \(\omega(D)\leq 8M_D\) if \(s\) is odd and \(M_D\leq c\sqrt{\Delta}(\log\Delta)^2\) for an absolute constant \(c>0\).
Furthermore, they establish a link between \(M_D\) and special values of \(L\)-functions in the spirit of Kronecker’s limit formula: \[M_D\sim {\sqrt{\Delta}\over \zeta^{(\Delta)}(2)} \Biggl(L(D)+{1\over h} L(1,\chi_\Delta)\log\sqrt{D}\Biggr).\]
The authors obtain lower and upper bounds for \(\omega(D)\), e.g., \(\omega(D)\leq 8M_D\) if \(s\) is odd and \(M_D\leq c\sqrt{\Delta}(\log\Delta)^2\) for an absolute constant \(c>0\).
Furthermore, they establish a link between \(M_D\) and special values of \(L\)-functions in the spirit of Kronecker’s limit formula: \[M_D\sim {\sqrt{\Delta}\over \zeta^{(\Delta)}(2)} \Biggl(L(D)+{1\over h} L(1,\chi_\Delta)\log\sqrt{D}\Biggr).\]
Reviewer: Meinhard Peters (Münster)
MSC:
| 11E12 | Quadratic forms over global rings and fields |
| 11R11 | Quadratic extensions |
| 11A55 | Continued fractions |
Keywords:
universal quadratic form;real quadratic form;number field;continued fraction;additively indecomposable integer;Kronecker’s limit formulaCite
References:
| [1] | M. Bhargava,On the Conway-Schneeberger Fifteen Theorem, Contemp. Math.272 (1999), 27-37. zbl 0987.11027; MR 1803359 ·Zbl 0987.11027 |
| [2] | M. Bhargava, J. Hanke,Universal quadratic forms and the 290-theorem, preprint |
| [3] | A. Bir\'o, A. Granville,Zeta functions for ideal classes in real quadratic fields, at \(s=0\), J. Number Theory132 (2012), 1807-1829. DOI 10.1016/j.jnt.2012.02.003; zbl 1276.11180; MR 2922348 ·Zbl 1276.11180 ·doi:10.1016/j.jnt.2012.02.003 |
| [4] | V. Blomer, G. Harcos, P. Michel,Bounds for modular \(L\)-functions in the level aspect, Ann. Sci. Ecole Norm. Sup.40 (2007), 697-740. DOI 10.1016/j.ansens.2007.05.003; zbl 1185.11034; MR 2382859 ·Zbl 1185.11034 ·doi:10.1016/j.ansens.2007.05.003 |
| [5] | V. Blomer, V. Kala,Number fields without universal \(n\)-ary quadratic forms, Math. Proc. Cambridge Philos. Soc.159 (2015), 239-252. DOI 10.1017/S030500411500033X; zbl 1371.11084; MR 3395370; arxiv 1503.05736 ·Zbl 1371.11084 ·doi:10.1017/S030500411500033X |
| [6] | W. K. Chan, M.-H. Kim, S. Raghavan,Ternary universal integral quadratic forms, Japan. J. Math.22 (1996), 263-273. MR 1432376 ·Zbl 0868.11020 |
| [7] | A. Dress, R. Scharlau,Indecomposable totally positive numbers in real quadratic orders, J. Number Theory14 (1982), 292-306. DOI 10.1016/0022-314X(82)90064-6; zbl 0507.12002; MR 0660374 ·Zbl 0507.12002 ·doi:10.1016/0022-314X(82)90064-6 |
| [8] | W. Duke, J. Friedlander, H. Iwaniec,The subconvexity problem for Artin \(L\)-functions, Invent. Math.149 (2002), 489-577. DOI 10.1007/s002220200223; zbl 1056.11072; MR 1923476 ·Zbl 1056.11072 ·doi:10.1007/s002220200223 |
| [9] | C. Friesen,On continued fractions of given period, Proc. Amer. Math. Soc.103 (1988), 8-14. DOI 10.1090/S0002-9939-1988-0938635-4; zbl 0652.10006; MR 0938635 ·Zbl 0652.10006 ·doi:10.1090/S0002-9939-1988-0938635-4 |
| [10] | E. Fogels,On the zeros of Hecke’s L-functions I, Acta Arith.7 (1962), 87-106. zbl 0100.03801; MR 0136585 ·Zbl 0100.03801 |
| [11] | G. H. Hardy, E. M. Wright,An introduction to the theory of numbers, 5th edition. The Clarendon Press, Oxford University Press, New York, 1979 ·Zbl 0423.10001 |
| [12] | Heath-Brown,Hybrid bounds for Dirichlet \(L\)-functions. II, Quart. J. Math. Oxford31 (1980), 157-167. zbl 0396.10030; MR 0576334 ·Zbl 0396.10030 |
| [13] | E. Hecke, \`“Uber die Kroneckersche Grenzformel f\"ur reelle quadratische K\"orper und die Klassenzahl relativ-abelscher K\'”orper, Verhandl. d. Naturforschenden Gesell. Basel28, 363-372 (1917). (Mathematische Werke pp.198-207). zbl 47.0144.01 ·JFM 47.0144.01 |
| [14] | G. Herglotz,\`“Uber die Kroneckersche Grenzformel f\"ur reelle, quadratische K\"orper I, II, Berichte Verhandl. S\'”achs. Akad. Wiss. Leipzig75 (1923), 3-14, 31-37. zbl 49.0125.03 ·JFM 49.0125.03 |
| [15] | S. W. Jang, B. M. Kim,A refinement of the Dress-Scharlau theorem, J. Number Theory158 (2016), 234-243. DOI 10.1016/j.jnt.2015.06.003; zbl 1331.11092; MR 3393549 ·Zbl 1331.11092 ·doi:10.1016/j.jnt.2015.06.003 |
| [16] | M. Jacobson, H. Williams,Solving the Pell equation, CMS books in mathematics, Springer-Verlag 2009. ISBN 978-0-387-84922-5. MR 2466979 ·Zbl 1177.11027 |
| [17] | V. Kala,Universal quadratic forms and elements of small norm in real quadratic fields, Bull. Aust. Math. Soc.94 (2016), 7-14. DOI 10.1017/S0004972715001495; zbl 1345.11025; MR 3539315 ·Zbl 1345.11025 ·doi:10.1017/S0004972715001495 |
| [18] | V. Kala,Norms of indecomposable integers in real quadratic fields, J. Number Theory166 (2016), 193-207. DOI 10.1016/j.jnt.2016.02.022; zbl 06572137; MR 3486273; arxiv 1512.04691 ·Zbl 1414.11131 ·doi:10.1016/j.jnt.2016.02.022 |
| [19] | B. M. Kim,Finiteness of real quadratic fields which admit positive integral diagonal septenary universal forms, Manuscr. Math.99 (1999), 181-184. DOI 10.1007/s002290050168; zbl 0961.11016; MR 1697212 ·Zbl 0961.11016 ·doi:10.1007/s002290050168 |
| [20] | B. M. Kim,Universal octonary diagonal forms over some real quadratic fields, Commentarii Math. Helv.75 (2000), 410-414. DOI 10.1007/s000140050133; zbl 1120.11301; MR 1793795 ·Zbl 1120.11301 ·doi:10.1007/s000140050133 |
| [21] | X. Li,Upper bounds on \(L\)-functions at the edge of the critical strip, IMRN2010, 727-755. zbl 1219.11136; MR 2595006 ·Zbl 1219.11136 |
| [22] | J. E. Littlewood,On the class-number of the corpus \(P(\sqrt{-k})\), Proc. Lond. Math. Soc27 (1928), 358-372. DOI 10.1112/plms/s2-27.1.358; zbl 54.0206.02; MR 1575396 ·JFM 54.0206.02 ·doi:10.1112/plms/s2-27.1.358 |
| [23] | H. Maa{\ss},\`“ Uber die Darstellung total positiver Zahlen des K\'” orpers \(R(\sqrt 5)\) als Summe von drei Quadraten, Abh. Math. Sem. Univ. Hamburg14 (1942), 185-191. DOI 10.1007/BF02940744; zbl 67.0103.02; MR 3069720 ·JFM 67.0103.02 ·doi:10.1007/BF02940744 |
| [24] | O. Perron,Die Lehre von den Kettenbr\" uchen, Chelsea, New York, 1958 |
| [25] | H. Sasaki,Quaternary universal forms over \(\mathbb Q[\sqrt{13}]\), Ramanujan J.18 (2009), 73-80. DOI 10.1007/s11139-007-9056-2; zbl 1193.11033; MR 2471617 ·Zbl 1193.11033 ·doi:10.1007/s11139-007-9056-2 |
| [26] | C. L. Siegel,Sums of \(m\)-th powers of algebraic integers, Ann. Math.46 (1945), 313-339. zbl 0063.07010; MR 0012630 ·Zbl 0063.07010 |
| [27] | P. Yatsyna,A lower bound for the rank of a universal quadratic form with integer coefficients in a totally real field, preprint ·Zbl 1471.11128 |
| [28] | D. Zagier,A Kronecker limit formula for real quadratic fields, Math. Ann.213 (1975), 153-184. DOI 10.1007/BF01343950; zbl 0283.12004; MR 0366877 ·Zbl 0283.12004 ·doi:10.1007/BF01343950 |
| [29] | D. Zagier,Zetafunktionen und quadratische K\"orper, Springer-Verlag 1981, ISBN 3-540-10603-0 MR 0631688 |
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