Special values of partial zeta functions of real quadratic fields at nonpositive integers and the Euler-Maclaurin formula
HTML articles powered by AMS MathViewer
- by Byungheup Jun and Jungyun Lee PDF
- Trans. Amer. Math. Soc. 368 (2016), 7935-7964 Request permission
Abstract:
We compute the special values at nonpositive integers of the partial zeta function of an ideal of a real quadratic field in terms of the positive continued fraction of the reduced element defining the ideal. We apply the integral expression of the partial zeta value due to Garoufalidis-Pommersheim (2001) using the Euler-Maclaurin summation formula for a lattice cone associated to the ideal. From the additive property of Todd series w.r.t. the (virtual) cone decomposition arising from the positive continued fraction of the reduced element of the ideal, we obtain a polynomial expression of the partial zeta values with variables given by the coefficient of the continued fraction. We compute the partial zeta values explicitly for $s=0,-1,-2$ and compare the result with earlier works of Zagier (1977) and Garoufalidis-Pommersheim (2001). Finally, we present a way to construct Yokoi-Byeon-Kim type class number one criterion for some families of real quadratic fields.References
- A. Baker, Imaginary quadratic fields with class number $2$, Ann. of Math. (2) 94 (1971), 139–152. MR 299583, DOI 10.2307/1970739
- András Biró, Yokoi’s conjecture, Acta Arith. 106 (2003), no. 1, 85–104. MR 1956977, DOI 10.4064/aa106-1-6
- András Biró, Chowla’s conjecture, Acta Arith. 107 (2003), no. 2, 179–194. MR 1970822, DOI 10.4064/aa107-2-5
- András Biró and Andrew Granville, Zeta functions for ideal classes in real quadratic fields, at $s=0$, J. Number Theory 132 (2012), no. 8, 1807–1829. MR 2922348, DOI 10.1016/j.jnt.2012.02.003
- Michel Brion and Michèle Vergne, Lattice points in simple polytopes, J. Amer. Math. Soc. 10 (1997), no. 2, 371–392. MR 1415319, DOI 10.1090/S0894-0347-97-00229-4
- Dongho Byeon and Hyun Kwang Kim, Class number $1$ criteria for real quadratic fields of Richaud-Degert type, J. Number Theory 57 (1996), no. 2, 328–339. MR 1382755, DOI 10.1006/jnth.1996.0052
- Dongho Byeon and Hyun Kwang Kim, Class number $2$ criteria for real quadratic fields of Richaud-Degert type, J. Number Theory 62 (1997), no. 2, 257–272. MR 1432773, DOI 10.1006/jnth.1997.2059
- Dongho Byeon, Myoungil Kim, and Jungyun Lee, Mollin’s conjecture, Acta Arith. 126 (2007), no. 2, 99–114. MR 2289410, DOI 10.4064/aa126-2-1
- Dongho Byeon and Jungyun Lee, Class number 2 problem for certain real quadratic fields of Richaud-Degert type, J. Number Theory 128 (2008), no. 4, 865–883. MR 2400045, DOI 10.1016/j.jnt.2007.02.006
- Dongho Byeon and Jungyun Lee, A complete determination of Rabinowitsch polynomials, J. Number Theory 131 (2011), no. 8, 1513–1529. MR 2793892, DOI 10.1016/j.jnt.2011.02.009
- Pierre Cartier, An introduction to zeta functions, From number theory to physics (Les Houches, 1989) Springer, Berlin, 1992, pp. 1–63. MR 1221100
- J. Coates and W. Sinnott, On $p$-adic $L$-functions over real quadratic fields, Invent. Math. 25 (1974), 253–279. MR 354615, DOI 10.1007/BF01389730
- Pierre Deligne and Kenneth A. Ribet, Values of abelian $L$-functions at negative integers over totally real fields, Invent. Math. 59 (1980), no. 3, 227–286. MR 579702, DOI 10.1007/BF01453237
- William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
- Stavros Garoufalidis and James E. Pommersheim, Values of zeta functions at negative integers, Dedekind sums and toric geometry, J. Amer. Math. Soc. 14 (2001), no. 1, 1–23. MR 1800347, DOI 10.1090/S0894-0347-00-00352-0
- C. F. Gauss, Disquisitiones Aritheticae, Göttingen (1801); English translation by A. Clarke, revised by W. Waterhouse, 1986 Springer-Verlag reprint of the Yale Univ. Press, New Haven, 1966 ed.
- Gerard van der Geer, Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 16, Springer-Verlag, Berlin, 1988. MR 930101, DOI 10.1007/978-3-642-61553-5
- Dorian M. Goldfeld, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 4, 624–663. MR 450233
- Dorian Goldfeld, Gauss’s class number problem for imaginary quadratic fields, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 1, 23–37. MR 788386, DOI 10.1090/S0273-0979-1985-15352-2
- Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI 10.1007/BF01388809
- Kurt Heegner, Diophantische Analysis und Modulfunktionen, Math. Z. 56 (1952), 227–253 (German). MR 53135, DOI 10.1007/BF01174749
- Byungheup Jun and Jungyun Lee, Caliber number of real quadratic fields, J. Number Theory 130 (2010), no. 11, 2586–2595. MR 2678863, DOI 10.1016/j.jnt.2010.05.010
- Byungheup Jun and Jungyun Lee, The behavior of Hecke $L$-functions of real quadratic fields at $s=0$, Algebra Number Theory 5 (2011), no. 8, 1001–1026. MR 2948469, DOI 10.2140/ant.2011.5.1001
- Byungheup Jun and Jungyun Lee, Polynomial behavior of special values of partial zeta functions of real quadratic fields at $s=0$, Selecta Math. (N.S.) 19 (2013), no. 1, 97–123. MR 3022753, DOI 10.1007/s00029-012-0095-1
- Byungheup Jun and Jungyun Lee, Equidistribution of generalized Dedekind sums and exponential sums, J. Number Theory 137 (2014), 67–92. MR 3157779, DOI 10.1016/j.jnt.2013.10.020
- Yael Karshon, Shlomo Sternberg, and Jonathan Weitsman, Euler-Maclaurin with remainder for a simple integral polytope, Duke Math. J. 130 (2005), no. 3, 401–434. MR 2184566, DOI 10.1215/S0012-7094-05-13031-9
- Yael Karshon, Shlomo Sternberg, and Jonathan Weitsman, Exact Euler-Maclaurin formulas for simple lattice polytopes, Adv. in Appl. Math. 39 (2007), no. 1, 1–50. MR 2319562, DOI 10.1016/j.aam.2006.04.003
- Nicholas M. Katz, Another look at $p$-adic $L$-functions for totally real fields, Math. Ann. 255 (1981), no. 1, 33–43. MR 611271, DOI 10.1007/BF01450554
- Jungyun Lee, The complete determination of wide Richaud-Degert types which are not 5 modulo 8 with class number one, Acta Arith. 140 (2009), no. 1, 1–29. MR 2557850, DOI 10.4064/aa140-1-1
- Jungyun Lee, The complete determination of narrow Richaud-Degert type which is not 5 modulo 8 with class number two, J. Number Theory 129 (2009), no. 3, 604–620. MR 2488592, DOI 10.1016/j.jnt.2008.09.011
- Yuri I. Manin and Matilde Marcolli, Modular shadows and the Lévy-Mellin $\infty$-adic transform, Modular forms on Schiermonnikoog, Cambridge Univ. Press, Cambridge, 2008, pp. 189–238. MR 2512362, DOI 10.1017/CBO9780511543371.012
- Curt Meyer, Die Berechnung der Klassenzahl Abelscher Körper über quadratischen Zahlkörpern, Akademie-Verlag, Berlin, 1957 (German). MR 0088510
- James E. Pommersheim, Barvinok’s algorithm and the Todd class of a toric variety, J. Pure Appl. Algebra 117/118 (1997), 519–533. Algorithms for algebra (Eindhoven, 1996). MR 1457853, DOI 10.1016/S0022-4049(97)00025-X
- Takuro Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 2, 393–417. MR 427231
- Carl Ludwig Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1969 (1969), 87–102 (German). MR 252349
- H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1–27. MR 222050
- H. M. Stark, A transcendence theorem for class-number problems. II, Ann. of Math. (2) 96 (1972), 174–209. MR 309878, DOI 10.2307/1970897
- Shuji Yamamoto, On Kronecker limit formulas for real quadratic fields, J. Number Theory 128 (2008), no. 2, 426–450. MR 2380329, DOI 10.1016/j.jnt.2007.05.010
- Hideo Yokoi, The fundamental unit and class number one problem of real quadratic fields with prime discriminant, Nagoya Math. J. 120 (1990), 51–59. MR 1086568, DOI 10.1017/S0027763000003238
- Maria Vlasenko and Don Zagier, Higher Kronecker “limit” formulas for real quadratic fields, J. Reine Angew. Math. 679 (2013), 23–64. MR 3065153, DOI 10.1515/crelle.2012.022
- Don Zagier, A Kronecker limit formula for real quadratic fields, Math. Ann. 213 (1975), 153–184. MR 366877, DOI 10.1007/BF01343950
- D. Zagier, Valeurs des fonctions zêta des corps quadratiques réels aux entiers négatifs, Journées Arithmétiques de Caen (Univ. Caen, Caen, 1976) Astérisque No. 41–42, Soc. Math. France, Paris, 1977, pp. 135–151 (French). MR 0441925
Additional Information
- Byungheup Jun
- Affiliation: Department of Mathematics, Yonsei University, Yonsei-ro 50, Seodaemun-gu, Seoul 120-749, Korea
- Address at time of publication: Department of Mathematical Sciences, UNIST, UNIST-gil 50, Ulsan 689-798, Korea
- Email: bhjun@unist.ac.kr
- Jungyun Lee
- Affiliation: Department of Mathematics, Ewha Womans University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul 120-750, Korea
- Email: lee9311@ewha.ac.kr
- Received by editor(s): October 29, 2013
- Received by editor(s) in revised form: February 2, 2015
- Published electronically: February 12, 2016
- Additional Notes: The work of the first author was supported by NRF grant (NRF-2015R1D1A1A09059083) and Samsung Science Technology Foundation grant BA1301-03
The work of the second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (2011-0023688) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 7935-7964
- MSC (2010): Primary 11E41, 11M41, 11R11, 11R29, 11R80
- DOI: https://doi.org/10.1090/tran/6679
- MathSciNet review: 3546789