Class numbers and representations by ternary quadratic forms with congruence conditions
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- Math. Comp. 91 (2022), 295-329 Request permission
Abstract:
In this paper, we are interested in the interplay between integral ternary quadratic forms and class numbers. This is partially motivated by a question of Petersson.References
- E. T. Bell, The class number relations implicit in the Disquisitiones Arithmeticae, Bull. Amer. Math. Soc. 30 (1924), no. 5-6, 236–238. MR 1560883, DOI 10.1090/S0002-9904-1924-03891-9
- Wai Kiu Chan and Byeong-Kweon Oh, Representations of integral quadratic polynomials, Diophantine methods, lattices, and arithmetic theory of quadratic forms, Contemp. Math., vol. 587, Amer. Math. Soc., Providence, RI, 2013, pp. 31–46. MR 3074801, DOI 10.1090/conm/587/11684
- Henri Cohen, Sums involving the values at negative integers of $L$-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271–285. MR 382192, DOI 10.1007/BF01436180
- Ilse Ebel, Analytische Bestimmung der Darstellungsanzahlen natürlicher Zahlen durch spezielle ternäre quadratische Formen mit Kongruenzbedingungen, Math. Z. 64 (1956), 217–228 (German). MR 76795, DOI 10.1007/BF01166569
- F. Hirzebruch and D. Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. Math. 36 (1976), 57–113. MR 453649, DOI 10.1007/BF01390005
- A. S. Mosunov and M. J. Jacobson Jr., Unconditional class group tabulation of imaginary quadratic fields to $|\Delta |<2^{40}$, Math. Comp. 85 (2016), no. 300, 1983–2009. MR 3471116, DOI 10.1090/mcom3050
- Michael J. Jacobson Jr., Shantha Ramachandran, and Hugh C. Williams, Numerical results on class groups of imaginary quadratic fields, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 87–101. MR 2282917, DOI 10.1007/11792086_{7}
- W. Jagy, I. Kaplansky, and A. Schiemann, There are 918 regular ternary forms, Mathematika 44 (1997), 332–341.
- Burton W. Jones, The Arithmetic Theory of Quadratic Forms, Carcus Monograph Series, no. 10, Mathematical Association of America, Buffalo, N.Y., 1950. MR 0037321, DOI 10.5948/UPO9781614440109
- W. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285–315.
- Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
- Hans Petersson, Über die Berechnung der Skalarprodukte ganzer Modulformen, Comment. Math. Helv. 22 (1949), 168–199 (German). MR 28426, DOI 10.1007/BF02568055
- Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI 10.2307/1970831
- Goro Shimura, Inhomogeneous quadratic forms and triangular numbers, Amer. J. Math. 126 (2004), no. 1, 191–214. MR 2033567, DOI 10.1353/ajm.2004.0007
- Liang Sun, The growth of class numbers of quadratic Diophantine equations, J. Number Theory 183 (2018), 133–145. MR 3715231, DOI 10.1016/j.jnt.2017.07.009
- F. van der Blij, On the theory of quadratic forms, Ann. of Math. (2) 50 (1949), 875–883. MR 31001, DOI 10.2307/1969584
- Don Zagier, Nombres de classes et formes modulaires de poids $3/2$, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), no. 21, Ai, A883–A886 (French, with English summary). MR 429750
Additional Information
- Kathrin Bringmann
- Affiliation: Department of Mathematics and Computer Science, University of Cologne, Weyertal 86–90, 50931 Cologne, Germany
- MR Author ID: 774752
- Email: kbringma@math.uni-koeln.de
- Ben Kane
- Affiliation: Department of Mathematics, University of Hong Kong, Pokfulam, Hong Kong
- MR Author ID: 789505
- ORCID: 0000-0003-4074-7662
- Email: bkane@hku.hk
- Received by editor(s): March 7, 2020
- Received by editor(s) in revised form: January 28, 2021
- Published electronically: September 7, 2021
- Additional Notes: The first author was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101001179). The research of the second author was supported by grants from the Research Grants Council of the Hong Kong SAR, China (project numbers HKU 17316416, 17301317 and 17303618)
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 91 (2022), 295-329
- MSC (2020): Primary 11E20, 11Y99, 11E25, 11E41, 11F37
- DOI: https://doi.org/10.1090/mcom/3648
- MathSciNet review: 4350541