Unlimited Lists of Quadratic Integers of Given Norm - Application to Some Arithmetic Properties

Year 2023, Volume: 6 Issue: 3, 148 - 176, 17.09.2023

Georges Gras

https://doi.org/10.33434/cams.1327372

Abstract

We use the polynomials $m_s(t) = t^2 - 4 s$, $s \in \{-1, 1\}$, in an elementary process giving unlimited lists of {\it fundamental units of norm $s$}, of real quadratic fields, with ascending order of the discriminates. As $t$ grows from $1$ to an upper bound $\textbf{B}$, for each {\it first occurrence} of a square-free integer $M \geq 2$, in the factorization $m_s(t) =: M r^2$, the unit $\frac{1}{2} \big(t + r \sqrt{M}\big)$ is the fundamental unit of norm $s$ of $\mathbb{Q}(\sqrt M)$, even if $r >1$ (Theorem 4.2). Using $m_{s\nu}(t) = t^2 - 4 s \nu$, $\nu \geq 2$, the algorithm gives unlimited lists of {\it fundamental integers of norm $s\nu$} (Theorem~4.6). We deduce, for any prime $p>2$, unlimited lists of {\it non $p$-rational} quadratic fields (Theorems 6.3, 6.4, 6.5) and lists of degree $p-1$ imaginary fields with {\it non-trivial $p$-class group} (Theorems 7.1, 7.2). All PARI programs are given.

Keywords

Fundamental units, Norm equations, PARI programs, $p$-class numbers, $p$-rationality, Real quadratic fields

References

• [1] The PARI Group, PARI/GP, version 2.9.0, Universit´e de Bordeaux (2016). http://pari.math.u-bordeaux.fr/ 148
• [2] J. Mc Laughlin, Polynomial Solutions of Pell’s equation and fundamental units in real quadratic fields, Jour. London Math. Soc., (2) 67(1) (2003), 16–28. https://doi.org/10.1112/S002461070200371X 149, 150, 161
• [3] J. Mc Laughlin, P. Zimmer, Some more Long continued fractions, I, Acta Arithmetica 127(4) (2007), 365–389. http://eudml.org/doc/278353
• [4] M. B. Nathanson, Polynomial Pell’s equations, Proc. Amer. Math. Soc., 56(1) (1976), 89–92. https://doi.org/10.2307/2041581
• [5] A. M. S. Ramasamy, Polynomial solutions for the Pell’s equation,Indian J. Pure Appl. Math., 25 (1994), 577–581. https://www.academia.edu/33430848/
• [6] H. Sankari, A. Abdo, On Polynomial solutions of Pell’s equation, Hindawi Journal of Mathematics 2021 (2021), 1–4. https://doi.org/10.1155/2021/5379284 149
• [7] H. Yokoi, On real quadratic fields containing units with norm 􀀀1, Nagoya Math. J. 33 (1968), 139–152. https://doi.org/10.1017/S0027763000012939 149, 153
• [8] J.B. Friedlander, H. Iwaniec, Square-free values of quadratic polynomials, Proc. Edinburgh Math. Soc., 53(2) (2010), 385–392. https://doi.org/10.1017/S0013091508000989 151
• [9] Z. Rudnick, Square-free values of quadratic polynomials, Lecture Notes, (2015). http://www.math.tau.ac.il/rudnick/courses/sieves2015/squarefrees.pdf 151
• [10] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98(1) (1976), 263–284. https://doi.org/10.2307/2373625 152, 164, 165
• [11] J-F. Jaulent, Classes logarithmiques des corps de nombres, J. Th´eorie des Nombres de Bordeaux 6 (1994), 301–325. https://doi.org/10.5802/jtnb.117 152
• [12] J.-F. Jaulent, Note sur la conjecture de Greenberg, J. Ramanujan Math. Soc., 34 (2019), 59–80. http://www.mathjournals.org/jrms/2019-034-001/2019-034-001-005.html 152, 164
• [13] K. Belabas, J.-F. Jaulent, The logarithmic class group package in PARI/GP, Pub. Math. Besanc¸on, Alg`ebre et theorie des nombres 2016, 5–18. https://doi.org/10.5802/pmb.o-1 152, 164
• [14] G. Gras, New characterization of the norm of the fundamental unit of Q( p M), 2023, arxiv:2206.13931 [math NT]. https://arxiv.org/pdf/2206.13931.pdf. 157, 158
• [15] G. Gras, Practice of the incomplete p-Ramification over a number Field – History of abelian p-Ramification, Commun. Adv. Math. Sci., 2(4) (2019), 251–280. https://doi.org/10.33434/cams.573729 163, 164, 165
• [16] J-F. Jaulent, S-classes infinit´esimales d’un corps de nombres alg´ebriques, Ann. Inst. Fourier 34(2) (1984), 1–27. https://doi.org/10.5802/aif.960 163
• [17] J-F. Jaulent, L’arithm´etique des `-extensions Th`ese de doctorat d’Etat, Pub. Math. Besanc¸on (Th´eorie des Nombres) (1986), 1–349. http://doi.org/10.5802/pmb.a-42 163, 164
• [18] T. Nguyen Quang Do, Sur la Zp-torsion de certains modules galoisiens, Ann. Inst. Fourier, 36(2) (1986), 27–46. https://doi.org/10.5802/aif.1045 163, 164.
• [19] A. Movahhedi, Sur les p-extensions des corps p-rationnels, Th`ese, Univ. Paris VII, 1988. http://www.unilim.fr/pages perso/chazad.movahhedi/These 1988.pdf Sur les p-extensions des corps p-rationnels, Math. Nachr. 149 (1990), 163–176. http://onlinelibrary.wiley.com/doi/10.1002/mana.19901490113/full 163.
• [20] A. Movahhedi et T. Nguyen Quang Do, Sur l’arithm´etique des corps de nombres p-rationnels, S´eminaire de Theorie des Nombres, Paris 1987–88, Progress in Math., 81, 1990, 155–200. https://link.springer.com/chapter/10.1007 2F978-1-4612-3460-9 9 163
• [21] G. Gras, Class Field Theory: From Theory to Practice, corr. 2nd ed. Springer Monographs in Mathematics, Springer, xiii+507 pages (2005). 163
• [22] G. Gras, The p-adic Kummer–Leopoldt Constant: Normalized p-adic Regulator, Int. J. Number Theory, 14(2) (2018), 329–337. https://doi.org/10.1142/S1793042118500203 163
• [23] G. Gras, Heuristics and conjectures in the direction of a p-adic Brauer–Siegel theorem, Math. Comp. 88(318) (2019), 1929–1965. https://doi.org/10.1090/mcom/3395 163, 164
• [24] J. Assim, Z. Bouazzaoui, Half-integral weight modular forms and real quadratic p-rational fields, Funct. Approx. Comment. Math., 63(2) (2020), 201–213. https://doi.org/10.7169/facm/1851 163
• [25] Y. Benmerieme, Les corps multi-quadratiques p-rationnels, Th`ese (2021LIMO0100), Universit´e de Limoges (2021). http://aurore.unilim.fr/ori-oai-search/notice/view/2021LIMO0100 163, 164
• [26] G. Boeckle, D.A. Guiraud, S. Kalyanswamy, C. Khare, Wieferich Primes and a mod p Leopoldt Conjecture (2018), arXiv.1805.00131 [math NT]. https://doi.org/10.48550/arXiv.1805.00131 163
• [27] Y. Benmerieme, A. Movahhedi, Multi-quadratic p-rational number fields, J. Pure Appl. Algebra, 225(9) (2021), 1–17. https://doi.org/10.1016/j.jpaa.2020.106657 163, 164
• [28] Z. Bouazzaoui, Fibonacci numbers and real quadratic p-rational fields, Period. Math. Hungar., 81(1) (2020), 123–133. https://doi.org/10.1007/s10998-020-00320-7 163, 164
• [29] R. Barbulescu, J. Ray, Numerical verification of the Cohen–Lenstra–Martinet heuristics and of Greenberg’s p-rationality conjecture J. Th´eorie des Nombres de Bordeaux 32(1) (2020), 159–177. https://doi.org/10.5802/jtnb.1115 163, 164
• [30] D. Byeon, Indivisibility of special values of Dedekind zeta functions of real quadratic fields, Acta Arithmetica, 109(3) (2003), 231–235. https://doi.org/10.4064/AA109-3-3 163
• [31] G. Gras, Les q-r´egulateurs locaux d’un nombre alg´ebrique : Conjectures p-adiques, Canadian J. Math., 68(3) (2016), 571–624.http://doi.org/10.4153/CJM-2015-026-3 163, 164, 165 English translation: arXiv.1701.02618 [math NT] https://doi.org/10.48550/arXiv.1701.02618
• [32] J. Koperecz, Triquadratic p-rational fields, J. Number Theory, 242 (2023), 402–408. https://doi.org/10.1016/j.jnt.2022.04.011 163, 164
• [33] C. Maire, M. Rougnant, Composantes isotypiques de pro-p-extensions de corps de nombres et p-rationalit´e, Publ. Math. Debrecen, 94(1/2) (2019), 123–155. https://doi.org/10.5486/PMD.2019.8281 163, 164
• [34] C. Maire, M. Rougnant, A note on p-rational fields and the abc-conjecture, Proc. Amer. Math. Soc., 148(8) (2020), 3263–3271. https://doi.org/10.1090/proc/14983 163
• [35] J. Chattopadhyay, H. Laxmi, A. Saikia, On the p-rationality of consecutive quadratic fields, J. Number Theory, 248 (2023), 14–26. https://doi.org/10.1016/j.jnt.2023.01.001 163
• [36] R. Greenberg, Galois representation with open image, Ann. Math. Qu´e. 40(1) (2016), 83–119. https://doi.org/10.1007/s40316-015-0050-6 164.
• [37] G. Gras, J.-F. Jaulent, Note on 2-rational fields, J. Number Theory, 129(2) (2009), 495–498. https://doi.org/10.1016/j.jnt.2008.06.012 164
• [38] G. Gras, On p-rationality of number fields. Applications–PARI/GP programs, Pub. Math. Besancon (Th´eorie des Nombres), Ann´ees 2018/2019. https://doi.org/10.5802/pmb.35 164, 166
• [39] F. Pitoun, F. Varescon, Computing the torsion of the p-ramified module of a number field, Math. Comp., 84(291) (2015), 371–383. https://doi.org/10.1090/S0025-5718-2014-02838-X 164
• [40] G. Gras, Algorithmic complexity of Greenberg’s conjecture, Arch. Math., 117 (2021), 277–289. https://doi.org/10.1007/s00013-021-01618-9 164, 165
• [41] G. Gras, Tate–Shafarevich groups in the cyclotomicbZ-extension and Weber’s class number problem, J. Number Theory, 228 (2021), 219–252. https://doi.org/10.1016/j.jnt.2021.04.019 165
• [42] C. Maire, Sur la dimension cohomologique des pro-p-extensions des corps de nombres, J. Th´eor. Nombres Bordeaux, 17(2) (2005), 575–606. https://doi.org/10.5802/jtnb.509 165.
• [43] G. Gras, Sur la norme du groupe des unit´es d’extensions quadratiques relatives, Acta Arith., 61 (1992), 307–317. https://doi.org/10.4064/aa-61-4-307-317 169