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

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,

PDF

___

[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

Communications in Advanced Mathematical Sciences-Cover

ISSN: 2651-4001
Yayın Aralığı: Yılda 4 Sayı
Başlangıç: 2018
Yayıncı: Emrah Evren KARA

Arşiv

Sayıdaki Diğer Makaleler

Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator

Yaşar ÇAKMAK

Ces\`{a}ro-Type Operator on the Dirichlet Space of the Upper Half Plane

Effie OYUGİ, Job BONYO, David AMBOGO

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

Georges GRAS

Dynamics and Stability of $\Xi$-Hilfer Fractional Fuzzy Differential Equations with Impulses

Ravichandran VIVEK, Kangarajan K., Dvivek VİVEK, Elsayed ELSAYED

The Fractional Integral Inequalities Involving Kober and Saigo–Maeda Operators

Kalpana RAJPUT, Rajshree MISHRA, Deepak Kumar JAİN, Altaf Ahmad BHAT, Farooq AHMAD