We present a density result for the norm of the fundamental unit in a real quadratic order that follows from an equidistribution assumption for the infinite Frobenius elements in the class groups of these orders.
Keywords: real quadratic fields, quadratic units, Pell equation
@article{JTNB_1995__7_1_121_0,
author = {Peter Stevenhagen},
title = {Frobenius distributions for real quadratic orders},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {121--132},
year = {1995},
publisher = {Universit\'e Bordeaux I},
volume = {7},
number = {1},
zbl = {0847.11010},
mrnumber = {1413571},
language = {en},
url = {https://jtnb.centre-mersenne.org/item/JTNB_1995__7_1_121_0/}
}
TY - JOUR AU - Peter Stevenhagen TI - Frobenius distributions for real quadratic orders JO - Journal de théorie des nombres de Bordeaux PY - 1995 SP - 121 EP - 132 VL - 7 IS - 1 PB - Université Bordeaux I UR - https://jtnb.centre-mersenne.org/item/JTNB_1995__7_1_121_0/ LA - en ID - JTNB_1995__7_1_121_0 ER -
Peter Stevenhagen. Frobenius distributions for real quadratic orders. Journal de théorie des nombres de Bordeaux, Tome 7 (1995) no. 1, pp. 121-132. https://jtnb.centre-mersenne.org/item/JTNB_1995__7_1_121_0/
[1] and , A numerical investigation of the Diophantine equation x2 - dy2 = -1, Proc. 3rd Southeastern Conf. on Combinatorics, Graph Theory and Computing, 1972, pp. 37-52. | Zbl | MR
[2] and , Density computations for real quadratic units, Math. Comp., to appear (1995). | Zbl | MR
[3] and , Jr., Heuristics on class groups of number fields, Number Theory Noordwijkerhout 1983 (H. Jager, ed.), Springer LNM 1068, 1984. | Zbl
[4] and , An introduction to the theory of numbers, Oxford University Press, 1938. | JFM
[5] , Über die Lösbarkeit der Gleichung x2 - Dy2 = -1, Arkiv för Mat., Astr., o. Fysik 23 (1932), no. B/6, 1-5. | JFM
[6] , Über die Pellsche Gleichung t2 - du2 = -1, J. reine angew. Math. 173 (1935), 193-221. | Zbl | JFM
[7] , Über einige Mittelwertfragen im quadratischen Zahlkörper, J. reine angew. Math. 174 (1936), 131-148. | Zbl
[8] , Über die Anzahl der als Summe von zwei Quadraten darstellbaren und in einer primen Restklasse gelegenen Zahlen unterhalb einer positiven Schranke. II, J. reine angew. Math. 217 (1965), 200-216. | Zbl | MR
[9] and , Some computational results on a problem of Eisenstein, Théorie des Nombres - Number Theory (J. W. M. de Koninck and C. Levesque, eds. ), de Gruyter, 1992, pp. 869-886. | Zbl | MR
[10] , On the 2-power divisibility of certain quadratic class numbers, J. of Number Theory 43 (1993), no. (1), 1-19. | Zbl | MR
[11] , The number of real quadratic fields having units of negative norm, Exp. Math. 2 (1993), no. (2), 121-136. | Zbl | MR | EuDML
[12] , On a problem of Eisenstein, Acta Arith., (to appear, 1995). | Zbl | MR