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2024 | OriginalPaper | Chapter

Connections of Class Numbers to the Group Structure of Generalized Pythagorean Triples

Authors : Thomas Jaklitsch, Thomas C. Martinez, Steven J. Miller, Sagnik Mukherjee

Published in: Class Groups of Number Fields and Related Topics

Publisher: Springer Nature Singapore

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Abstract

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves, for the first we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently Yekutieli discussed a connection between these two problems, and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. In [5] we generalized these methods and results to Pell’s equation. We find a similar group structure and count on the number of solutions for a given z to \(x^2 + Dy^2 = z^2\) when D is 1 or 2 modulo 4 and the class group of \(\mathbb {Q}[\sqrt{-D}]\) is a free \(\ensuremath {\mathbb {Z}}_2\) module, which always happens if the class number is at most 2. In this paper we discuss the main results of [5] using some concrete examples in the case of \(D=105\).

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previous chapter On the Mod p Iwasawa Theory for Elliptic Curves

next chapter The Partition Function Modulo 4

Mazur [6, 7] proved that there are only 15 possibilities for \(\mathbb {T}\): \(\ensuremath {\mathbb {Z}}/N\ensuremath {\mathbb {Z}}\) for \(N \in \{1,\dots , 10, 12\}\) and \(\ensuremath {\mathbb {Z}}_2 \oplus \ensuremath {\mathbb {Z}}/2N\ensuremath {\mathbb {Z}}\) with \(N \in \{1, \dots , 4\}\). The possible values of the rank are still a mystery. In 1938, Billing found an elliptic curve with rank 3. The largest known rank increased slowly over the years, with the current record due to Elkies in 2006, and is rank at least 28 (see [3] for a more comprehensive historical data on elliptic curve records). While originally it was conjectured that the rank can be arbitrarily large, now some models (see [12]) suggest that there may only be finitely many curves with rank exceeding 28.

We would have \(2a^2 = c^2\) and thus \(\sqrt{2} = c/a \in \mathbb {Q}\).

Note that given any solution \((x_0,y_0,z_0)\) to (1.6) we can generate infinitely many solutions out of this particular solutions, for example \((dx_0,dy_0,dz_0)\), \((ey_0,ex_0,ez_0)\) for any arbitrary integer d and e. Thus by introducing this notion of normalized solution we are just considering the non-trivial generators of the solution space of (1.6).

Note that such \(x_0\) and \(y_0\) always exist for every prime p satisfying \(p\equiv 1\)(mod 4). In fact this choice of \(x_0\) and \(y_0\) is unique.

Note the similarity of this result with the Mordell-Weil theorem for elliptic curves.

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Title: Connections of Class Numbers to the Group Structure of Generalized Pythagorean Triples
Authors: Thomas Jaklitsch
Thomas C. Martinez
Steven J. Miller
Sagnik Mukherjee
Publisher: Springer Nature Singapore
Book: Class Groups of Number Fields and Related Topics
Print ISBN: 978-981-9769-10-0

Electronic ISBN: 978-981-9769-11-7

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-981-97-6911-7_3

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