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The arithmetic of curves defined by iteration

Wade Hindes (2015)

Acta Arithmetica

We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...

The Cohen-Lenstra heuristics, moments and p j -ranks of some groups

Christophe Delaunay, Frédéric Jouhet (2014)

Acta Arithmetica

This article deals with the coherence of the model given by the Cohen-Lenstra heuristic philosophy for class groups and also for their generalizations to Tate-Shafarevich groups. More precisely, our first goal is to extend a previous result due to É. Fouvry and J. Klüners which proves that a conjecture provided by the Cohen-Lenstra philosophy implies another such conjecture. As a consequence of our work, we can deduce, for example, a conjecture for the probability laws of p j -ranks of Selmer groups...

The Diophantine Equation X³ = u+v over Real Quadratic Fields

Takaaki Kagawa (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

Let k be a real quadratic field and let k and k × be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v ( X ∈ k , u , v ∈ k × ) is developed.

The integral points on elliptic curves y 2 = x 3 + ( 36 n 2 - 9 ) x - 2 ( 36 n 2 - 5 )

Hai Yang, Ruiqin Fu (2013)

Czechoslovak Mathematical Journal

Let n be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if n > 1 and both 6 n 2 - 1 and 12 n 2 + 1 are odd primes, then the general elliptic curve y 2 = x 3 + ( 36 n 2 - 9 ) x - 2 ( 36 n 2 - 5 ) has only the integral point ( x , y ) = ( 2 , 0 ) . By this result we can get that the above elliptic curve has only the trivial integral point for n = 3 , 13 , 17 etc. Thus it can be seen that the elliptic curve y 2 = x 3 + 27 x - 62 really is an unusual elliptic curve which has large integral points.

The Ljunggren equation revisited

Konstantinos A. Draziotis (2007)

Colloquium Mathematicae

We study the Ljunggren equation Y² + 1 = 2X⁴ using the "multiplication by 2" method of Chabauty.

The Mordell-Weil bases for the elliptic curve y 2 = x 3 - m 2 x + m 2

Sudhansu Sekhar Rout, Abhishek Juyal (2021)

Czechoslovak Mathematical Journal

Let D m be an elliptic curve over ℚ of the form y 2 = x 3 - m 2 x + m 2 , where m is an integer. In this paper we prove that the two points P - 1 = ( - m , m ) and P 0 = ( 0 , m ) on D m can be extended to a basis for D m ( ℚ ) under certain conditions described explicitly.

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