Terms in elliptic divisibility sequences divisible by their indices
The arithmetic of curves defined by iteration
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 Arithmetic of Elliptic Curves.
The arithmetic of Fermat curves.
The average rank of elliptic curves I. With Appendix "The Explicit formula for elliptic curves over function fields" by Oisín McGuinness.
The boundary of the Eisenstein symbol.
The boundary of the Eisenstein symbol (Erratum).
The Cohen-Lenstra heuristics, moments and -ranks of some groups
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 -ranks of Selmer groups...
The D(1)-extensions of D(-1)-triples 1,2,c and integer points on the attached elliptic curves
The Diophantine Equation X³ = u+v over Real Quadratic Fields
Let k be a real quadratic field and let and be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v (, ) is developed.
The equation xyz=x+y+z=1 in integers of a quartic field
The existence of infinitely many supersingular primes for every elliptic curve over Q.
The integral points on elliptic curves
Let be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if and both and are odd primes, then the general elliptic curve has only the integral point . By this result we can get that the above elliptic curve has only the trivial integral point for etc. Thus it can be seen that the elliptic curve really is an unusual elliptic curve which has large integral points.
The least prime which does not split completely.
The level of distribution of the special values of L-functions
The Ljunggren equation revisited
We study the Ljunggren equation Y² + 1 = 2X⁴ using the "multiplication by 2" method of Chabauty.
The "main conjectures" of Iwasawa theory for imaginary quadratic fields.
The modular degree and the congruence number of a weight 2 cusp form
The Mordell-Weil bases for the elliptic curve
Let be an elliptic curve over of the form , where is an integer. In this paper we prove that the two points and on can be extended to a basis for under certain conditions described explicitly.
The number of elliptic curves over Q with conductor N.