Elementary 3-descent with a 3-isogeny
Elliptic Carmichael numbers and elliptic Korselt criteria
Elliptic curves and class fields of real quadratic fields: Algorithms and evidence.
Elliptic curves and continued fractions.
Elliptic curves and special values of L-series.
Elliptic curves associated with simplest quartic fields
We are studying the infinite family of elliptic curves associated with simplest cubic fields. If the rank of such curves is 1, we determine the whole structure of the Mordell-Weil group and find all integral points on the original model of the curve. Note however, that we are not able to find them on the Weierstrass model if the parameter is even. We have also obtained similar results for an infinite subfamily of curves of rank 2. To our knowledge, this is the first time that so much information...
Elliptic curves of high rank with nontrivial torsion group over .
Elliptic curves over function fields with a large set of integral points
We construct isotrivial and non-isotrivial elliptic curves over with an arbitrarily large set of separable integral points. As an application of this construction, we prove that there are isotrivial log-general type varieties over with a Zariski dense set of separable integral points. This provides a counterexample to a natural translation of the Lang-Vojta conjecture to the function field setting. We also show that our main result provides examples of elliptic curves with an explicit and arbitrarily...
Elliptic curves with and counterexamples to local-global divisibility by 9
We give a family of elliptic curves, depending on two nonzero rational parameters and , such that the following statement holds: let be an elliptic curve and let be its 3-torsion subgroup. This group verifies if and only if belongs to .Furthermore, we consider the problem of the local-global divisibility by 9 for points of elliptic curves. The number 9 is one of the few exceptional powers of primes, for which an answer to the local-global divisibility is unknown in the case of such...
Elliptic curves with all quadratic twists of positive rank
Elliptic curves with complex multiplication and Galois module structure.
Elliptic Curves With Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer.
Elliptic curves with non-trivial 2-adic Iwasawa μ-invariant
Equivalences between elliptic curves and real quadratic congruence function fields
In 1994, the well-known Diffie-Hellman key exchange protocol was for the first time implemented in a non-group based setting. Here, the underlying key space was the set of reduced principal ideals of a real quadratic number field. This set does not possess a group structure, but instead exhibits a so-called infrastructure. More recently, the scheme was extended to real quadratic congruence function fields, whose set of reduced principal ideals has a similar infrastructure. As always, the security...
Erratum: "On the number of prime divisors of the order of elliptic curves modulo p" (Acta Arith. 117 (2005), 341-352)
Étude de la courbe elliptique et monogénéité de certains anneaux d’entiers
Euler systems obtained from congruences between Hilbert modular forms
Euler's concordant forms
Existence of nonelliptic mod Galois representations for every .
Experimental study of the rank of families of elliptic curves over . (Étude expérimentale du rang de familles de courbes elliptiques sur .)