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On a conjecture of Watkins

Neil Dummigan (2006)

Journal de Théorie des Nombres de Bordeaux

Watkins has conjectured that if R is the rank of the group of rational points of an elliptic curve E over the rationals, then 2 R divides the modular parametrisation degree. We show, for a certain class of E , chosen to make things as easy as possible, that this divisibility would follow from the statement that a certain 2 -adic deformation ring is isomorphic to a certain Hecke ring, and is a complete intersection. However, we show also that the method developed by Taylor, Wiles and others, to prove...

On a family of elliptic curves of rank at least 2

Kalyan Chakraborty, Richa Sharma (2022)

Czechoslovak Mathematical Journal

Let C m : y 2 = x 3 - m 2 x + p 2 q 2 be a family of elliptic curves over , where m is a positive integer and p , q are distinct odd primes. We study the torsion part and the rank of C m ( ) . More specifically, we prove that the torsion subgroup of C m ( ) is trivial and the -rank of this family is at least 2, whenever m ¬ 0 ( mod 3 ) , m ¬ 0 ( mod 4 ) and m 2 ( mod 64 ) with neither p nor q dividing m .

On arithmetic progressions on Edwards curves

Enrique González-Jiménez (2015)

Acta Arithmetica

Let m > 0 and a,q ∈ ℚ. Denote by m ( a , q ) the set of rational numbers d such that a, a + q, ..., a + (m-1)q form an arithmetic progression in the Edwards curve E d : x ² + y ² = 1 + d x ² y ² . We study the set m ( a , q ) and we parametrize it by the rational points of an algebraic curve.

On congruent primes and class numbers of imaginary quadratic fields

Nils Bruin, Brett Hemenway (2013)

Acta Arithmetica

We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate-Shafarevich groups of the associated elliptic curves. We also present a related criterion for primes p such that 16 divides the class number of the imaginary quadratic field ℚ(√-p)....

On elementary equivalence, isomorphism and isogeny

Pete L. Clark (2006)

Journal de Théorie des Nombres de Bordeaux

Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These results are applied to deduce new instances of “elementary equivalence implies isomorphism”: for all genus zero...

On elliptic curves and random matrix theory

Mark Watkins (2008)

Journal de Théorie des Nombres de Bordeaux

Rubinstein has produced a substantial amount of data about the even parity quadratic twists of various elliptic curves, and compared the results to predictions from random matrix theory. We use the method of Heegner points to obtain a comparable (yet smaller) amount of data for the case of odd parity. We again see that at least one of the principal predictions of random matrix theory is well-evidenced by the data.

On families of 9-congruent elliptic curves

Tom Fisher (2015)

Acta Arithmetica

We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over ℚ, i.e. pairs of non-isogenous elliptic curves over ℚ whose 9-torsion subgroups are isomorphic as Galois modules.

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