On a special case of Watkins' conjecture

@article{Kazalicki2016OnAS,
  title={On a special case of Watkins' conjecture},
  author={Matija Kazalicki and Daniela Kohen},
  journal={arXiv: Number Theory},
  year={2016}
}
Watkins' conjecture asserts that for a rational elliptic curve $E$ the degree of the modular parametrization is divisible by $2^r$, where $r$ is the rank of $E$. In this paper we prove that if the modular degree is odd then $E$ has rank $0$. Moreover, we prove that the conjecture holds for all rank two rational elliptic curves of prime conductor and positive discriminant. 
3 Citations

Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor

. Watkins’ conjecture asserts that the rank of an elliptic curve is upper bounded by the 2-adic valuation of its modular degree. We show that this conjecture is satisfied when E is any quadratic twist

A conjecture of Watkins for quadratic twists

Watkins conjectured that for an elliptic curve E over Q of Mordell-Weil rank r, the modular degree of E is divisible by 2. If E has non-trivial rational 2-torsion, we prove the conjecture for all the

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References

SHOWING 1-9 OF 9 REFERENCES

Elliptic curves of odd modular degree

The modular degree mE of an elliptic curve E/Q is the minimal degree of any surjective morphism X0(N) → E, where N is the conductor of E. We give a necessary set of criteria for mE to be odd. In the

Modular Abelian Varieties of Odd Modular Degree

In this paper, we will study modular Abelian varieties with odd congruence numbers by examining the cuspidal subgroup of $J_0(N)$. We will show that the conductor of such Abelian varieties must be of

Supersingular zeros of divisor polynomials of elliptic curves of prime conductor

For a prime number p, we study the zeros modulo p of divisor polynomials of rational elliptic curves E of conductor p. Ono (CBMS regional conference series in mathematics, 2003, vol 102, p. 118) made

Modular Parametrizations of Neumann-Setzer Elliptic Curves

Suppose $p$ is a prime of the form $u^2+64$ for some integer $u$, which we take to be 3 mod 4. Then there are two Neumann--Setzer elliptic curves $E_0$ and $E_1$ of prime conductor $p$, and both have

Computing the Modular Degree of an Elliptic Curve

  • M. Watkins
  • Mathematics, Computer Science
    Exp. Math.
  • 2002
A new method is presented, based upon the computation of a special value of the symmetric square L-function of the elliptic curve, which is sufficiently fast to allow large-scale experiments to be done and show two interesting phenomena on the arithmetic character of the modular degree.

Rational Points on Modular Elliptic Curves

Elliptic curves Modular forms Heegner points on $X_0(N)$ Heegner points on Shimura curves Rigid analytic modular forms Rigid analytic modular parametrisations Totally real fields ATR points

Heights and the central critical values of triple product $L$-functions

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La méthode des graphes

Exemples et applications

  • Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata)
  • 1986