Under GRH the average analytic rank of y²=x³-dx over odd fourth-power-free d is at most 13/6, and at most 3/2 assuming a quartic Gauss-sum conjecture.
Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves
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We show that a positive proportion of Hecke $L$-functions attached to the quartic residue symbols $\big( \frac{\cdot}{q} \big)_4$ for squarefree $q \in \mathbb{Z}[i]$ do not vanish at the central point. Our method also extends to the Hecke characters associated to quartic twists of the congruent number curve $E : y^2 = x^3 - x$. In particular, we prove that the elliptic curve $E^{(q)} : y^2 = x^3 - qx$ has Mordell-Weil rank $0$ over $\mathbb{Q}(i)$ for a positive proportion of squarefree $q \in \mathbb{Z}[i]$ ordered by norm.
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Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$
Under GRH the average analytic rank of y²=x³-dx over odd fourth-power-free d is at most 13/6, and at most 3/2 assuming a quartic Gauss-sum conjecture.