math.NT — Number Theory

A zero-free strip for the $L$-functions of the elliptic curves $y^2=x^3+d$

Contributed by Michael Rubinstein, Mateo Restrepo Borrero

For a nonzero integer $d$ let $E_d$ be the elliptic curve $y^2=x^3+d$ over $\mathbb Q$. Part I adapts OpenAI's $11/12$ argument to prove that $L(E_d,s)\ne0$ for $\Re s>17/12$. Part II adapts OpenAI's $7/8$ argument to prove $L(s,\overline\alpha\eta)\ne0$ for $\Re s>7/8$, where $\eta$ is any finite-order Hecke character of $\mathbb Q(\sqrt{-3})$ and $\alpha(n)=n/|n|$ on primary generators. Consequently $L(E_d,s)\ne0$ for $\Re s>11/8$. The analytic normalization is $L(E_d,s+\frac12)$, so the two boundaries are $11/12$ and $7/8$ within $0<\Re s<1$. The curve characters have infinite order, and require new transformation and moment arguments. These are proved here; steps identical to the original arguments are cited. Part II also uses the ideas from OpenAI's finite-order $7/8$ theorem. Higher-weight and symmetric-power extensions are discussed; more general CM forms are the subject of a second paper.