math.NT — Number Theory

Infinitely Many Elliptic Curves of Exact Ranks Five, Six, and Seven

Contributed by P. M. Aronow, Patrick Lopatto

We prove that for each $r\in\{5,6,7\}$, there exist infinitely many elliptic curves over $\mathbb{Q}$, pairwise nonisomorphic over $\overline{\mathbb{Q}}$, whose Mordell–Weil rank over $\mathbb{Q}$ is exactly $r$. Zywina recently proved the analogous statement for $0\le r\le 4$ over every number field, and our proof follows his strategy. For each $r$, we give an explicit pair of $2$-isogenous elliptic curves whose discriminants are, up to constants, products of powers of twelve binary linear forms in two integer parameters. The families are obtained from Zywina's curves of rank four, or from a related pair of curves, by base change along a conic. Explicit rational points give lower bounds for the images of the two descent maps. We specialize at parameters in a fixed congruence class at which the twelve linear forms, divided by fixed constants, take prime values. There, necessary local conditions in the descent become linear equations over $\mathbb{F}_2$, and their solution spaces have the same dimensions as the lower bounds. The coefficients of these equations are determined by the congruence class up to a single sign, and both values of this sign give the same bound. The Green–Tao–Ziegler theorem on linear equations in primes then gives infinitely many such specializations, with infinitely many $j$-invariants.