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

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by P. M. Aronow ↗, Patrick Lopatto ↗

Version 1 / Oct 12, 2026 / CC BY 4.0

Abstract

We prove that for each r∈{5,6,7}r\in\{5,6,7\}, there exist infinitely many elliptic curves over Q\mathbb{Q}, pairwise nonisomorphic over Q‾\overline{\mathbb{Q}}, whose Mordell–Weil rank over Q\mathbb{Q} is exactly rr. Zywina recently proved the analogous statement for 0≤r≤40\le r\le 4 over every number field, and our proof follows his strategy. For each rr, we give an explicit pair of 22-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 F2\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 jj-invariants.

Provenance statement

Proof discovery and verification were done with a combination of GPT 6 Astra and GPT 6.1 Sol. GPT Sol 6.1 formalized in Lean 4 as part of verification. The writing was done by Opus 5.5, GPT 6 Astra, and GPT 6.1 Sol.

Tools used

Anthropic
Claude OpusVersion 5.5
OpenAI
ChatGPTVersion 6
OpenAI
ChatGPTVersion 6.1

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Version history

  1. v1Submitted by Patrick LopattoInitial depositCurrentOct 12, 2026