# Elliptic curves and abelian varieties > 7 AI-made results on elliptic curves and abelian varieties, each explained in plain English. Curves and higher-dimensional shapes defined by polynomial… Canonical: https://closedform.nishachay.tech/topic/elliptic-curves-and-abelian-varieties/ Last updated: 2026-10-14 Curves and higher-dimensional shapes defined by polynomial equations that also carry a way to add points. 7 AI-made results on elliptic curves and abelian varieties, from 18 papers in the OpenAI math collection, each explained in plain English. 1 has a formal Lean proof listed; 6 are claimed in a paper and not yet machine-checked. They span 2 fields, led by number theory (5) and algebraic and complex geometry (2). The oldest question here dates back to 1950: [The rational Hodge conjecture for CM abelian varieties](https://closedform.nishachay.tech/e/openai-math-2026-032/). Results rated Landmark include [The full BSD formula from low Selmer corank](https://closedform.nishachay.tech/e/openai-math-2026-002/) and [Modularity of elliptic curves over imaginary quadratic fields](https://closedform.nishachay.tech/e/openai-math-2026-030/). ## Results (7) - [Milne’s rationality conjecture and algebraic specialization](https://closedform.nishachay.tech/e/openai-math-2026-001/) - Entry 001 · After reduction mod p, Hodge classes on abelian varieties are claimed to give one rational number in every cohomology. (Claimed, not checked) - [The full BSD formula from low Selmer corank](https://closedform.nishachay.tech/e/openai-math-2026-002/) - Entry 002 · The full Birch–Swinnerton-Dyer formula is claimed for every elliptic curve with a small Selmer group at one prime. (Claimed, not checked) - [Goldfeld’s conjecture: densities and mean analytic rank](https://closedform.nishachay.tech/e/openai-math-2026-006/) - Entry 006 · Among the quadratic twists of any elliptic curve, half are claimed to have analytic rank 0 and half rank 1. (Claimed, not checked) - [Zilber–Pink in abelian varieties and the Siegel threefold](https://closedform.nishachay.tech/e/openai-math-2026-016/) - Entry 016 · Unlikely intersections in abelian varieties are claimed to be finite in number, in every dimension. (Claimed, not checked) - [Modularity of elliptic curves over imaginary quadratic fields](https://closedform.nishachay.tech/e/openai-math-2026-030/) - Entry 030 · Every elliptic curve over every imaginary quadratic field is claimed to match an automorphic form, place by place. (Claimed, not checked) - [The rational Hodge conjecture for CM abelian varieties](https://closedform.nishachay.tech/e/openai-math-2026-032/) - Entry 032 · The rational Hodge conjecture is claimed for every abelian variety with complex multiplication, in every dimension. (Claimed, not checked) - [Semialgebraic universal covers and bounded domains](https://closedform.nishachay.tech/e/openai-math-2026-058/) - Entry 058 · Unrolled shapes with a polynomial description split into just three kinds of pieces. (Formal proof listed) ## Related - [Claimed, not yet checked](https://closedform.nishachay.tech/trust/claimed/) - Trust · 130 results - [Landmark results](https://closedform.nishachay.tech/significance/landmark/) - Significance · 49 results - [Erdős problems](https://closedform.nishachay.tech/topic/erdos-problems/) - Topic · 13 results - [Named conjectures](https://closedform.nishachay.tech/topic/named-conjectures/) - Topic · 85 results - [Ramsey theory](https://closedform.nishachay.tech/topic/ramsey-theory/) - Topic · 5 results - [Prime numbers](https://closedform.nishachay.tech/topic/prime-numbers/) - Topic · 12 results - [Graph theory](https://closedform.nishachay.tech/topic/graph-theory/) - Topic · 40 results - [Distances and packing](https://closedform.nishachay.tech/topic/distances-and-packing/) - Topic · 5 results - [Number theory](https://closedform.nishachay.tech/science/mathematics/number-theory/) - Field · 5 here - [Algebraic and complex geometry](https://closedform.nishachay.tech/science/mathematics/algebraic-and-complex-geometry/) - Field · 2 here - [All topics](https://closedform.nishachay.tech/topics/) - Every way to browse