Elliptic curves and abelian varieties
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. Results rated Landmark include The full BSD formula from low Selmer corank and Modularity of elliptic curves over imaginary quadratic fields.
Number theory5
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Milne’s rationality conjecture and algebraic specialization
Proves Milne's rationality conjecture for abelian varieties over with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-p and crystalline realizations, for every residue characteristic p.
Claimed, not checked -
The full BSD formula from low Selmer corank
Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one for some prime q, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.
Claimed, not checked -
Goldfeld’s conjecture: densities and mean analytic rank
Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.
Claimed, not checked -
Zilber–Pink in abelian varieties and the Siegel threefold
Proves the abelian Zilber–Pink conjecture over : every irreducible subvariety has finitely many maximal atypical subvarieties relative to its smallest containing torsion coset. It also proves the full curve case in the Siegel threefold for Hodge-generic curves defined over , without boundary or reduction assumptions.
Claimed, not checked -
Modularity of elliptic curves over imaginary quadratic fields
Proves the modularity conjecture for elliptic curves over imaginary quadratic fields: every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place.
Claimed, not checked
Algebraic and complex geometry2
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The rational Hodge conjecture for CM abelian varieties
Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension. Through Milne's theorems, this also gives the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic.
Claimed, not checked -
Semialgebraic universal covers and bounded domains
Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products , with D bounded symmetric and F simply connected, normal, and projective. A universal cover is quasi-projective exactly when the bounded symmetric factor is absent. In particular, a smooth projective variety covered by ℂn has a finite étale cover by an abelian variety.
Formal proof listed