Browse by result family, then open its individual manuscripts. Plain-language guides are marked; other entries show OpenAI's source summary. All results are attributed claims.
Number theory#001
OpenAI source summary
Proves Milne's rationality conjecture for abelian varieties over \overline{\mathbb Q} 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.
1 manuscriptNo Lean link listed
Number theory#002
OpenAI source summary
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.
3 manuscriptsNo Lean link listed
Number theory#003
OpenAI source summary
Proves that every Dirichlet L-function, including \zeta(s), is zero-free in \Re s> 7/8, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over \mathbb Q(\sqrt{-3}). A companion gives a different proof of the zero-free half-plane \Re s> 11/12.
3 manuscriptsLean scope linked
Number theory#004
OpenAI source summary
Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.
2 manuscriptsNo Lean link listed
Number theory#005
OpenAI source summary
Proves that Catalan's constant G=\sum_{j\ge0}(-1)^j/(2j+1)^2 is irrational.
1 manuscriptLean scope linked
Number theory#006
OpenAI source summary
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.
2 manuscriptsNo Lean link listed
Number theory#007
Plain-language guide
Count whether each integer has an even or odd number of prime factors, including repeats. This paper claims a precise cancellation result for paired patterns in that sequence, together with a wider result for related arithmetic functions.
1 manuscriptLean scope linked
Number theory#008
OpenAI source summary
Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture.
1 manuscriptLean scope linked
Number theory#009
OpenAI source summary
Reconstructs function fields of transcendence degree at least two over algebraically closed constants from K^{\mathrm M}_1/\ell, K^{\mathrm M}_2/\ell, and their product. These data recover the perfect closure and constants when ℓ differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-ℓ Galois data away from the characteristic.
3 manuscriptsLean scope linked
Number theory#010
OpenAI source summary
Every continuous odd absolutely irreducible two-dimensional 2-adic representation of G_{\mathbb Q} unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at 2 with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.
3 manuscriptsNo Lean link listed
Number theory#011
OpenAI source summary
Proves that the normalized ordered logarithms of the prime factors of p-1, counted with multiplicity, converge jointly to the Poisson–Dirichlet law PD(1) as p ranges uniformly over primes up to x and x→∞. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than n^{1-ε} totient preimages, for every ε > 0.
3 manuscriptsNo Lean link listed
Number theory#012
OpenAI source summary
Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and n+1 are asymptotically independent in ordinary natural density. In particular, the integers satisfying P^+(n)< P^+(n+1) have density 1/2.
1 manuscriptLean scope linked
Number theory#013
Plain-language guide
Could the primes be made by adding every member of one set of nonnegative integers to every member of another? The paper claims this is impossible when both sets contain at least two numbers, even after changing finitely many primes.
1 manuscriptLean scope linked
Number theory#014
OpenAI source summary
Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over \overline{\mathbb F}_q under the four stated Lie-theoretic characteristic hypotheses. Over arbitrary algebraically closed fields of characteristic p > 0, the same conclusion holds assuming additionally that p is very good for the group and p\nmid |W_G|. Over global function fields, proves Ramanujan at every place for globally generic cuspidal representations of split adjoint absolutely simple exceptional groups, without characteristic or ramification-depth restrictions, and at every unramified place for cuspidal representations of split adjoint absolutely simple groups with a generic unramified component. Assuming the finite-level Ramanujan–Arthur decomposition, constructs global Arthur enhancements of occurring cuspidal excursion parameters for split connected semisimple groups at full finite level, recovering the given parameters by diagonal specialization on the entire Weil group, including inertia.
8 manuscriptsNo Lean link listed
Number theory#015
OpenAI source summary
Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity.
3 manuscriptsLean scope linked
Number theory#016
OpenAI source summary
Proves the abelian Zilber–Pink conjecture over \overline{\mathbb Q}: 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 \mathcal A_2 for Hodge-generic curves defined over \overline{\mathbb Q}, without boundary or reduction assumptions.
4 manuscriptsNo Lean link listed
Number theory#017
Plain-language guide
Fractions can get very close to pi, but how close depends on their denominator. This manuscript claims to pin down the best possible long-run rate of approximation, settling the irrationality exponent of pi at two.
1 manuscriptLean scope linked
Number theory#018
OpenAI source summary
Proves the Margulis–Platonov conjecture over every global field, including function fields of characteristic two. For an absolutely almost simple simply connected algebraic group G over k, every noncentral abstract normal subgroup of G(k) is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.
2 manuscriptsNo Lean link listed