A modern mathematical library
Explore open, AI-resilient conjectures suggested by patterns in the LMFDB and robust under extensive numerical testing—problems selected for their mathematical depth and potential consequences in number theory.
Each paper is an invitation to investigate—not a claim of proof, novelty, or truth.
Selected for potential impact
Conjecture 009
AI selectionRetainedWe propose a rank-conditioned distribution for ordinary and signed cyclotomic Iwasawa λ-invariants of elliptic curves over ℚ. After removing the zero forced by Mordell–Weil rank, the remaining half-difference is conjectured to be geometrically distributed, with asymptotic independence across primes and between the two signed invariants at a supersingular prime.
Conjecture 017
AI selectionRetainedFor a fixed imaginary quadratic field, we conjecture that finite-order twists of cyclic base change have density zero among non-CM rational Bianchi newforms of parallel weight two and trivial central character, ordered by the norm of the level. We record the observed decay in the deepest available fixed-field datasets and isolate the counting problem required for a proof.
Conjecture 023
AI selectionRetainedFix an imaginary quadratic field K. We consider non-CM rational Bianchi newforms that arise by exact cyclic base change from rational classical newforms of weight two. Ordered by the norm of their Bianchi level and counted once as Bianchi Galois orbits, we conjecture that their global root numbers are equidistributed. Artin formalism reduces the problem to a fixed quadratic-character cancellation problem for elliptic curves ordered essentially by conductor.
Conjecture 024
AI selectionRetainedLet C/Q be a genus-two curve whose Jacobian has geometric endomorphism ring Z. We conjecture that its residual Galois representation is surjective onto GSp₄(F_(ℓ)) for every prime ℓ > 31. The value 31 is compatible with the current large-scale computations and is known to be a necessary lower boundary, but even the existence of an absolute uniform bound remains open.
Conjecture 025
AI selectionRetainedFix an imaginary quadratic field K. Among genuine, non-CM, rational Bianchi newforms of parallel weight two and trivial central character, we conjecture that the forms realized by elliptic curves over K have density one when ordered by level norm. The complementary quaternionic-multiplication branch is infinite in some fields but is constrained to squarefull levels; proving density zero nevertheless requires new conductor-aspect counting results.
Conjecture 026
AI selectionRetainedFor a genus-two curve C/Q with $\operatorname{End}_{\overline{\mathbf Q}}(J_C)=\mathbf Z$, let Exc (C) be the set of primes at which the residual Galois representation is not surjective. We conjecture the uniform cardinality bound |Exc (C)| ≤ 3. This bound is attained in the current data, while no example with four exceptional primes is known.
Reading the collection
Every paper keeps its proposed statement close to the evidence, mathematical context, and unresolved analytic or arithmetic gap.
Each claim names its objects, hypotheses, and predicted relation as precisely as the source record allows.
Numerical tables and finite checks are reproduced from the paper with their scope and limitations; they are evidence, not proof.
Read the complete discussion of known results, proof strategies, obstructions, what would still need to be established, and the consequences these conjectures could have in number theory.
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