conjecturesAI-generated conjectures in number theory, based on LMFDB data. Read the statements, numerical evidence, and full papers.
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Conjecture 009Iwasawa Theory
We 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 017Modular and Automorphic Forms
For 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 023Modular and Automorphic Forms
Fix 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 024Arithmetic Geometry
Let 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 025Modular and Automorphic Forms
Fix 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 026Arithmetic Geometry
For a genus-two curve C/Q with , 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.
Conjecture 004Arithmetic Geometry
We formulate a conjectural power-law distribution for local Tamagawa numbers at marked split multiplicative primes of semistable elliptic curves over ℚ, ordered by conductor and conditioned by Mordell–Weil rank and the sign of the minimal discriminant. We record the numerical evidence motivating the conjecture and explain the gap between the proposed conductor-ordered statement and known fixed-prime, height-ordered local density results.
Conjecture 012Algebraic Number Theory
We conjecture that monogenicity and unit-signature rank are asymptotically independent in the discriminant-ordered family of totally real non-Galois cubic fields. We present the numerical ratios motivating the conjecture and discuss why existing results on monogenic fields and on narrow class groups do not presently imply it.
Conjecture 015Algebraic Number Theory
We study pairs of distinct totally real S₃-cubic fields with the same discriminant. For every prime ℓ ≥ 5, we conjecture that divisibility of the two class numbers by ℓ is asymptotically independent under the natural ordered-pair weighting. The exclusion of ℓ = 3 is essential and reflects the 3-primary class-field-theoretic mechanism producing common-discriminant multiplets.
Conjecture 016Algebraic Number Theory
For ordered pairs of distinct totally real S₃-cubic fields with the same discriminant, we conjecture that the two unit-signature ranks are asymptotically independent. The conjecture separates the 2-primary archimedean invariant from the 3-primary ring-class mechanism producing common-discriminant multiplets.
Conjecture 019Modular and Automorphic Forms
We conjecture that the two global root numbers occur with equal limiting frequency among genuine non-CM rational Bianchi newforms of parallel weight two and trivial central character over a fixed imaginary quadratic field, ordered by level norm. We describe the finite-level bias visible in current data and explain the obstruction created by constant-sign quadratic-twist families.
Conjecture 021Galois Representations
We consider faithful irreducible two-dimensional Artin representations of G_(Q) with image SL₂(F₃), trivial determinant, and Frobenius–Schur indicator −1. Ordered by Artin conductor, we conjecture that this family is infinite and that its global root numbers are equidistributed. We also record a proof of infinitude by quadratic twisting and explain why this construction does not itself produce sign cancellation.
Conjecture 022Arithmetic Geometry
We study singleton non-CM isogeny classes of elliptic curves over Q with prime conductor. For each fixed rank 0 ≤ r ≤ 3, we conjecture a two-term asymptotic for the proportion having negative minimal discriminant. The leading constant is the classical archimedean ratio, while the secondary constants exhibit a strict rank-dependent ordering suggested by the available data.
Conjecture 027Arithmetic Geometry
Fix an odd prime ℓ and prescribe whether ℓ divides the conductor. Among typical genus-two curves whose mod-ℓ representation is nonsurjective and whose Jacobian has no rational ℓ-torsion, we conjecture root-number equidistribution when curves are ordered by absolute minimal discriminant. The exclusions isolate the residual-image condition from the most immediate sources of finite-range sign bias.
Conjecture 028Modular and Automorphic Forms
Fix an imaginary quadratic field K. We pair genuine rational non-CM Bianchi newforms with exact rational-source base-change newforms having the same level ideal, and weight all ordered pairs equally. We conjecture that the two global root numbers become asymptotically independent. The statement admits an exact reformulation as the vanishing of a level-weighted covariance.
Conjecture 031Algebraic Number Theory
We formulate conjectural first moments for the ordinary and narrow class-group 2-torsion of discriminant-ordered S₅-quintic fields. The fields are stratified simultaneously by signature, squarefreeness of the discriminant, and the set of common index divisors. The proposed constants are the usual fixed-signature Cohen–Lenstra–Martinet–Malle and Dummit–Voight moments, and the conjecture asserts that the additional local restrictions do not change them.
Conjecture 033Arithmetic Geometry
For a fixed quadratic field, we consider non-CM elliptic-curve isogeny classes that are not ℚ-curves and impose an exact finite pattern of rational cyclic isogenies. We conjecture that every infinite such conductor-ordered stratum satisfies the minimalist rank law: asymptotic mass one half in each of ranks zero and one, and zero density in higher rank. We record the available numerical evidence and explain the additional root-number and Selmer-distribution inputs required beyond the usual minimalist conjecture.
Conjecture 034Modular and Automorphic Forms
We formulate a level-aspect Hilbert analogue of a Maeda-type giant-orbit conjecture. Over a fixed real quadratic field of narrow class number one, and separately in each prime-level functional-equation-sign sector, the largest non-CM, non-base-change Hecke orbit is conjectured to occupy asymptotically all of the sector. The numerical evidence is strong in the deepest available fields, but the corresponding classical prime-level statement is itself open.
Conjecture 035Algebraic Number Theory
We propose a joint Cohen–Lenstra–Martinet law for the good-primary parts of class groups of discriminant-ordered S₅-quintic fields. The signature and the set of common index divisors are fixed. For primes not dividing |S₅|, the primary class groups are conjectured to follow the standard u-probability measure and to be asymptotically independent across distinct primes.
Conjecture 036Arithmetic Geometry
For a fixed quadratic field, we conjecture that non-CM ℚ-curves have density zero among all non-CM elliptic-curve isogeny classes ordered by conductor norm. The assertion includes base changes from ℚ as well as strict ℚ-curves. Numerical data over the deepest imaginary quadratic fields show a consistent decline in both components, while the main theoretical difficulty is obtaining conductor-aspect counts uniform over all modular degrees and twist families.
Conjecture 037Algebraic Number Theory
We conjecture that, after fixing the signature and the squarefree or nonsquarefree discriminant regime, the limiting unit-signature distribution of discriminant-ordered S₅-quintic fields is independent of the exact set of common index divisors. The conjecture separates an archimedean 2-primary invariant from finite local splitting conditions at 2 and 3.
Conjecture 038Modular and Automorphic Forms
Over a fixed real quadratic field of narrow class number one, we consider non-CM, non-base-change Hilbert newforms of parallel weight two and order their Hecke-Galois orbits by level norm. We conjecture that, when every orbit receives equal weight, those with bounded Hecke-field degree have density zero. This is stronger than known eigenform-weighted rationality-field results and is naturally related to generalized Maeda heuristics.
Conjecture 039Algebraic Number Theory
We record a conjectural ramification bias in the unit-signature ranks of discriminant-ordered S₅-quintic fields. After fixing the signature and the exact set of common index divisors, fields of squarefree discriminant appear more likely to have a deficient unit-signature rank than fields of nonsquarefree discriminant. The observed inequalities are stable in the largest available height shells, but no existing random-space model predicts their strict direction.
Conjecture 040Modular and Automorphic Forms
We conjecture joint equidistribution of prescribed local Atkin–Lehner signs among genuine parallel-weight-two Hilbert newform orbits of a fixed Hecke-field degree. Ambient Plancherel equidistribution gives the corresponding statement without the rationality-degree restriction. The fixed-degree condition is global and thin, and no known trace-formula projector isolates it.
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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.
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