arXiv subjects

Ken Ono

Publications and source records attributed to Ken Ono.

At least 19 recordsLinked to original sources

arXiv2026

Rogers--Ramanujan identities from the geometry of $X^a=Y^b$

We prove the conjecture of Huang, Jiang, and Oblomkov (HJO) giving a geometric extension of the Rogers--Ramanujan and Andrews--Gordon identities for every torus-knot singularity $X^a=Y^b$ with coprime $1<a<b.$ For a prime power $q$, let $\mathcal{NC}_n^{a,b}(\mathbb F_q)$ denote the set of pairs of commuting nilpotent $n\times n$ matrices $(A,B)$ over $\mathbb F_q$ satisfying $A^a=B^b$. We establish the threefold equality between their normalized counts, the HJO $q$-series $Z_{a,b}$, and the explicit infinite product $P_{a,b}$: \[ \underbrace{\vphantom{\Bigg|} \prod_{m\geq1}(1-q^{-m}) \Biggl(\sum_{n=0}^{\infty} \frac{\lvert\mathcal{NC}_n^{a,b}(\mathbb F_q)\rvert} {\lvert\operatorname{GL}_n(\mathbb F_q)\rvert}\Biggr) }_{\text{point count}} = \underbrace{\vphantom{\Bigg|}Z_{a,b}(q^{-1}) }_{\text{$q$-series}} = \underbrace{\vphantom{\Bigg|}P_{a,b}(q^{-1}) }_{\text{infinite product}}. \] Our main result is a stronger finite identity: the rank $N$ HJO sum equals $(q;q)_N$ times the generating function for balanced cylindric partitions with entries bounded by $N$. Taking $N\to\infty$ yields the HJO conjecture. The proof combines the compositional rational shuffle theorem of Bergeron--Garsia--Leven--Xin and Mellit with a multiplicativity theorem for slope operators and a determinantal model for bounded cylindric partitions, linked by a common $q$-difference equation. The finite identity and the HJO conjecture have been formalized in Lean by AxiomProver, conditional on two stated literature inputs.

math.NT↗

arXiv2026

On a conjecture of Browning and Sawin on random hypersurfaces with sign coefficients

Browning and Sawin conjectured that random hypersurfaces with sign coefficients are smooth with probability tending to one as the degree grows. We prove this conjecture and obtain a quantitative bound. For each $n\geq1$, a degree $d$ form in $n+1$ variables, with independent uniform coefficients in $\{-1,1\}$, defines a singular complex hypersurface with probability $O_n(d^{-1/2})$. The positive-dimensional singular loci occur with exponentially small probability. For $n\geq3$, the same exponential bound holds for failure of absolute irreducibility. These results have been formalized in Lean by AxiomProver assuming existing literature.

math.NT↗

arXiv2026

Parity of the partition function in quadratic progressions

The parity of the partition function $p(n)$ is one of the most stubborn problems in number theory. In 2010, the first author conjectured, for square-free $1<D\equiv 23\pmod{24}$, that the values $p\!\left(\frac{Dm^2+1}{24}\right)$, as $m$ ranges over positive integers with $(m,6)=1$, include infinitely many even and infinitely many odd terms. We prove this conjecture. The key new idea is geometric. Logarithmic derivatives of twisted Borcherds products built from Ramanujan's third-order mock theta functions recast the problem in terms of CM points of discriminant $-D$ on $X_0(6)$. The uniqueness of canonical lifts from characteristic $2$ to characteristic zero shows that the CM points supporting the poles remain distinct after reduction modulo $2$. This fact, combined with an Eisenstein series comparison and a Galois representation argument, rules out constant parity and gives infinitely many values of each parity. The method applies to the coefficients of analogous generalized twisted Borcherds products. These results imply that \[\#\{0\leq n\leq X: p(n)\text{ is odd}\}\gg\sqrt X,\] which is now the best known lower bound for odd values of $p(n)$. The algebraic identities at the heart of this paper, as well as the improved lower bound for odd partition numbers, have been formalized in Lean by AxiomProver.

math.NT↗

arXiv2026

On the paucity of lattice triangles

A rational triangle $T$ (one whose angles are rational multiples of $π$) unfolds to a translation surface ${X_T}$. The lattice triangle problem asks to classify those $T$ for which ${X_T}$ is a Veech (lattice) surface, which means that the $\operatorname{SL}_2(\mathbb R)$-orbit of ${X_T}$ is closed in its stratum (so its projection to moduli space is a Teichmüller curve). The most mysterious regime is the "hard obtuse window" (largest angle in $(π/2,2π/3]$), where it is conjectured that no lattice triangles exist. Using an arithmetic reformulation of the Mirzakhani-Wright rank obstruction, we prove a quantitative theorem that rules out all but a proportion $n^{-1+o(1)}$ of the triangles in this window with denominator $n$. The main technical result in our proof was autoformalized by AxiomProver in Lean (using mathlib).

math.DS↗

arXiv2026

Proof of the Lyons--White Conjecture

Let $D_n$ be the dihedral group of order $2n$. Consider a continuous-time random walk on $D_n$ driven by arbitrary symmetric rates whose support generates $D_n$. For $p\in[1,\infty]$, we say the pair $(D_n,p)$ is rate-monotonic if for each fixed time $t$, the $\ell^p$-distance between the random walk's distribution at time $t$ and the uniform distribution is monotonically decreasing as a function of the rates. Lyons and White proved that $(D_n,2)$ and $(D_n,\infty)$ are rate-monotonic. Somewhat counterintuitively, they found several pairs $(D_n,p)$ with ${p\in[1,1.997]\cup[2.001,3.999]\cup[4.001,5.995]}$ that are not rate-monotonic, and they asked whether any such pairs exist with $p=4$ or $p=6$. We resolve their question, proving that $(D_n,2m)$ is rate-monotonic for all positive integers $m$ and $n$. In fact, we prove a generalization of this result to a broader family of groups that includes generalized dihedral groups, dicyclic groups, and generalized quaternion groups. In the other direction, we prove that for every real $p\geq 1$ that is not an even integer, there exists a positive integer $n$ such that $(D_n,p)$ is not rate-monotonic. The results of this paper were formally verified in Lean by AxiomProver assuming standard literature.

math.PR↗

arXiv2026

Zero-free columns in character tables of symmetric groups

The rows and columns of the character table of the symmetric group $S_n$ are both naturally indexed by partitions of $n$. Let $D(n)$ denote the number of conjugacy classes of $S_n$ whose column contains no zero entry. The identity column is always zero-free, so $D(n)\geq 1$. It is known that $D(n)\ll n^2$. We prove that $D(n)\ll n^{3/4}$. Second, we prove for almost all positive integers $n$ that $D(n)\ll_B n^{1/2}(\log n)^B$ for every $B>5/6$, with a quantitative bound for the exceptional set, using work of Matomäki and Radziwill. Finally, we offer a heuristic supporting our conjecture that $D(n)\ll_{\varepsilon} n^{\varepsilon}$. AxiomProver formalized the results in this paper in Lean assuming preexisting literature.

math.CO↗

arXiv2026

Algebraic geometric framework of Rogers--Ramanujan identities

The Rogers--Ramanujan identities equate a $q$-series whose exponents are governed by a quadratic form with an infinite product supported on two residue classes modulo~$5$. Identities of this shape are scarce, and a central problem is to identify the structures that produce them in families. Huang, Jiang, and Oblomkov have proposed a source of a new kind: to each pair of coprime integers $a,b>1$ they attach an infinite-rank $q$-series $Z_{a,b}(q)$, assembled from counts of commuting nilpotent matrix pairs $(A,B)$ with $A^a=B^b$ over finite fields, and they conjecture that it equals an explicit product of $(a-1)(b-1)/2$ modular units of level $a+b$. The $a=2$ cases are the Andrews--Gordon identities; no case with $a>2$ was known. We prove the conjecture for $(a,b)=(3,4)$, $(3,5)$, $(3,7)$, and $(3,8)$. Our proofs pass through a finer sum-to-sum identity, which we conjecture for all $b$ coprime to $3$ and establish for all $b$ when $q=1$. Lau and Ono have since proved that identity in general, and with it the full $a=3$ case. These identities have been formalized and verified in Lean by AxiomProver.

math.NT↗

arXiv2026

Formalized $q$-series: The Rogers-Ramanujan Identities and Beyond

The theory of $q$-series and basic hypergeometric series plays a crucial role at the intersection of combinatorics, number theory, and representation theory. From the classical partition identities of Euler and Jacobi to modern developments in class field theory, vertex operator algebras, and the Monstrous Moonshine conjecture, $q$-series provide the analytic framework for a wide range of profound applications. In this paper, we discuss the formalization of this theory in the Lean proof assistant, a process that requires careful design of scalable and versatile structures to reconcile formal algebraic identities with analytic convergence properties. We address these foundational challenges by focusing on the construction of $q$-Pochhammer symbols, $q$-binomial coefficients, Bailey's Lemma and similar primitives. To demonstrate the utility of this work, we provide fully verified proofs of the Jacobi Triple Product formula and the celebrated Rogers-Ramanujan identities, which serve as both historical and technical benchmarks for the field. This work establishes a rigorous computational foundation for the future formalization of mock theta functions, modular forms, and the diverse algebraic structures that underpin their applications across mathematics and physics. AxiomProver was used to produce the formalizations in this paper.

math.NT↗

arXiv2026

Four-digit Kaprekar dynamics in odd bases

Start with four digits, arrange them in both descending and ascending order, subtract, and repeat. This simple process is known as the Kaprekar routine, famous in base ten for sending every nonconstant four-digit string to $6174$. We show that in every odd base $B>3$, the four-digit Kaprekar map has an unexpectedly rigid structure. After at most three iterations, every nonconstant orbit enters an explicit triangular region $\mathcal{T}_B$, and on this region the map is conjugate to projective doubling: \[ \{[r],[s]\}\longmapsto \{[2r],[2s]\}. \] This gives a complete finite description of all nonconstant terminal cycles, including an explicit formula for their lengths and counts. In particular, the longest terminal cycle has length at most $(B-1)/2$, and equality can occur only when $B$ is prime. For primes $p>5$, equality occurs precisely when the least positive $m$ with $2^m\equiv\pm1\pmod p$ is $m=(p-1)/2$. The results proved here were first formulated by Schwartz and Thakur. As a test case for AI-assisted formal mathematics, AxiomProver produced Lean/mathlib formalizations of these results.

math.NT↗

arXiv2026

Modularity of Point Counts for the Curves $X^a=Y^b$: New Rogers--Ramanujan Identities

For coprime $1<a<b$, let $M_n^{a,b}(\mathbb{F}_q)$ be the set of commuting pairs of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ with $X^a=Y^b$. Huang, Jiang, and Oblomkov assembled their orders as an Eulerian $q$-series $Z_{a,b}(q)$. They conjectured that it is an explicit product $P_{a,b}(q)$ involving Jacobi's theta function and Dedekind's eta-function, implying the threefold equality $$\underbrace{\prod_{n\ge1}(1-q^n)\cdot\Biggl(\sum_{n=0}^{\infty}\frac{|M_n^{a,b}(\mathbb{F}_q)|}{|\mathrm{GL}_n(\mathbb{F}_q)|}\Biggr)\Biggr|_{q\mapsto q^{-1}}}_{\text{point count}}\;=\;\underbrace{Z_{a,b}(q)}_{q\text{-series}}\;=\;\underbrace{P_{a,b}(q)}_{\text{theta quotient}}$$ If true, the point count on $X^a=Y^b$ is essentially a modular function on $Γ(a+b)$. The conjecture is layered in $a$, with an identity for each $b$. The $a=2$ layer is classical, including identities of Rogers--Ramanujan and Andrews--Gordon. For $a\geq3,$ nothing was known. We prove the $a=3$ layer in full: a new infinite family of Rogers--Ramanujan identities, and a geometric origin for Warnaar's products. AxiomProver verified these new identities in Lean assuming existing literature.

math.NT↗

arXiv2026

A bijective proof of a partition theorem of Berkovich and Uncu

In 2016, Berkovich and Uncu proved that, for all nonnegative integers $i$, $j$, and $n$, the number of strict partitions of $n$ with $i$ odd-indexed odd parts and $j$ even-indexed odd parts equals the number of strict partitions of $n$ with $i$ parts congruent to $1$ modulo $4$ and $j$ parts congruent to $3$ modulo $4$. Their proof used generating functions, and they asked for a combinatorial proof. We answer their question with an explicit bijection, assembled from three classical ingredients: $2$-modular diagrams, an insertion algorithm of Chen, Gao, Ji, and Li, and Glaisher's bijection. AxiomProver autonomously formalized and verified the proof of the main theorem in Lean.

math.CO↗

arXiv2026

On a conjecture of Han and Xiong for fractional Gaussian binomial coefficients

Han and Xiong recently extended the Gaussian binomial coefficient $\genfrac{[}{]}{0pt}{}{r+k}{k}_{q}$ to positive rational $r$ and conjectured that its integer trace, the integer-exponent part of the resulting power series, is coefficientwise largest at $r=1/2$. We prove a support-dominance theorem comparing rational parameters under an explicit divisibility condition. It settles the conjecture for every $r\geq 1/2$ and reduces the full conjecture to the unit fractions $r=\frac{1}{2m}$, only finitely many of which are nontrivial for each fixed $k$. A computer computation then verifies the conjecture for every positive rational $r$ and every $k\leq 200$. The theoretical results were autonomously produced and verified in Lean by AxiomProver.

math.CO↗

arXiv2026

Record compositions of alternating permutations and noncommutative symmetric functions

Amdeberhan, Shareshian, and Stanley recently proved that a function $φ$ arising in the theory of partition Eisenstein series counts the alternating permutations of $\{1,\dots,2n\}$ with a given `record' partition, and they asked whether there is a similar theory for record compositions, suggesting a role for noncommutative symmetric functions. Here we solve their open problem by showing that the number of alternating permutations of $\{1,\dots,2n\}$ with record composition $(α_1,\dots,α_\ell)$ is \[ \prod_{j=1}^{\ell}\binom{2s_j-1}{2α_j-1}E_{2α_j-1}, \] where $s_j=α_1+\dots+α_j$, $E_k$ is an Euler number, and the record composition of $w=a_1a_2\dots a_{2n}$ (so $a_1>a_2 \dotsb$) lists the factor lengths obtained by cutting $a_1a_3\dots a_{2n-1}$ before each left-to-right maximum other than the first. These numbers are the coefficients of a natural lift of the degree-$n$ sprout symmetric function with seed $\sec(\sqrt{t}\,)$ to noncommutative symmetric functions, expanded in products of noncommutative power sums of the first kind. An analogous refinement holds for every sprout sequence whose seed is given by the exponential formula. AxiomProver autonomously produced and verified the results in this paper in Lean.

math.CO↗

arXiv2026

Beyond Mock Modularity: Elliptic Corrections for Higher Dyson Ranks

When $m = 1$, the Dyson rank generating function is a classical bridge between partition theory, Ramanujan's mock theta functions, and the theory of harmonic Maass forms and nonholomorphic Jacobi forms. The rank is a statistic on partitions, and the higher Dyson systems, for $m \geq 2$, are a natural multivariable refinement of it, combining $m$ graded rank contributions. Unlike the classical case, these higher systems are not expected to fit the mock-modular framework, which raises the question of what analytic structure governs them. We show that their root-of-unity specializations carry a hidden elliptic structure. A finite $q$-difference recurrence produces an explicit polynomial obstruction to the expected index $m$ elliptic transformation law, and because the obstruction is finite, its partial fractions canonically determine finitely many Appell--Lerch correction terms that remove it. The corrected functions satisfy a twisted index $m$ elliptic law; a natural translation removes the twist, and their holomorphic finite parts admit finite theta decompositions. Thus, the natural analogue of Dyson's mock-modular phenomenon at higher $m$ is not mock modularity but a finite theta decomposition governed by an index $m$ elliptic transformation law. These results grew out of a human--AI collaboration, and the key new formulas were formalized and machine-verified in Lean/Mathlib by AxiomProver.

math.NT↗

arXiv2026

Integer values of $\tan(\arctan 1+\arctan 2+\cdots+\arctan n)$ are rare

For $n\ge1$, we let $$x_n:=\tan\bigl(\sum_{k=1}^{n}\arctan k\bigr).$$ In 2008, Amdeberhan, Medina, and Moll conjectured that $x_n\not \in \mathbb{Z}$ for every $n\ge5$. This was known for a set of positive integers of density $\tfrac{120}{817}\approx0.1469$. We prove that an integer value $x_n=m$ satisfies $|m|\ge e^{(1/2+o(1))\,n\log n}$, which we use to deduce that $$\#\{\,1\leq n\le N:x_n\in\mathbb{Z}\,\}=O(\log N). $$ In particular, the conjecture holds for a density-one set of $n$. The results in this note were formalized in Lean/Mathlib and produced autonomously by AxiomProver from natural-language statements.

math.NT↗

arXiv2026

The partition function and elliptic curves

The Bruinier-Ono formula expresses the partition number $p(n)$ as a trace of `non-holomorphic' singular moduli of discriminant $Δ_n:=1-24n$ CM points on $X_0(6).$ We interpret this trace through the geometry of CM points. Each nonholomorphic contribution is the value of the weight-two completion $E_2^*$ at a CM point, which is a canonical invariant of the underlying elliptic curve, determined by the diagonal `tangent' of the CM isogeny relation. This turns the trace into a quantity that can be reduced to the supersingular locus that is organized by Deuring-Eichler multiplicities and a Brandt-module pairing. For primes $ \ell\geq 5$ that are nonsplit in $\mathbb{Q}(\sqrt{Δ_n})$, we obtain a supersingular trace formula on $X_0(6)$ over $\overline{\mathbb{F}}_{\ell}$. For the special primes $\ell=5,7,11$, this sheds new light on Ramanujan's classical partition congruences. These primes are special because they are the only ones for which the supersingular locus of $X_0(6)$ lies over $j\in \{0, 1728\}.$ This perspective offers a moduli-theoretic framework for Ramanujan's congruences modulo powers of these primes, organized through elliptic curves. The two new algebraic identities at the heart of this framework, as opposed to the classical results it builds on, were formalized and verified in Lean by AxiomProver.

math.NT↗

arXiv2026

Modular forms for chromatic homotopy: Supersingular congruences

We prove a conjecture of Larson in Behrens' program on congruences of modular forms attached to the divided beta family in the Adams--Novikov spectral sequence for the stable homotopy groups of spheres. The conjecture gives a sharp criterion for when the modular form associated to a divided beta element can be represented by a pure power of the discriminant modular form. Writing $i=rp^{n}$ with $(r,p)=1$ and $t=i(p^2-1)/12$, Larson's conjecture asserts that the Behrens form $f_{i/j}$ (which is well defined modulo $p$) may be taken to be the pure power $Δ^{t}$ precisely when $1\le j\le p^{n}$, and admits no such representative otherwise. We prove this for all primes $p\ge5$. The proof reduces the decisive congruence condition to a geometric statement on supersingular points of modular curves. Namely, that for every prime $\ell\ne p$, the value of the modular function $V_\ell(Δ)/Δ$ at each supersingular point of $X_0(\ell)$ is an $(p^2-1)/12$-th root of unity.

math.AT↗

arXiv2026

Parity of $k$-differentials in genus zero and one

Here we completely determine the spin parity of $k$-differentials with prescribed zero and pole orders on Riemann surfaces of genus zero and one. This result was previously obtained conditionally by the first author and Quentin Gendron assuming the truth of a number-theoretic hypothesis Conjecture A.10. We prove this hypothesis by reformulating it in terms of Jacobi symbols, reducing the proof to a combinatorial identity and standard facts about Jacobi symbols. The proof was obtained by AxiomProver and the system formalized the proof of the combinatorial identity in Lean/Mathlib (see the Appendix).

math.NT↗