arXane › Mathematics › Mathematics
Wednesday, 4 November 2026: 16 new papers, 8 cross-lists and 22 replacements in Number Theory (math.NT), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We study quantitative density of orbits of the free semigroup $\Gamma$ generated by two maps of $(0,\infty)$. For the M\"obius pair $x\mapsto 1/x$, $x\mapsto x+1$ we show that every orbit is $O(n^{-1})$-dense in $[0,\infty)$, that every irrational number $u$ is approached at a rate governed by the continued fraction of $u$ (at least $n^{-2}$, exponential when $u$ has bounded partial quotients, and $\exp(-cn/\log n)$ for almost every $u$), and that for rational $u$ and rational starting points the rate $n^{-1}$ is optimal. For the affine pair $T_0(x)=x/a$, $T_1(x)=bx+1$ with $a,b>1$, which was studied by Bergelson, Misiurewicz and Senti, we prove that for every $x>0$ and $u\ge 0$ there are infinitely many $\gamma\in\Gamma$ with \[ |\gamma(x)-u|<C\,\lambda^{-|\gamma|},\qquad \lambda=\exp\!\Big(\frac{\log a\,\log\frac{a}{a-1}}{\log\frac{a^2b}{a-1}}\Big)>1, \] where $|\gamma|$ is the word length of $\gamma$ and $C$ is independent of $\gamma$. This exponential rate holds whether or not $\log_a b$ is rational, and the exponential order cannot be improved for almost every $u$. We also give a corrected version of the Kronecker-type estimate $|\gamma(x)-u|<(u+1)(t^{1/|\gamma|}-1)$. Finally, we discuss the approximation of real numbers by ratios of primes and by ratios of logarithms of primes, updated in light of recent results on gaps between primes.
We determine all the groups that appear as the torsion group of an elliptic curve over a number field of degree $6$. Apart from the $44$ groups that occur infinitely often, which were determined by Derickx and Sutherland, exactly four groups occur: $\mathbb{Z}/25\mathbb{Z}$, $\mathbb{Z}/37\mathbb{Z}$, $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}$. The first three were known to occur, while the group $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}$ is new. We also show that, up to isomorphism, each of the groups $\mathbb{Z}/37\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}$ is the torsion group of a single elliptic curve over a single sextic field. The methods build on those developed by Derickx and Najman for quartic fields and by the author for quintic fields. Among the new ingredients are a formal immersion criterion at the prime $2$ which uses the Cartier operator, and arguments with several diamond operators and with cuspidal divisor classes on quotients of modular curves.
Let $F$ be a totally real field, abelian over $\Q$, in which an odd prime $p$ splits completely, and let $\rho:G_F\to\GL_2(\overline\Q_p)$ be an odd irreducible representation which is de Rham with distinct Hodge--Tate weights at the places above $p$ and whose residual representation is reducible. We consider the \emph{mixed} case of the Fontaine--Mazur conjecture, in which $\rho$ is reducible at some places above $p$ and irreducible at the others. This case is not covered by the work of Pan and Zhang. We prove some instances of modularity in this situation. Our main result is that $\rho$ is modular if it is non-split at every place above $p$ where it is reducible, by combining Zhang's pro-modularity theorem with a classicality criterion based on Emerton's ordinary parts. Together with a theorem of Zhao on local indecomposability, this gives the modularity of the $\lambda$-adic representations attached to abelian varieties of $\GL_2$-type in the mixed case, and, using a result of Snowden, of potentially modular representations of parallel weight two under a mild additional condition. It also gives the modularity of representations whose local type at every place above $p$ is of discrete series type, and, through a member of the compatible system with interchanged Hodge-Tate weights, of potentially modular representations of weight $(2,k)$ over real quadratic fields with Steinberg type at one place above $p$, under conditions on that member.
In a famous paper of Andrews, he defined and studied the smallest parts function $\operatorname{spt}(n)$. He showed that it satisfies the three Ramanujan-type congruences: \[ \operatorname{spt}(5n+4)\equiv0\pmod5,\qquad \operatorname{spt}(7n+5)\equiv0\pmod7,\qquad \operatorname{spt}(13n+6)\equiv0\pmod{13}. \] In a recent preprint, the author showed that these are the only "canonical'' congruences. Here, we complete the classification of Ramanujan-type congruences for $\operatorname{spt}$, showing that if $\operatorname{spt}(\ell n+b)\equiv0\pmod\ell$ for all $n\geq0$, where $\ell\geq5$ is prime, then $(\ell,b)\in\{(5,4),(7,5),(13,6)\}.$ That is, Andrews' congruences are the only Ramanujan-type congruences for $\operatorname{spt}(n)$. These results have been verified in Lean by AxiomProver.
We establish an analogue of the Hilbert irreducibility theorem for all K3 surfaces over number fields that admit an automorphism of positive entropy. Our method draws ingredients from both algebraic geometry and complex dynamics.
For $\alpha\in\mathbb{R}$ and $k\in\mathbb{N}$, let $S_{\alpha,k}(x):=\#\{n\in\mathbb N:n\le x,\ \gcd(n,\lfloor\alpha n^k\rfloor)=1\}$. Roth conjectured, and Watson proved, that $S_{\alpha,1}(x)\sim 6x/\pi^2$ for every irrational $\alpha$. Whether the same asymptotic holds for every fixed $k\ge2$ is a special case of a question of Bergelson and Richter, and was asked explicitly by Champagne. We answer this for $k=2$, with no Diophantine condition on $\alpha$: for every irrational $\alpha$, $S_{\alpha,2}(x)\sim x/\zeta(2)$ as $x\to\infty$. More precisely, for every fixed $A>0$ we obtain an error term $O_A(x/(\log q(x))^A)$, where $q(x)$ is the largest continued-fraction convergent denominator of $\alpha$ not exceeding $x$. The proof sieves $S_{\alpha,2}(x)$ over divisors, splitting at the parameter $R=q(x)^{1/8}$. Divisors larger than $R$ are handled by reducing the condition $d\mid\gcd(n,\lfloor\alpha n^2\rfloor)$ to M\"obius sums over arithmetic progressions and applying Zhan's uniform estimate for exponential sums twisted by $\mu(n)$ over short intervals; divisors at most $R$ are handled instead through the Erd\H{o}s-Tur\'an inequality together with Weyl-type bounds for quadratic exponential sums of the form $\sum e(\beta n^2)$.
Recently, Ono established a new connection between partition congruences and the arithmetic of elliptic curves by expressing the partition function $p(n)$ as a CM trace on the modular curve $X_0(6)$. This approach yields new proofs of Ramanujan-type congruences for the partition function modulo powers of $5$, $7$, and $11$, revealing how the geometry of supersingular elliptic curves and the arithmetic of oriented optimal embeddings govern these congruence properties. Motivated by Ono's work, we develop an analogous passage from CM traces to congruences modulo $5$, $7$, and $13$ for the for Andrews's smallest-parts partition function $spt(n)$, employing the established theta-lift and modular-trace methods suited to this setting. With $P$ the Bruinier--Ono weight-zero weak Maass function and $f$ an explicit weakly holomorphic modular function on $\Gamma_0(6)$, the Ahlgren--Andersen formula expresses $spt(n)$ as one twelfth of the discriminant $1-24n$ CM trace of $f-P$. We explain the theta-lift normalization, the contribution of the two orientations, and the harmonic-Maass-form uniqueness argument. Following Garvan's work, we construct integral forms $G_\ell$ of weight $\ell+1$ on $\Gamma_0(\ell)$, compute their Fricke transformations, and prove the exact identities $$ \mathcal T_5(G_5)=-5E_6,\qquad \mathcal T_7(G_7)=-7E_8,\qquad \mathcal T_{13}(G_{13})=13E_{14}. $$ Their reductions yield the three congruences on the Andrews's smallest-parts partition function. We also give a supersingular specialization of the trace of $f$, with actual reduction multiplicities, and explain the role of imprimitive forms, the conductor restriction in CM reduction, and the distinction between the primes $11$ and $13$. These results have been verified in Lean using AxiomProver.
Let $f\in\mathbb Z[x]$ be a polynomial of degree $n$ with integer coefficients of absolute value less than $2^b$. Using Pan's approximate factorization theorem, we give a deterministic algorithm that finds all distinct integer roots of $f$ in $\widetilde O(nb)$ bit operations. This matches the bit size of the input up to logarithmic factors and does not require $f$ to be square-free. Since we were unable to verify parts of the running time analysis of Pan's theorem, we also give a self-contained deterministic algorithm with running time $\widetilde O(nb+\min(n,b)^3b^2)$, which is $\widetilde O(nb)$ whenever $b\leq n^{1/4}$. This answers a question of Harvey and Hittmeir (2022), who asked whether integer roots can be found deterministically in quasilinear time, conditionally on Pan's theorem, and unconditionally for $b\leq n^{1/4}$. The fastest previous deterministic algorithm, due to Nir, takes $\widetilde O(n^{3/2}b)$ bit operations and works only for square-free polynomials. Instead of $p$-adic lifting, we combine approximate complex root finding with exact polynomial evaluation. We first approximate the complex roots of the polynomial formed by the top $\min(n,b)+1$ coefficients of $f$. Every integer root $r$ of $f$ with $|r|\geq 2$ lies within distance $2$ of some complex root of this truncated polynomial. Rounding the real parts of the approximate roots therefore gives a list of $O(\min(n,b))$ candidate integers that is guaranteed to contain all integer roots of $f$. Although a single candidate can have up to $b$ bits, a bound via the Mahler measure shows that the bit lengths of all candidates sum to $O(b)$. This allows us to test all candidates by exact multipoint evaluation in $\widetilde O(nb)$ bit operations.
We give a counting formula for the coefficients $a\in\F_{p^h}$ for which $x^{2n}+ax^n+1$ is irreducible in $\F_{p^h}[x;\sigma]$, for arbitrary positive integers $n$ and $h$, where $\sigma(a)=a^p$ and $p$ is an arbitrary prime power. The proof reduces the problem to quadratic reduced norms, establishes a uniformity statement for every Frobenius generator, and counts the remaining coefficients by M\"obius inversion.
In this paper, we study binary 3 x 3 matrices with determinant one that generate the Tribonacci recurrence. Starting from a specific generating matrix, we obtain explicit expressions for its powers and inverse powers in terms of the Tribonacci sequence T_n and two associated sequences B_n and R_n. This matrix framework yields identities involving traces, determinants, and products, including addition formulas and Cassini-type relations linking T_n, B_n, and Rn. Moreover, we derive new identities from the matrix representation of inverse powers and establish structural relations between these sequences. As an application, we investigate arithmetic properties of Tribonacci numbers and determine the exact 2-adic valuation of T_2^k for sufficiently large k. Finally, we provide a classification of 3 x 3 matrices associated with the Tribonacci recurrence via similarity over Q.
In a recent paper of the same title, we considered nondistinct partitions of integers over the Fibonacci sequence and related linearly recurrent sequences. In this paper, continuing this story, we apply the same method, but consider distinct partitions. In particular, let $q_F(m)$ denote the number of ways of writing $m$ as the sum of distinct Fibonacci numbers. It is well-known that, while $q_F(m)$ behaves wildly, its partial sums behave in a more structured way. In particular, Kempton showed that the function $N^{-\log 2/\log\varphi}\sum_{m\leqslant N}q_F(m)$ is asymptotically logarithmically-periodic. In this paper, we improve on this result by providing an explicit error term. Additionally, we prove the analogous result for a class of more general linear recurrent sequences.
Let $C>3$ be an odd square-free integer, $q$ an odd prime not dividing the class number of $\mathbb Q(\sqrt{-C})$, and $p_1,\dots,p_k$ distinct odd primes not dividing $C$. We describe all solutions of $Cx^2+p_1^{2m_1}\cdots p_k^{2m_k}=y^q$ in positive integers with $\gcd(x,y)=1$ and $y$ odd. Writing $N=p_1^{m_1}\cdots p_k^{m_k}$, every solution comes from a unique factorization $x\sqrt C+N\sqrt{-1}=\pm(u\sqrt C\pm v\sqrt{-1})^q$, or from a half-integral analogue when $q=3$, in which each $p_i\ne q$ divides exactly one of $v$ and $N/v$. Using primitive divisors of Lehmer sequences, we show that for $q\ge5$ some $p_i$ satisfies $p_i\equiv\leg{-C}{p_i}\pmod q$; in particular, there is no solution when $q\ge\max(5,p_1+2,\dots,p_k+2)$. For $q=3$ the equation reduces to four explicit auxiliary equations.
We study the asymptotic behaviour of the mean square of long truncations of the Dirichlet series for $\bigl(\zeta'/\zeta\bigr)^{k}$ on the critical line; we establish the connection between this problem and the short-interval variance of the convoluted von Mangoldt function $\Lambda^{*k}$ in short intervals and in arithmetic progressions. Under a shifted moments conjecture and the Riemann Hypothesis (RH), we prove that the short-interval variance leading coefficient is the derivative of the mean-square leading coefficient. A unitary trace-sum average governs the variance of sums of $\Lambda^{*k}$ over monic polynomials in $\mathbb F_q[T]$; we denote its leading coefficient by $\delta_k(c)$, where the total trace degree and matrix size grow with a fixed ratio $c$. Under a weaker shifted moments conjecture and RH, we recover this $\delta_k(c)$ as the leading coefficient of the variance of short-interval sums of $\Lambda^{*k}$ over the integers. It is the counterpart of $\gamma_k(c)$, the piecewise polynomial from the variance of divisor sums, with the Dirichlet coefficients of $\zeta(s)^k$ replaced by those of $\bigl(-\zeta'/\zeta\bigr)^k$.
Andrews and El Bachraoui (\textit{RACSAM} \textbf{119}, \textit{Article Number 4 (2025)}) established identities relating $4$-regular partitions to partitions with distinct even parts whose largest part is odd. As an extension to their work, in this paper we define partitions of a positive integer $n$ with bounded multiplicity restriction on even parts and whose largest part is odd. We prove some identities relating our partition functions and $2r$-regular partition function, and show that identities due to Andrews and El Bachraoui follow as particular cases.
A cubic partition of a positive integer $n$ is a partition of $n$ in which even parts may appear in two distinct colors. The purpose of this paper is to study the arithmetic functions $COS(n)$ and $CES(n)$, which denote the sums of all odd and even parts, respectively, in all cubic partitions of $n$. We obtain their generating functions and divisor-sum formulas, and establish Ramanujan-type congruences modulo $3$, $4$, $5$, and $7$. To prove these congruences, we use Lambert series, classical theta-function identities, dissections, coefficient extraction, and weighted Euler identities.
We prove a power-saving form of the van der Corput property for cubes. As a consequence, we obtain a power-saving bound $|A|\ll_\varepsilon N^{59/60+\varepsilon}$ for sets $A\subseteq\{1,\ldots,N\}$ with no nonzero cube differences.
The ensemble average of Narain conformal field theories over their moduli space is a real-analytic Eisenstein series, whose Poincar\'e expansion can be interpreted as a sum over lens spaces weighted by the partition functions of Abelian Chern-Simons theory. In this paper we extend this correspondence from lens spaces to Seifert manifolds, which gives an example of a restricted sum over topologies of the kind that arises in three-dimensional quantum gravity. A Seifert manifold is obtained from a fixed core, a punctured surface times a circle, by gluing solid tori to its boundary tori, and we place a copy of the Narain theory on each of these tori. The ensemble average supplies the sum over the solid tori, or more precisely over marked presentations with fixed base genus, number of fibers and integer Seifert invariant. The resulting sum converges absolutely and is a polynomial in the components of the vector-valued Eisenstein series. When all the boundary tori share a single modulus, the average acquires connected corrections, which we identify with wormholes gluing the core to itself; for two boundary tori, the terms with rational Lagrangian sublattices of the Siegel-Eisenstein series are realized by wormholes containing an exceptional fiber. After reduction along the fiber, the construction closely parallels the two-dimensional topological gravity of Marolf and Maxfield, with the base of the fibration playing the role of their spacetime. We also discuss Dehn fillings of 3-manifolds with torus boundary, non-orientable bases, and the surgery formula. Our results suggest that modularity could be an organizing principle in the summation of geometries in quantum gravity.
We investigate the structure of locally analytic representations of $p$-adic groups, and establish a sufficient condition for the irreducibility of locally analytic vectors in certain Banach representations of $D^\times$, where $D$ is the quaternion division algebra over $\mathbb{Q}_p$. As an application, we prove that $\mathrm{JL}(\rho)^{\mathrm{an}}/\mathrm{JL}(\rho)^{1\text{-}\mathrm{alg}}$ is a topologically irreducible locally analytic representation of $D^\times$ under certain genericity conditions on $\rho$. Here $\mathrm{JL}(\rho)$ is the $\rho(-1)$-multiplicity space in the image of $\Pi(\rho)$ under Scholze's functor, and $\Pi(\rho)$ is the representation associated to $\rho$ by the $p$-adic local Langlands correspondence.
For $n\ge2$, a good slope of a permutation $f$ of $\mathbb{Z}_n=\mathbb{Z}/n\mathbb{Z}$ is an element $m\in\mathbb{Z}_n$ for which $x\mapsto f(x)+mx$ is a permutation. We call $f$ congruence-preserving if it preserves congruences modulo every divisor of $n$. For odd prime powers $n$, we classify all permutations with more than $n/2$ good slopes. Among permutations that are not congruence-preserving, we determine the maximum number of good slopes for $n=3^e$ with $e\ge2$, and for $n=25$, $125$ and $625$. Every permutation of $\mathbb{Z}_n$ has at most $\varphi(n)$ good slopes, where $\varphi$ is Euler's totient function. For several families of moduli with at least two distinct prime divisors, we prove a sharp upper bound for the number of good slopes of the permutations with fewer than $\varphi(n)$ good slopes. These families include $9q$ and $3q^e$ for every prime $q\ge5$ and every $e\ge1$. Together with a companion paper, this settles the bound for every odd composite modulus below $105$.
For every \(c>1\) sufficiently close to \(1\), we prove that there is \(\kappa>0\) such that every sufficiently large \(A\subset[N]\) of density at least \((\log N)^{-\kappa}\) contains \[ x,\ x+(p-1),\ x+(p-1)^2,\ldots,\ x+(p-1)^k, \qquad p=\lfloor n^c\rfloor\in\mathbb P. \] We also prove pointwise almost-everywhere convergence of the corresponding multiple ergodic averages.
Let $d \in \mathbb{N}$ and let $A \subseteq \mathbb{R}^d$ be a finite, non-empty set not contained in any translate of a hyperplane. We show that there exists a subset $S \subseteq A$ such that $d \ll |S| \ll d$ and \[E(A,S) := |\{(a_1,a_2,s_1,s_2)\in A\times A\times S\times S:a_1+s_1=a_2+s_2\}|\ll\lvert A\rvert\lvert S\rvert.\] This can be interpreted as a robust, analytic generalisation of the classical Freiman's lemma, and recovers Freiman's lemma up to a constant multiplicative factor. In the process of proving this result, we also show that for any integer $1 \leq m \leq d$ and any set $A \subset \mathbb{R}^d$ with $|A| \leq \binom{d}{m}$, one has \[ E(A, B) \leq 4(m + 1/m) |A||B|,\] where $B$ is the canonical basis of $\mathbb{Z}^d$. This gives new estimates for multiplicative energies between finite sets of primes and arbitrary subsets of positive integers, improving a previous result of Shkredov. Using related ideas, we also prove new bounds for an edge-isoperimetric problem for certain graphs on the boolean cube, improving an earlier result of Kirshner and Samorodnitsky. We give two different proofs of the first result described in this abstract and three different proofs of the second one.
We describe an algorithm and a computer program computing the relative Tate-Shafarevich kernel $Sha^S(K,G)$ of a reductive group $G$ over a global field $K$ with respect to a finite set $S$ of places of $K$ containing no archimedean places. We assume that we know the algebraic fundamental group $\Lambda=\pi_1^{alg}(G)$, the Galois group $\Theta=Gal(L/K)$ acting on $\Lambda$, where $L$ is a finite Galois extension of $K$ having no real places, the decomposition subgroups $\Theta_v\subseteq \Theta$ for all places $v$ in $S$, and all non-cyclic decomposition subgroups $\Theta_v$ for $v$ outside $S$. We also compute $Sha^S(K,G)$ in some cases where $S$ contains archimedean places.
Hecke operators can be used to relate characters of rational conformal field theories with different central charges, but it has remained a mystery how to interpret these relations physically. We offer a simple explanation in terms of self-duality under asymmetric orbifolds. Self-dual orbifolds also give more general identities beyond the familiar Harvey-Wu construction, which we illustrate through examples.
Using adelic intersection theory and following the method envisioned by Parshin and Moret-Bailly, we prove a conjecture of Vojta for curves of genus two and a weak form of the $abc$ conjecture