Papers

Let $K$ be a number field. We give new practical algorithms that determine whether $K$ is $p$-rational. For real cyclotomic fields $K=\mathbb{Q}(\zeta_{n})^{+}$, our method avoids computing either the class number or the full unit group. We also use the notion of $p$-rationality to explain the practical efficiency of an algorithm for determining whether a subgroup of the unit group $\mathcal{O}_{K}^{\times}$ is $p$-saturated. This in turn yields a new algorithm for the unconditional verification of unit groups of number fields that substantially outperforms existing unconditional algorithms in practice. Finally, by combining our algorithms for determining $p$-rationality with work of Greenberg, we construct certain Galois representations $\rho : \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathrm{GL}_{n}(\mathbb{Z}_{p})$ with open image and further prescribed properties.

For an integer $n > 1$, let $P^+(n)$ be the largest prime factor of $n$, and let $P_y^+(n)$ denote the largest prime factor of $n$ not exceeding $y$. One of Erdős and Turán's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is $1/2$. Furthermore, Wang conjectures that for $x\rightarrow\infty$, one has $\#\{p\leq x:P^+(p-1)<P^+(p+1)\}\sim \pi(x)/2$. In this paper, we show the following results. For any $3\leq y\ll x^{o(1)}$, for $x\rightarrow\infty$, we have \begin{align*} &\#\{n\leq x:P^+_y(n)<P^+_y(n+1)\}\sim \frac{1}{2}x, \&\#\{p\leq x:P^+_y(p-1)<P^+_y(p+1)\}\sim \frac{1}{2}\pi(x). \end{align*} For any $0<\alpha<17/32$, let $y=x^\alpha$. Then there exists $h(\alpha)>0$ such that \begin{align*} \#\{p\leq x:P_y^+(p-1)<P_y^+(p+1)\}\geq(h(\alpha)+o(1))\pi(x). \end{align*} In particular, the function $h$ satisfies $\lim_{\alpha\rightarrow 0^+}h(\alpha)=1/2$. Similar result also holds for $\#\{n\leq x:P_y^+(n)<P_y^+(n+1)\}$. These improve Rivat's result (2001) and Wang's result (2019).

We prove that $B_kA_6$ is stably rational over every field $k$ in which $2$ and $-3$ are nonzero squares. Combining this with a theorem of Plans, we deduce that $B_kA_7$ is stably rational over every field $k$ of characteristic zero in which $2$ and $-3$ are squares.

We establish the dynamical Mordell–Lang conjecture over \(\mathbb C\) for endomorphisms that are relatively étale over arbitrary endomorphisms of finite products of smooth projective curves. In particular, we establish the conjecture for arbitrary endomorphisms of product of smooth projective curves and relatively étale skew products. The arithmetic input is our unconditional theorem for split endomorphisms over \(\overline{\mathbb Q}\), proved by reducing to a simultaneous Hasse principle for the local periods of critical points and analyzing local inertia in joint arboreal towers. We combine this arithmetic input with relative étaleness, simultaneous algebraic specialization preserving wandering base coordinates, prescribed-reduction \(p\)-adic embeddings, and \(p\)-adic interpolation to obtain the result in the complex case.

In this paper, we derive thresholds and fluctuations for arithmetic progressions with prescribed color patterns in sparse random colorings of $[n]:=\{1, 2, \ldots, n\}$, where each element of $[n]$ is colored independently according to a given probability vector. For any admissible ordered palette of colors, we determine the full multi-parameter threshold region for the appearance of a colorful arithmetic progression. The threshold is governed by two competing mechanisms: a global first-moment condition and a local color-availability condition, resulting in a polyhedral satisfiability region, with a piecewise-polyhedral threshold surface. In the satisfiability region we establish asymptotic normality for the number of colorful arithmetic progressions of a given length, with an explicit rate of convergence in Wasserstein distance. On the threshold surface, we identify three distinct asymptotic regimes: Poisson, compound Poisson with mixed Poisson jumps, and compound Poisson with uniform jumps, after an appropriate normalization. These results provide a complete description of the threshold and fluctuation behavior of general colored arithmetic progressions under sparse random colorings, in a unified framework that interpolates between classical uncolored/monochromatic progressions in binomial random subsets and multicolored, including rainbow, arithmetic progressions.

One of the remarkable contributions of Don Zagier was the Kronecker limit formula for a real quadratic field, where he connects the double series $\mathcal{Z}(s,w,w^\prime)$ to the Dedekind zeta function associated to a real quadratic field. Later, Ishibashi determined all the Laurent coefficients of $\mathcal{Z}(s,w,w^\prime)$ at $s=1$. Recently, Choie and kumar have studied the analytic behaviour of the analogous double series $\tilde{\mathcal{Z}}(s,w,w^\prime)$. In this article, we derive all the Laurent coefficients of $\tilde{\mathcal{Z}}(s,w,w^\prime)$, akin to Ishibashi. These Laurent coefficients involve an interesting function $\mathfrak{F}_k^0(x)$, which was earlier studied by Dixit et. al. (Ramanujan for $k=1$), where they obtained a beautiful symmetric relation for $\mathfrak{F}_k^0(x)$. We establish both the two term and the three term functional equation of $\mathfrak{F}_k^0(x)$, derive the action of the period-like Hecke operator on $\mathfrak{F}_k^0(x)$ and connect an important integral with $\mathfrak{F}_k^0(x)$.

We give an explicit combinatorial criterion for an Ekedahl-Oort (EO) stratum in the moduli stack of principally polarized abelian varieties of dimension $g$ in characteristic $p > 2$ to meet the supersingular locus. The criterion is obtained by translating a non-emptiness criterion for basic affine Deligne-Lusztig varieties, into an explicit condition on the elementary sequences indexing the EO strata. Furthermore, we give explicit bounds for the number of EO strata of $p$-rank zero which are disjoint from the supersingular locus. As a result, we prove that asymptotically almost every EO stratum of $p$-rank zero meets the supersingular locus. As an application, we also give a lower bound for the number of EO strata meeting the supersingular locus on unitary PEL Shimura varieties of signature $(a,b)$.

We study the arithmetic constraints that two-term Rogers dilogarithm relations impose on rank-two Nahm systems with mixed denominator steps. Monomial complement equations give a uniform conversion to quadratic data in every algebraic degree. Within a specified ten-record simplest-cubic family, positive coupling, positive definiteness, and integer-valued exponents leave, for m > 2, exactly the steps (1,3), (1,13), and (1,31). At the smallest compatible modulus 31, the required support cannot be a multiplicative subgroup coset. Quadratic examples include known product identities and two systems for which the first radial correction excludes every individual modular sum with rational linear terms. For the index-13 system, we prove a two-term Rogers identity and recover its arguments from the matrix of a previously proved two-dimensional Fricke transformation. Its cusp growth agrees with the resulting saddle action, and an independent calculation reproduces all three leading Fricke amplitudes exactly. The associated two-summand product evaluations remain conjectural; the small rank and dimension contrast with the classical rank-five, six-component Andrews-Gordon construction at the same modulus. A mixed-signature cubic family illustrates the limits of the arithmetic restriction

Let $\mathcal G\to S$ be a semiabelian scheme over a smooth complex algebraic base, with abelian relative dimension $g$ and toric rank $r$. We prove that a dominating irreducible subvariety not contained in a proper relative special subvariety has Zariski-dense fibrewise torsion if and only if its mixed Betti rank is $2(g+r)$. In particular, its dimension is at least $g+r$. For every prime, primary torsion is then dense even among points of maximal rank. The exceptional loci include generalized Ribet loci. Over algebraic numbers, we prove that generic small points force full mixed rank whenever the abelian Betti rank is maximal. This yields a transfer from abelian relative Bogomolov dimension theorems to arbitrary toric extensions, together with toric height gaps above abelian torsion and order-linear gluing-height bounds for nodal Pell equations. The proof combines algebraic quotients of marked $1$-motives, quadratic nef compensation, a central-monodromy calculation, and algebraicity of mixed Betti strata.

Let $\mathcal{H}_k$ denote the set of normalized holomorphic Hecke cusp forms of weight $k$ for the full modular group $ \mathrm{SL}_2(\mathbb Z)$. For each $f\in\mathcal{H}_k$, let $\lambda_f(n)$ denote the corresponding eigenvalue of the normalized Hecke operator $T_n$. We prove nontrivial upper bounds for \[\sum_{f \in \mathcal{H}_k}\omega_f \left|\sum_{n \leq x} \lambda_f(n)\right|^{2q},\] where $k$ is a positive even integer, $1\leqslant x\leqslant k$, $0\leqslant q \leqslant 1$ and $ \omega_f = \frac{\Gamma(k-1)} {(4\pi)^{k-1}\langle f,f\rangle} = \frac{2\pi^2} {(k-1)L(1,\mathrm{Sym}^2 f)}. $ Our estimates match the conjecturally sharp upper bound for $1\leqslant x\leqslant k^{0.99}$. In particular, whenever both $x$ and $k/x$ tend to infinity with $k$, we obtain \[\sum_{f \in \mathcal{H}_k}\omega_f \left|\sum_{n \leq x} \lambda_f(n)\right|=o(\sqrt{x}).\] We also prove upper bounds for low moments of sums of Hecke eigenvalues associated with Hecke–Maass cusp forms for $\mathrm{SL}_2(\mathbb Z)$. We study these sums through the probabilistic model proposed by Cogdell–Michel. We also determine the order of magnitude of low moments of this probabilistic model, which is equivalent to determining the order of magnitude of low moments of Hecke eigenvalue sums in the limit.

We prove Sylvester's conjecture, originating in his 1879 study of ternary cubic equations, that every prime $p\equiv4,7,8\pmod9$ is a sum of two rational cubes. Elkies announced a proof for the classes $4$ and $7$ in 1994, and Yin recently supplied a complete proof. For the remaining class $p\equiv8\pmod9$, we prove that the elliptic curve $E_p:y^2=x^3+p^2/4$ has analytic rank one, as predicted by the Birch and Swinnerton-Dyer conjecture, and so $p$ is a sum of two rational cubes. The proof begins by adapting the auxiliary Rankin–Selberg construction from the authors' work on the rank one converse for CM elliptic curves. The Rankin–Selberg $L$-function factors as the $L$-function of $E_p$ times a complementary $L$-function. Chan's $3$-isogeny descent and the rank zero converse show that the complementary central $L$-value is non-zero, and so it suffices to prove that a cubic component of the associated Heegner point is non-torsion. A basic difficulty is that the unweighted Hecke trace of the underlying CM orbit vanishes. Our decisive idea is to take $\lambda$-division before taking the trace, where $\lambda=1-\omega$ and $\omega$ is a primitive cube root of unity. We prove that the resulting division boundary is non-zero by analysing Frobenius at $p$. The Galois action on the CM orbit and ramification theory then transfer this non-vanishing to the cubic component.

We show that there exist infinitely many polynomials $F\in\mathbb{Q}[x]$ with exactly $m$ nonzero coefficients such that every irreducible factor of \(F\) over \(\mathbb{Q}\) has \(\exp\) \(\!\bigl(\Omega(\sqrt{m/\log m})\bigr)\) nonzero coefficients. This places multiplication in a sharply different category from powers, composition, and other algebraic operations for which reverse-sparsity principles are known. It also gives an unconditional, degree-free obstruction to sparse factorization, complementing positive factor-sparsity results that impose hypotheses on coefficient height, exponent positions, reciprocal structure, or degree bounds.

We prove the transcendence over $\mathbb{Q}$ of the image of Montgomery reduction factor $R'$ in the ring $\mathscr{A}$ of integers modulo infinitely large primes. Here, for an odd prime number $p$, $R'$ is defined as the modular inverse of a power $R$ of $2$ modulo $p$ satisfying $2^{-k} R \leq p < R$ for a fixed constant $k \in \mathbb{N}_{> 0}$ typically given as the standard bit size $32$ of an integer type, and is the element of $\mathbb{Z}/p \mathbb{Z}$ representing Montgomery reduction regarded as a $\mathbb{Z}/p \mathbb{Z}$-linear homomorphism.

Lee and O'Clarey recently conjectured that every integer $n>24$ can be written as the sum of a prime and a positive integer that is not square-free. They proved this for odd $n$, and for all $n>24$ assuming the generalised Riemann hypothesis for Dirichlet $L$-functions. We prove their conjecture unconditionally for every integer $n>24$ that is not divisible by $997\#$, where $997\#=\prod_{p\leq 997}p$. In particular, any counterexample must be divisible by every prime up to $997$, and hence exceeds $10^{415}$. The proof uses an elementary congruence argument together with finite computation.

Let $G$ be a countable locally finite abelian group, and let \[ G_1\leq G_2\leq\cdots, \qquad \bigcup_{i\geq1}G_i=G, \] be any filtration of $G$ by finite subgroups. For $A\subseteq G$, let $\mathcal P(A)$ denote the set of all finite subset sums of elements of $A$, and let $2G:=\{2g:g\in G\}$. We prove that $|2G|=\infty$ if and only if for every filtration and every interval $[\alpha,\beta]\subseteq[0,1]$, there exists $A\subseteq G$ such that the set of limit points of \[ \left( \frac{|\mathcal P(A)\cap G_i|}{|G_i|} \right)_{i\geq1} \] is exactly $[\alpha,\beta]$. We also prove a positive-density result that does not require $|2G|=\infty$: if $G$ is infinite and $\mathcal P(A)$ has positive upper density along the given filtration, then there is an infinite set $B\subseteq \mathcal P(A)$ such that $B+B\subseteq \mathcal P(A)$. Combining this with the realization theorem, we show that, when $|2G|=\infty$, every interval $[\alpha,\beta]$ with $0\leq\alpha\leq\beta\leq1$ and $\beta>0$ can be realized by a set $A$ with this additional property.

Let $p$ be an odd prime, and write $A\RS B=\{a+b:a\in A,\ b\in B,\ a\ne b\}$. We classify all pairs $A,B\subseteq\Fp$ satisfying $|A|+|B|=p$, $|A|>|B|=\ell\ge1$, and $|A\RS B|=p-2$. After normalizing the two missing sums to $\{0,1\}$, we obtain exactly $\ell+1$ explicit models. The classification forces $B\subseteq A$ and yields, for fixed $p,\ell$, exactly $\binom p2(\ell+1)$ ordered pairs and $1+\lfloor\ell/2\rfloor$ equivalence classes under simultaneous affine transformations. We determine when either set is an arithmetic progression and compute exact difference-set cardinalities. For $\ell\ge3$ and $p\ge4\ell-5$, the cardinality $|B-B|$ determines the equivalence class at fixed $p,\ell$. The proof uses punctured translates and cyclic component counting. We also exhibit a critical pair in $\mathbb F_{13}$, with size gap three and $|A|+|B|=p-2$, that contradicts a proposed inverse statement below the boundary.

Let $\bar{p}_{s,t}(N)$ be the number of overpartitions of $N$ into parts that are not divisible by $s$ or $t$. In this paper, we obtain asymptotic formulas for $\bar{p}_{s,t}(N)$ for relatively prime integers $s>t\ge 10$ via the saddle-point method. Our results cover different ranges of the parameters $s$ and $t$ relative to $N$. As an application, we show that $\bar{p}_{s,t}(N)$ satisfies higher-order Turán inequalities for all sufficiently large $N$ by using a result of Griffin, Ono, Rolen, and Zagier.

Recently, René Peschmann proved perfect-cuboid nonexistence on 1,072 explicit master-tuple fibers with $\max(m,n)\le100$, leaving 968 fibers in the same bounded parameter range. In this paper, I eliminate all 968 remaining fibers. Most fibers fall to fairly short rank-zero or covering arguments, while the final cases need the stronger local-global sieves. Together with Peschmann's theorem this excludes perfect cuboids on all 2,040 admissible fibers with $m\le100$; the result is still a bounded-fiber theorem rather than a proof of the full perfect-cuboid conjecture.

In 1965, Chowla conjectured that the signs of the Liouville function become asymptotically uncorrelated at any fixed collection of distinct shifts. In 2015, Matomäki, Radziwiłł and Tao proved an averaged form of Chowla's conjecture. In 2022, Lichtman proved a variant of this conjecture over primes on average. In the same year, Lichtman and Teräväinen proved the Hardy–Littlewood–Chowla conjecture on average. In this article, motivated by the recent work of Bergelson and Richter on the dynamical generalizations of the prime number theorem, we will establish the dynamical generalizations of these three results related to Chowla's conjecture. In the proofs, we will show a uniform pretentious-distance estimate on the prime Omega function. Then exponential sum estimates are used to establish the averaged shift invariance of the distributions related to the shifted sum of the prime Omega function.

For every negative fundamental discriminant $-f$ and every $k\geq1$, we construct a rational function $R_{f,2k}\in\mathbb{Q}(x_1,\ldots,x_{2k})$ and a constant $r_{f,2k}\in\mathbb{Q}^\times$ such that \[ m(R_{f,2k})=r_{f,2k}L'(\chi_{-f},1-2k), \] where $m$ denotes the logarithmic Mahler measure and $\chi_{-f}$ is the quadratic Dirichlet character associated with $-f$. This proves the weak Chinburg conjecture. An independent construction using Bloch cycles yields a stronger result in two variables: there exists a nonzero polynomial $R_f\in\mathbb{Q}[x,y]$ satisfying \[ m(R_f)=4wL'(\chi_{-f},-1), \] where $w$ is the number of roots of unity in $\mathbb{Q}(\sqrt{-f})$.

Assume the Generalised Riemann Hypothesis. If $A\subseteq \{1,...,N\}$ contains no two elements differing by $p+1$, $p$ prime, then $|A|\ll N^{\frac{89}{90} + o(1)}$.

Let $q$ be odd. The trace conic in $\mathbb{F}_{q^3}$ determines a Singer difference set in $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^\times$ and a natural square-class lift to $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^{\times 2}$. We study the odd multiplicative Fourier coefficients of this lift and prove that they are bounded in absolute value by $2\sqrt{q}$. The proof improves the naive six-puncture Weil bound by exploiting a projective Klein-four symmetry of the associated rank-one local system. The resulting nontrivial cocycle produces a quaternionic action on its four-dimensional cohomology, while Frobenius symmetry reduces the relevant trace to two Weil-scale eigenvalues. As an application, the oriented conic yields an explicit Singer-invariant signing of the point-line incidence graph of $\mathrm{PG}(2, q)$. The corresponding dihedral Cayley graph is a connected Ramanujan double cover. Thus a conic lift already known in finite-geometric constructions has an additional Ramanujan spectral property governed by its odd multiplicative character sums.

Let $\mathbb{F}_q$ be a finite field of odd characteristic with $q \equiv 2 \pmod3$. We study the directed graph defined by the Borwein–Borwein cubic arithmetic–geometric mean (AGM) over $\mathbb{F}_q$. We prove that this graph is a disjoint union of directed cycles. We associate Hessian curves with this AGM. We also show that each edge corresponds to a $3$-isogeny defined over $\mathbb{F}_q$ between the curves associated with its initial and terminal vertices. We then use a counting formula for Hessian curves to derive a lower bound for the number of cycles.

We develop an analytic framework to study hyperbolicity of Jensen polynomials and related inequalities for arithmetic sequences with general asymptotic growth. Our results provide partial converses to the Hermite–Jensen phenomenon of Griffin, Ono, Zagier, and the third author. As applications we prove several conjectures concerning log-concavity, higher-order Turan inequalities, Laguerre inequalities, infinite log-concavity, and Toeplitz determinants for partition-type functions.

For each fixed integer $K\ge2$, we consider the Enots–Wolley sequence in which every term after the initial $1,2$ is required to have between two and $K$ distinct prime divisors. We prove that every integer satisfying this restriction occurs. If an exact prime support $T$ were selected only finitely often, then after a finite cutoff the terms meeting $T$ would form short episodes, and every full term would force an earlier proper term at comparable numerical height. The case $|T|=K$ is then ruled out directly. In the remaining case $|T|<K$, at a record proper value $H$, greediness forces every unblocked rank-$K$ integer below $H$ containing exactly one prime of $T$ to have occurred earlier. Fixed-order Landau estimates show that this proper population has order $H(\log\log H)^{K-2}/\log H$, while the entire possible full population on the same scale has strictly smaller logarithmic order. This contradiction proves surjectivity. The theorem concerns each fixed rank cap and does not settle surjectivity of the unrestricted Enots–Wolley sequence.

We study the distribution of the number of coprime lattice points in randomly translated disks. We prove that its spatial variance is asymptotic to $\sigma^2r\log r$ and establish a central limit theorem under the corresponding normalization, where $\sigma^2$ is an explicit positive constant. By truncating the numerators of rational frequencies in lowest terms, the proof reduces higher-moment estimates to counting squarefree denominators in zero-sum relations.

Let $\mathscr L$ denote the set of Liouville numbers. We prove a local arithmetic rigidity theorem for real-analytic functions. If $U\subset\mathbb R$ is an open interval and $f:U\to\mathbb R$ is real-analytic with $f(\mathscr L\cap U)\subseteq\mathscr L$, then $f$ is the restriction to $U$ of a rational function in $\mathbb R(x)$. Quantitatively, there exists an absolute constant $\tau>2$ such that, for every nonrational real-analytic function $f$ and every nonempty open subinterval $V\subset U$, the set of $\xi\in V\cap\mathscr L$ satisfying $\mu(f(\xi))\leq\tau$ contains a Cantor set. As a consequence, every entire function $F:\mathbb C\to\mathbb C$ satisfying $F(\mathscr L)\subseteq \mathscr L$ is a polynomial with real coefficients. This gives a negative answer to a problem posed by Mahler in 1984. The proof develops a two-height counting estimate for rational approximation to analytic graphs, treating source and target denominators separately, and combines it with a nested construction producing Liouville inputs whose images have uniformly bounded irrationality exponent.

Let N be a sufficiently large odd integer. We study the representation of a large integer m N^k as a sum of m almost equal k-th powers of primes, where the common distribution scale is at least N^theta. Assuming the Generalized Riemann Hypothesis (GRH), we prove that for any epsilon > 0 one may take theta > 1/2, provided that m < k(k+1)/2 and that m, k are odd integers sufficiently large. This is achieved by employing a new tool closely related to earlier work. For sufficiently large m, by refining the discussion of the minor arcs, we further show that under GRH the equation has a solution provided m > 16k log k + 4k + 2. Moreover, we give an asymptotic formula for the number of solutions, with main term involving N^{2k theta (1 - (1 - 1/k)^{m_1}) + theta(m-1) + 1 - k}, where m_1 = 4k. Under GRH and for sufficiently large m, k, the short-interval Waring-Goldbach problem is essentially settled.

Let $s(m)$ be the Conway subprime function define the binary operation on the natural numbers $x \circ y= s(x + y)$, and denote by $C_n$, $n \ge 0$, the sequence of subsets of natural numbers defined by $C_0 = \{1\}$, and $C_{n+1} = C_n \cup (C_n \circ C_n)$. We prove the conjecture by Caragiu, Vicol and Zaki that $$\lim_{n\to \infty} \frac{ | C_{n+1}| }{ | C_n |}= \frac{1+\sqrt{5}}{2}.$$ The underlying mathematical proof in this paper was constructed with some algorithmic assistance from GPT-6 Astra and its correctness has been formally verified using the Lean 4 proof assistant.

In this article, we first prove that the Hecke operator $T_t$ has no eigenform in $\Sla$ with eigenvalue $-t^{k/2}$, when the characteristic of the base field is odd. Furthermore, if $\dim \Slt$ is even, we show that $T_t$ has no eigenform in $\Sla$ with eigenvalue $t^{k/2}$. As a consequence, we prove that the direct sum decomposition $\Slt=\Sold \oplus \Snew$ holds when $\dim \Slt$ is even. This proves the conjecture \cite[Conjecture 1.1(3)]{BV19a} of Bandini and Valentino for an infinite family of cusp forms. In particular, for any weight $k$, there exists at least one type $m$ (there are only two possible non-trivial values of $m$) for which the conjecture \cite[Conjecture 1.1(3)]{BV19a} is true.