Papers

In this paper, we establish the arithmetic purity of strong approximation for smooth geometrically integral cubic hypersurfaces $X\subset \mathbb{P}^n$ over number fields $k$, provided that $n\ge 323$. For $k=\mathbb{Q}$, the bound can be lowered to $n \ge 30$.

Let $K$ be a field of characteristic zero, and let $\phi(x)\in K[x]$ be a polynomial of degree at least 2. Denote the $n$-th iterate of $\phi$ by $\phi^n$. The $n$-th dynatomic polynomial of $\phi$ is defined by $$\Phi_{\phi,n}(x) := \prod_{k\mid n}(\phi^k(x)-x)^{\mu(n/k)}.$$ In this paper, we specialize to the case $\phi(x) = x^d$. We first establish several properties of $\Phi_{\phi,n}$ that are analogous to those of cyclotomic polynomials. We then combine these properties with known results on resultants of cyclotomic polynomials to determine the resultants of dynatomic polynomials. In particular, for $1\leq n< m$, we show that $\operatorname{Res}(\Phi_{\phi,n},\Phi_{\phi,m}) = 1$ if and only if $n\nmid m$. When $n\mid m$, we obtain the explicit formula $$\operatorname{Res}(\Phi_{\phi,n},\Phi_{\phi,m}) = \Phi_{m/n}(d^n)^{\nu_d(n)},$$ where $\Phi_j$ is the $j$-th cyclotomic polynomial, $\nu_d(1) = d-1$, and $\nu_d(n) = \operatorname{deg}\Phi_{\phi,n}$ for $n>1$.

We construct a three-parameter family of rational approximations to values of $q$-hypergeometric series. Using these approximations, we prove that, for every integer $x$ with $|x|\geq2$, the values at $r=x^{-1}$ of Ramanujan's theta function $\psi(r)=\sum_{n\geq0}r^{n(n+1)/2}$, the generating function $\Delta(r)=\sum_{m\geq0}d(2m+1)r^m$ of the divisor function restricted to odd integers, and the generating function $B_4(r)=\sum_{n\geq0}b_4(n)r^n$ for $4$-regular partitions are irrational. We further obtain the upper bounds $18/7$, $18\pi^2/(7\pi^2-24)$, and $3$, respectively, for their irrationality measures. We also show that one of the constructed approximations coincides with the Padé approximation to a Lambert series due to Coussement–Smet.

We determine the Hausdorff and packing dimensions of Cartesian products of one-dimensional exact approximation sets. Our main result establishes the exact-approximation counterpart of the recent product theorem of Wang and Wu (2024) for limsup approximation sets, showing that passing to the substantially smaller exact approximation sets (liminf sets) does not reduce the Hausdorff dimension of the Cartesian product. One of the key ingredients is a refinement of the well-distributed-system framework of Bandi–Ghosh–Nandi (2023) by exploiting the fine arithmetic distribution of rational points which then gives the Hausdorff dimension of the product set under a weaker convergence condition.

Let $f,g:\mathbb{N}\to\mathbb{T}$ be completely multiplicative functions with dense images in the complex unit circle $\mathbb{T}$. We prove that, for every non-empty open set $U \subset \mathbb{T}^2$, the set of integers $n$ such that $(f(n),g(n+1)) \in U$ has positive lower logarithmic density, unless the pair $(f,g)$ is of a special form. This result strengthens earlier theorems of Klurman and Mangerel, as well as of Charamaras, Mountakis, and Tsinas, and yields substantially simpler proofs.

Let $p>3$ be a prime, $E/\mathbb Q$ be an elliptic curve and $K$ an imaginary quadratic field satisfying the hypotheses of Burungale-Castella-Skinner, and let $L/K$ be the unique $\mathbb{Z}_p^2$-extension. In this note, by combining their integral two-variable main conjecture with the specialisation formula of the first-named author, we obtain a characteristic-ideal identity over every $\mathbb{Z}_p$-line in $L/K$, involving an explicit local factor. With the characteristic ideal of a non-torsion module defined to be zero, this identity also applies when the two-variable Perrin-Riou element specialises to zero. We call such lines exceptional and prove that only finitely many occur. We also prove that the cyclotomic line is non-exceptional with trivial local factor, and that the anticyclotomic line is exceptional. Finally, we bound the number of exceptional lines by the cyclotomic augmentation order and prove that if the $p$-primary Tate-Shafarevich group over $K$ is finite and the cyclotomic $p$-adic height pairing is non-degenerate, then this number is at most ${\rm rank}(E(K))$. In particular, when $E(K)$ has rank one, the anticyclotomic line is the unique exceptional line.

We prove that the generic part of the cohomology of Siegel modular varieties with torsion coefficients is concentrated above the middle degree, under a suitable notion of genericity that is optimal in the unramified case. Our method relies on the semi-perversity of the relative cohomology of the Igusa stack and on a trace formula computation that is made possible by a congruence technique introduced by Scholze and Fintzen–Shin. Compared with the work of Yang–Zhu, which handles Shimura varieties of abelian type via categorical local Langlands, our method is adapted to Siegel modular varieties but handles coefficients of arbitrarily small characteristic.

Let $\mathcal{Z}_w^{(\mathbb{F}_q)}$ be the $\mathbb{F}_q$-linear subspace of $\mathbb{F}_q(\!(\theta^{-1})\!)$ spanned by Thakur's multiple zeta values $\zeta_A(\mathfrak{s})$ of weight $w$. We prove that $\sum_{w=1}^{\infty} \left(\dim_{\mathbb{F}_q} \mathcal{Z}_w^{(\mathbb{F}_q)}\right) x^w = \frac{x(1-x^q)(1-2x+x^q)}{(1-2x+x^{q+1})^2}$. Moreover, we construct an explicit $\mathbb{F}_q$-basis of $\mathcal{Z}_w^{(\mathbb{F}_q)}$, and prove that any $\mathbb{F}_q$-linear relation among Carlitz multiple polylogarithm values $\operatorname{Li}_A(\mathfrak{s})$ is an $\mathbb{F}_q$-linear combination of quadruple-carry relations. This result can be regarded as an $\mathbb{F}_q$-analogue of the corresponding $\mathbb{F}_q(\theta)$-theorem proved by Chang–Chen–Mishiba and independently by Im–Kim–Le–Ngo Dac–Pham. Our discovery of the quadruple-carry relations is inspired by the recent work of Im–Kim–Ngo Dac. These relations may be viewed as $\mathbb{F}_q$-analogues of the double-shuffle relations among classical multiple zeta values $\zeta(\mathfrak{s})$.

For fixed $k\ge2$, Garvan's modified $k$-rank moments are encoded by odd elliptic Taylor coefficients of an odd-level Appell function. The completed coefficient of order $2n+1$ has weight $2n+3/2$, so the family cannot be realized as a fixed-weight vector-valued modular form. Nevertheless, all non-holomorphic parts arise from scalar contractions of canonical higher Serre derivatives of a single weight-$3/2$ vector-valued harmonic Maass form. Among lifts with the prescribed unary-theta shadow, the first $d=k-1$ Appell corrections are coordinates on the weakly holomorphic ambiguity and select a unique lift; the next correction is nonzero. The initial Taylor data recover the principal part without using positive Fourier coefficients of the modified moment series, and the principal part determines the shadow. The positive coefficients of the corresponding Maass–Poincaré lift have convergent Rademacher expansions; the canonical lift differs from it by a unique cusp form. This yields exact recursions for all even modified moments. For $k=2$, the cusp-form adjustment vanishes, giving an exact formula for the numbers of $2$-marked Durfee symbols involving a convergent Kloosterman–Bessel series.

We prove that for any given frequency resonance exponent and phase resonance exponent, there exist a frequency and a phase exactly realizing these values such that the Almost Mathieu operator exhibits Anderson localization for $\ln|\lambda|>\max\{\beta(\alpha),\delta(\alpha,\theta)\}$. This resolves the conjecture in \cite{MR4686650}.

We construct period polynomial relations for finite systems of theta products stable under modular transformations and use them to derive identities among critical $L$-values. Our main example is a three-component theta system of weight $5$, whose coupled period polynomials are determined explicitly and yield new cross-form relations among critical values of the associated eta products. These relations are not consequences of the functional equations of the individual forms. We also develop a weight-$4$ system arising from a quadratic twist of conductor $3$ and obtain relations linking critical values of the original and twisted modular forms. Through modular parametrizations by complete elliptic integrals, these $L$-value identities yield corresponding moment identities. The method gives a unified modular-symbol explanation of several previously known elliptic integral moment relations while producing new critical-value relations between distinct modular forms.

For any given integer $p$, we show that primality of $p$ is equivalent to $\lceil \frac{1}{2}\log_{2}p \rceil$ congruences of binomial coefficients modulo $p$. We also give extensions of this result to several other congruences that are similar in nature.

We prove a quantitative neighboring-denominator variant of the Erdős-Mahler conjecture. Let $p_n/q_n$ be the convergents of an irrational real number $\xi$. If $p_nq_nq_{n+1}$ is $S$-smooth for infinitely many $n$, where $S$ is a fixed finite set of primes, then there exists an effectively computable constant $c=c(S)>0$ such that \[ \log q_{n+1}\gg_{\xi,S} q_n^c \] along those indices. Consequently, $\xi$ is a Liouville number. The proof uses the determinant identity for consecutive convergents and a fixed-base consequence of Yu's theorem on $p$-adic logarithmic forms.

We prove a binomial identity relating the $k$-Lucas and $k$-Fibonacci sequences: for real $k>0$ and integers $m\ge1$, $n\ge1$, a binomial-weighted sum of $k$-Lucas numbers reduces to a multiple of $L_{k,mn}$ when $n$ is even, but to a multiple of $F_{k,mn}$ when $n$ is odd. This parity dependence traces back to the $k$-Lucas Binet formula lacking the normalising factor $1/(\alpha-\beta)$ present for $k$-Fibonacci numbers. The identity supplies, for each parity of $n$ and each $m$, an explicit family of positive weights summing to one; combined with the general annulus principle of Dalal and Govil, this yields a $k$-Lucas annulus containing all the zeros of a complex polynomial of degree $n$, complementing the $k$-Fibonacci-based bounds of Díaz-Barrero, Bidkham-Shashahani, and Kaur.

We prove that for a connected, geometrically unibranch variety $X$ over a perfect field of positive characteristic and a dense open subset $U \subset X$, the canonical homomorphism between the Tannaka duals of the categories of overconvergent isocrystals is a quotient map. We also show that the analogous statement for convergent isocrystals, and for all isocrystals, fails. The counterexample is based on Crew's construction of a unit-root $F$-isocrystal that is convergent but not overconvergent.

This paper relates algebraic K-theory of fields to polylogarithms via general linear groups. We focus on the case of number fields and prove that the motivic realisation map from the Goncharov Lie coalgebra to the motivic Lie coalgebra is an isomorphism. This implies the Goncharov universality conjecture and a structural result for special values of Dedekind zeta functions. We also construct explicit polylogarithmic cocycles representing nonzero multiples of the Borel classes.

We study Eratosthenes sieve as a discrete dynamic system. At each stage of the sieve there is a cycle of gaps ${\mathcal G}(p^\#)$ of length $\phi(p^\#)$ and span $p^\#$. There is a recursion ${\mathcal G}(p_k^\#)\longrightarrow {\mathcal G}(p_{k+1}^\#)$ that creates the next cycle from the current one. If we take initial conditions from the cycle ${\mathcal G}(p_0^\#)$, then for all constellations of span $|s| < 2p_1$, including gaps $g < 2p_1$, the driving terms of various lengths form Markov chains. These yield exact models for the populations $n_s(p_k^\#)$ for all further stages of the sieve. If $s$ is an admissible constellation of length $J$, then its population $n_{s,J}(p^\#)$ grows as $\Theta \left( \prod (q-J-1)\right)$. So we factor out the superexponential growth to obtain the exact model for the relative population $w_s(p_k^\#)$ of the constellation $s$ across all further stages of the sieve. $$ w_{s,J}(p_k^\#) \; = \; n_{s,J}(p_k^\#) \, / \, \prod_{J+1 < p \le p_k} (p-J-1) $$ The asymptotic value of the relative population is a constant ${w_{s,J}(\infty) \ge 1}$ that depends only on the odd prime factors that divide a span in $s$. Assuming that the instances of a constellation $s$ are approximately uniformly distributed in ${\mathcal G}(p_k^\#)$, we develop first-order estimates of the number of instances $s$ that would occur in the interval of survival $\Delta H(p_k) = (p_k^2, p_{k+1}^2]$. We define a statistic $\eta_s(p_k)$, the quadratic density of the constellation $s$ over the interval $\Delta H(p_k)$. We show that the first-order estimates $\widehat{\eta_g}(p)$ for prime gaps agree with samples up to $5.677\,E14$.

For a fixed integer $n\ge 2,$ we consider the homogeneous polynomial $$ P(x_1, x_2, \ldots, x_{n+2})=\sum_{i=1}^{n} (x_2-x_1)^{i-1} x_1^{n-i} x_{i+2}. $$ We prove that, for any finite set $A$ of complex numbers, $$ \Bigl|\bigl\{P(x_1,x_2,\ldots,x_{n+2}): \, x_i\in A\bigr\}\Bigr|\gg |A|^{n}. $$ The implicit constant in $\gg$ may depend only on $n.$

Let $M$ be a Weil-Petersson random hyperbolic surface of genus $g$, and let $\Gamma \subset \mathbb{Z}^{2g}$ be a lattice of prime index $q$. We study the distribution of primitive closed geodesics in homology classes mod $\Gamma$ in the large genus limit. Averaging over all lattices of index $q$, with $q \to \infty$, we compute all the centered moments of the corresponding weighted counting functions, and exhibit a transition between Poisson and Gaussian regimes (depending on whether $\frac{X}{q\log X}$, the expected number of primitive geodesics in a given homology class mod $\Gamma$, tends to $\lambda>0$ or $\infty$). We also study the unnormalized variance $G_M(X,\Gamma)$ of the counts among homology classes, and show that as $X \to \infty$, averaged over all lattices of prime index $q$, it is asymptotic to $X\log X$ in the large genus limit. These results are analogous to phenomena arising in the distribution of primes in arithmetic progressions.

We compute the local integrals appearing in the refined Gan–Gross–Prasad conjecture for Fourier–Jacobi periods of $\mathrm{Sp}_4$ in new ramified cases and use this to formulate an explicit conjectural identity relating Petersson norms of degree 2 Siegel cusp forms and associated half-integral weight forms. We note consequences of our identity for the growth of Petersson norms, the size of Fourier coefficients, and non-vanishing of central $L$-values.

We give a brief and partially informal introduction to the Birch and Swinnerton-Dyer conjecture. After a short review on basic Diophantine equations, we define the natural abelian group structure on an elliptic curve, leading to the definition of the corresponding algebraic rank. Afterwards, we sketch how to define the analytic rank of an elliptic curve and we state the conjecture, that is, the equality of these two ranks.

Let $s_2(n)$ be the binary sum-of-digits function and let $c_t$ be the natural density of the integers $n\ge0$ for which $s_2(n+t)\ge s_2(n)$. Earlier work of the author proved the universal exponential bound $$c_t-\frac12\ge 2^{-2s_2(t)-1},$$ thereby resolving Cusick's conjecture for every $t$. This estimate, however, does not reflect the true size of the smallest possible bias at a given large Hamming weight. In this paper, we determine this extremal scale sharply: $$\inf_{s_2(t)=k}\left(c_t-\frac12\right) \sim \frac{1}{2\sqrt\pi} \left(\frac{\log_2 k}{k}\right)^{3/2} \qquad(k\to\infty).$$ Thus the optimal fixed-weight gap is polynomial-logarithmic rather than exponential, with the explicit sharp leading constant $1/(2\sqrt\pi)$. The proof combines the five-cumulant Edgeworth expansion of Spiegelhofer and Wallner with a new extremal rigidity mechanism for near-extremal binary block patterns. We also prove a stability theorem for asymptotic extremizers and give a separate shadow-energy interpretation of the same constant.

We prove an asymptotic formula with a power-saving error term for the second moment $$ \sum_{q \in \mathcal{Q}} \; \; \sideset{}{^\flat}\sum_{\chi \bmod q} \left| L\big( \tfrac{1}{2} , \chi \big) L\big(\tfrac{1}{2},f\otimes \chi)\right|^2 $$ of the twisted ${\rm GL}(2)$ {$L$}-function and the Dirichlet {$L$}-function at the central point under the assumption of Selberg's eigenvalue conjecture. Here $f$ is a fixed Hecke holomorphic cusp form for $\mathrm{SL}(2,\mathbb{Z})$ and the sum over $\chi$ runs over all primitive even Dirichlet characters modulo $q$, and $$\mathcal{Q}=\left\{q=q_1 q_2\asymp Q: q_1 \leq Q_1, q_2 \leq Q_2,\ Q_2\asymp Q^{\delta_1},(q,6)=1,(q_1,q_2)=1 \right\}$$ with $0<\delta_1<0.0004$.

We define a reformulation of the Collatz $3n+1$ map which allows us to express it as an operator on the set of sequences of Boolean polynomials. We express this operator in terms of certain carry sequences that arise from addition in base-2, and then we simplify them and find explicit formulas for them.

We obtain a uniform pointwise estimate for short Weyl sums of degree $n\geq3$ in the intermediate rational-approximation range $$ \frac{1}{qx^{n-2}y}\ll|\lambda| \ll\frac{1}{qy^{n-1}}. $$ This range arises from a second application of Dirichlet's rational approximation theorem in estimating the residual integral that occurs in the derivation of an asymptotic formula for Waring's problem with almost proportional summands. For $r=2^n+1$ and fixed positive numbers $\mu_1,\ldots,\mu_r$ satisfying $$ \mu_1+\cdots+\mu_r=1, $$ we derive an asymptotic formula for the number of representations $$ x_1^n+\cdots+x_r^n=N, \qquad |x_i^n-\mu_iN|\leq H,\qquad 1\le i \le r, $$ valid for $$ N^{1-\theta(n,r)+\varepsilon}\le H \le \frac{N}{\ln N}, \quad \theta(n,r)= \frac{2}{n\bigl((r-1)(n-1)+2\bigr)}. $$ The resulting admissible lower bound for $H$ improves the previously known bound for every $n\geq3$.

We establish an asymptotic formula for the mixed second moment involving the second derivative of the completed $L$-function, when averaged over monic, irreducible polynomials in the function field setting.

We construct a categorical framework in which we can view the direct integral as a functor taking as input objects in a certain category of Abstract Banach bundles and outputting a Banach space. We do this by constructing a quasi-abelian category of abstract Banach bundles, and we prove certain foundational and theoretical results about this category and its objects. As a consequence, we are able to define a notion of direct integral of sheaves.

A positive integer $n$ is called a practical number if every positive integer less than or equal to $n$ can be expressed as a sum of distinct positive divisors of $n$. In this paper, we study an open conjecture proposed by Wang and Sun concerning the quadratic representations of practical numbers. Specifically, we provide a short proof of the second part of the conjecture, demonstrating that for any positive integers $b$ and $c$ with $2 \nmid b$ and $2 \mid c$, there exists an integer $n$ satisfying $1 < n \le \max\{b, c\}$ such that $n^2 + bn + c$ is a practical number. Combined with the work of Somu, Li, and Kukla, this completely settles the conjecture of Wang and Sun.

We introduce two new arithmetic invariants associated with Hermitian line bundles on arithmetic varieties, which play the role of the Bergman distortion function in the arithmetic setting. As a first step, we show that these two functions are asymptotically equivalent. This leads to the introduction of an intrinsic measure $ \widehat{\mu}^{\sup}_{\mathrm{eq}},$ which we show to be a natural arithmetic analogue of the equilibrium measure. We prove that $\widehat{\mu}^{\sup}_{\mathrm{eq}}$ serves as a detector of arithmetic positivity. In particular, we determine it for weakly nef Hermitian line bundles and, in full generality, in the toric case. Moreover, $\widehat{\mu}^{\sup}_{\mathrm{eq}}$ governs the arithmetic volume function and its variational properties. More precisely, we establish weak integral representations of the arithmetic volume and of its first variation in terms of $\widehat{\mu}^{\sup}_{\mathrm{eq}}$. These formulas provide a measure-theoretic interpretation of the volume derivative of Yuan–Zhang and reveal a structural parallel with variational formulas in complex pluripotential theory. Our approach differs fundamentally from previous treatments based on Harder–Narasimhan filtrations or arithmetic Okounkov bodies. It places the arithmetic volume at the center of the theory and suggests that asymptotic volume invariants, rather than heights, provide the natural analytic framework for problems in equidistribution and arithmetic dynamics.

Let $K$ be a $p$-adic field. We study the arithmetic theory of abeloid varieties over $K$. Our aims are twofold. First, we study the theory of rigid analytic 1-motives, which will be viewed as a tool to describe degeneration of abeloid varieties, similarly as in the classical algebraic setting. Our key new results are the equivalence between formal (resp. log formal) 1-motives over $\mathcal{O}_K$ and rigid analytic 1-motives with good (resp. semi-stable) reduction over $K$, and the Néron-Ogg-Shafarevich criterion for the good (resp. semi-stable) reduction of rigid analytic 1-motives. In particular, we construct log formal 1-motives and log $p$-divisible groups over $\mathcal{O}_K$ from semi-stable abeloid varieties over $K$. Next, we study the conjugate uniformization of an arbitrary abeloid variety $A$ over $K$. This is a type of $p$-adic uniformization initiated by Iovita–Morrow–Zaharescu in case of abelian varieties with good reduction. Our approach here is based on Fargues' theory of $p$-divisible rigid analytic groups. In fact, we view the theory of conjugate uniformization as a study of rational points of dualizable $p$-divisible rigid analytic groups in terms of their classification Hodge–Tate triples. Along the way, we construct $p$-divisible rigid analytic groups from rigid analytic 1-motives.