Papers

Let $b_1,\dots,b_d\ge2$ be fixed integers, and let $k$ be their multiplicative rank. We study representations $n=a+b_1^{u_1}\cdots b_d^{u_d}$, where $a$ belongs to a set $\mathcal{A}$ of positive integers and $u_1,\dots,u_d$ are positive integers, counting distinct tuples of exponents separately. Under density and correlation assumptions on $\mathcal{A}$, we obtain a lower bound for the number of integers with many such representations, in terms of $d$ and $k$. In particular, when $\mathcal{A}$ is the set of primes or the set of positive integers representable as a sum of two squares, a positive proportion of the integers $n\le x$ have at least $c_1(\log x)^{d-1}$ or $c_1(\log x)^{d-1/2}$ representations, respectively, for some $c_1>0$ and all sufficiently large $x$. No multiplicative independence or coprimality assumptions on the bases are required.

In this paper, we obtain an exact sequence connecting $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$ to $H_2(\rm{SL}_2(\mathcal{O}_{K,T}), \mathbb{Z})$, where $\mathcal{O}_{K,S}$ is a ring of $S$-integers and $T$ is a set of primes containing $S$. We apply this sequence to establish a relation between $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$ with the second $K$-group $K_2(\mathcal{O}_{K,S})$, for $S$ large enough. This leads to a description of the rank and size of the torsion of $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$. As an application, we propose a homological version of the Birch-Tate formula (conjecture) under these assumptions.

For nonsquare $d\ge 2$, we construct fixed straight-line programs for the least non-trivial solution $(X_1,Y_1)$ of $x^2-dy^2=1$, using addition, truncated subtraction, multiplication, integer division, exponentiation, and remainder. A geometric-sum identity recovers $(X_1,Y_1)$ from the coordinate sums of an initial segment of Pell solutions, without knowing how many solutions were summed. Weighted binary encodings make these sums accessible to arithmetic computation. Counting operations with reuse of computed values, one program uses $94$ operations, with intermediate bit lengths bounded by $2^{2^{2^{O(d)}}}$. Four additional operations give a $98$-operation program with the bound $2^{2^{O(d)}}$. Hua's bound gives corresponding programs with $103$ and $107$ operations and replaces $O(d)$ by $O(\sqrt d\log d)$ in these size bounds. Once the fundamental solution is known, generating-function formulas compute the $n$-th positive solution in $21$ further operations by extracting its coordinates as base-$b$ digits. We prove that $2X_1(X_1+1)-1$ is the least integer base for which both formulas are defined and correct for every $n\ge1$.

This paper is devoted to studying permutation binomial $F_{r,a}(X)=X^r(X^{q-1}+a)\in\mathbb F_{q^e}[X]$ with $a\in\mathbb F_{q^e}^*$. We present a complete characterization for $F_{r,a}$ to be a permutation of $\mathbb{F}_{q^e}$. This yields a complete proof of the conjecture proposed by Masuda–Rubio–Santiago \cite{masuda2022permutation}. More precisely, we show that \(F_{r,a}\) permutes \(\mathbb F_{q^e}\) if and only if $(-a)^{(q^e-1)/(q-1)}\ne1, \gcd(r,q-1)=1$, and there exists \(1\le h<e\) with \(\gcd(h,e)=1\) such that $r(q^h-1)\equiv q-1\pmod{q^e-1}.$

A sequence of integers $1=a_0<a_1<\cdots<a_k=n$ is called an addition chain of length $k$ if $a_j=a_s+a_t$ with $0\le s,t<j$ for all integers $j\in \{1,2,\ldots,k\}$. We denote by $\ell(n)$ the minimal length of an addition chain leading to $n$. Here we investigate the distribution of the function $\ell$ through the counting function $$ F(m,r):=\#\{n\in [2^m,2^{m+1}):\ell(n)\le m+r\} $$ and show that, for every fixed $0<c<\log 2$, there exist positive constants $K_1$ and $K_2$ such that $$ K_1^r m^r\le F(m,r)\le K_2^r m^r $$ for all sufficiently large $m$ and all integers $m^{0.9}<r\le cm/\log m$. The upper bound also holds for every integer $r>m^{0.9}$. Moreover, denoting by $G(m,r)$ the number of distinct addition chains of length $m+r$ leading to an integer $n\in [2^m, 2^{m+1})$, we show that there exist positive constants $K_3$ and $K_4$ such that $$ K_3^r \left(\frac{m^2}{r}\right)^r\le G\left(m,r\right)\le K_4^r \left(\frac{m^2}{r} \right)^r $$ provided $m^{0.9}<r<m$. This improves and generalizes previous results on the minimal length of addition chains and addresses a question raised by Paul Erdős.

Given an abelian scheme $A$ over a $p$-adic ring $R$, Borger and Saha constructed a filtered module $\{\mathbf{H}_\delta(A) \supset \mathbf{X}_{\mathrm{prim}}(A)\supset \{0\}\}$ with a semilinear operator $\mathfrak{f}^*$ on $\mathbf{H}_\delta(A)$ using the theory of arithmetic jet spaces. The above object admits a canonical map $\Phi$ to the Hodge sequence $\{\mathbf{H}^1_{\mathrm{dR}}(A) \supset H^0(A,\Omega_A)\supset \{0\}\}$ of $A$ in the category of filtered modules. As a result, by restricting $\Phi$, we obtain a natural $R$-linear map $\Upsilon: \mathbf{X}_{\mathrm{prim}}(A) \rightarrow H^0(A,\Omega_A)$. In this paper, we show that the map $\Upsilon$ is an isomorphism of vector spaces over $K$, the field of fractions of $R$. As a consequence, we will show that for all abelian schemes $A$, the operator $\mathfrak{f}^*$ on $\mathbf{H}_\delta(A)_K$ is a bijection, and our object $\{\mathbf{H}_\delta(A)_K \supset \mathbf{X}_{\mathrm{prim}}(A)_K\supset \{0\}\}$ becomes a filtered isocrystal. In fact, the above results admit a generalization to the setting of semi-abelian schemes. The elements of $\mathbf{X}_{\mathrm{prim}}(A)$ are represented by primitive additive characters of the arithmetic jet spaces attached to $A$. Hence, our isomorphism given by $\Upsilon$ provides an interesting character-theoretic interpretation of $H^0(A,\Omega_A)$ in terms of primitive delta characters. As a result, to any $1$-form $\omega$, the above isomorphism associates a canonical numerical invariant that depends on deformation theoretic data of $A$. Furthermore, we also extend a comparison theorem between $\mathbf{H}_\delta(A)_K$ and the first crystalline cohomology $\mathbf{H}_{\mathrm{cris}}^1(A)_K$ in the general case when the elliptic curve $A$ is defined over the ring of integers of a $p$-adic field $K$ that is a finite extension of $\mathbb{Q}_p$.

In this paper, we investigate the values of zeta functions of root systems at non-positive integer points. In particular, we focus on the zeta functions of type $\mathrm{A}_r$. Since non-positive integer points are included in the set of singularities of zeta functions of root systems, we introduce the ordered limit values to define those values. We study four particular ordered limit values and give explicit formulas for them, or recurrence formulas that allow us to calculate them inductively. We then prove that three ordered limit values of zeta functions of type $\mathrm{A}_r$ vanish under some conditions that can be regarded as a multivariable analogue of the trivial zeros of the Riemann zeta function. Finally, we discuss the other zeros of zeta functions of type $\mathrm{A}_r$ at non-positive integer points and a possible connection of such zeros and non-trivial identities among classical Eisenstein series.

It is well known that an Eisenstein series $E_k$ cannot be written as a product of two lower-weight Eisenstein series, except for $E_{14}=E_4 E_{10}= E_4^2 E_6= E_6 E_8$. A similar result is also known for the Hecke eigenforms. It is also known that equalities among the monomials of the Eisenstein series can be reduced to the above-mentioned identities. Recently, the present author, along with E. Saha studied the possible monomial relations of the Poincaré cusp forms of index $1$. So far, the problem of studying the monomial relations has been restricted to the setup of holomorphic modular forms. For an even integer $k\ge 4$ and a negative integer $m$, the Poincaré series $G_{k}(z,m)$ of weight $k$ and index $m$ is a weakly holomorphic modular form having a pole of order $-m$ at $\iota\infty$. It is immediate that the Poincaré series $G_k(z,m)$ for $m<0$ can not be written as a product of two lower-weight Poincaré series of the same index $m$. In this article, we investigate the possible equalities among the monomials of the Poincaré series $G_k(z,m)$ for arbitrary $m<0$. In particular, we show that for any $m<0$ such that $0.06\le\{-2m\pi\}\le0.99$, two monomials composed of $G_k(z,m)$ of the weights $k\ge 50$ are never equal, where $\{x\}$ denotes the fractional part of a real number $x$. In view of Weyl's equidistribution criterion, at least $93\%$ of $m<0$ satisfies the above inequality.

We establish a Petersson-Rankin-Selberg identity for Hilbert Rankin-Cohen brackets over totally real fields of degree greater than one and arbitrary narrow class number. This extends results of Zhang-Zhang and Zhang-Zhou. For even $k,\ell\ge 2$, $r\in\mathbb{Z}_{\ge 0}$, $K=k+\ell+2r$, and normalized full-level cuspidal Hecke eigentuples $f\in S_K(\omega_f)$ and $h\in S_\ell(\omega_h)$, we show $\langle f,[E_k,h]_r\rangle=C_{F,k,\ell,r}L^S(k/2,\Pi(f)\times\Pi(h)^\vee)/L_F(k,\omega_f\omega_h^{-1})\ne 0$, where $C_{F,k,\ell,r}>0$. Specializing to $k=2$, we remove the remaining GRH assumption in the product classification of Hao-Qin-Zhou. Hence, over real quadratic fields of narrow class number one, the only full-level product identities among Hecke eigenforms of even parallel weights at least two, up to interchanging the factors, are $E_4^{\mathrm{eig}}=60(E_2^{\mathrm{eig}})^2$ and $h_8=120E_2^{\mathrm{eig}}h_6$ over $\mathbb{Q}(\sqrt{5})$.

Let $F=(f_1(x_1,\ldots,x_k),\ldots, f_m(x_1,\ldots,x_k))$ be a system of nonconstant polynomials of $k$ variables with integer coefficients and let \[ {\gcd}_F(x_1,\ldots,x_k)= \gcd(f_1(x_1,\ldots,x_k),\ldots, f_m(x_1,\ldots,x_k)). \] We obtain an unconditional asymptotic formula for the sum \[ \sum_{1\le x_1,\ldots,x_k\le x} h({\gcd}_F(x_1,\ldots,x_k)), \] where $F$ is a system of $m\ge 2$ polynomials of $k\ge 2$ variables subject to certain general properties, and $h$ is a bounded strongly multiplicative function. In particular, we deduce an asymptotic formula with error term concerning the density of $k$-tuples of positive integers $(x_1,\ldots,x_k)$ such that ${\gcd}_F(x_1,\ldots,x_k)=1$, given by the Ekedahl-Poonen formula.

The precise and efficient evaluation of large-scale lattice sums involving power-law kernels is a fundamental computational problem in the simulation of classical and quantum systems with long-range interactions. While methods for spatially isotropic kernels, some based on Epstein zeta functions, have advanced considerably in recent years, the anisotropic case has lagged behind, despite its broad relevance to both fundamental and effective interactions such as the dipole interaction in magnetic materials. In this work, we solve this issue by defining and analyzing anisotropic Epstein zeta functions for which we derive stably computable representations obtained from wave vector derivatives of lattice sums over isotropic interaction kernels. These functions find direct application in the analytical and numerical study of anisotropically interacting lattice systems. Going further, they provide the correction term in an exact equivalence between discrete lattices and their continuous analogs in a recent generalization of the classical Euler-Maclaurin summation formula to lattices and summands involving power-law kernels. Their connection to high-order derivatives of zeta functions can be used to improve convergence rates of numerical algorithms, for instance in micromagnetics, or to provide rapidly convergent expansions suitable for precomputations of generalized zeta functions. We derive a stably computable representation of anisotropic Epstein zeta functions, including the possibility for analytically removing Rayleigh–Wood singularities, and we develop a numerical algorithm for their stable evaluation for any lattice, power-law decay exponent and anisotropy order. We benchmark the algorithm against closed-form identities, direct summation and multi-precision results, obtaining machine precision across various lattices, power-law exponents, and anisotropy orders.

We realize the $\mathrm{SU}(3)$ Verlinde sums $V_n(m)$, up to an explicit factor, as the values at $n$ of the spectral zeta function of a higher-order Laplace operator on the triangular discrete torus on $m^2$ vertices. For fixed $m$, we express their generating function in terms of the logarithmic derivative of an associated even spectral polynomial and express this polynomial as an explicit iterated resultant. Exploiting permutation symmetry, we prove that this polynomial is a cube over $\mathbb{Q}$, apart from an explicit quadratic factor when $3\mid m$. This factorization yields shorter linear recurrences satisfied by $V_n(m)$ with constant coefficients. We also derive Binet-type formulas expressing $V_n(m)$ as finite linear combinations of powers of rescaled inverse squares of the roots of the spectral polynomial. These results provide an efficient algorithm for computing these Verlinde sums. Several fully developed examples demonstrating computational efficiency of the method are given, including expressions in terms of Fibonacci and Lucas numbers.

We give an elementary exposition of the tilted sieve introduced by GPT-5.6~Sol \cite{GPT2026}, showing that one residue class modulo each prime $p \le x$ can cover an interval of length \begin{equation*} \gg \frac{x\log x}{(\log_{2} x)\log_{3} x}. \end{equation*} This improves the classical Erdős–Rankin bound by a factor of $(\log_{2} x)/(\log_{3} x)^{2}$. Although weaker than the strongest known bounds, it shows what the tilt and its associated covering of composite survivors achieve without Maynard sieve weights or a hypergraph covering theorem. The proof uses the prime number theorem, Mertens' reciprocal-prime formula, and elementary probability. The appendices provide a historical survey and self-contained proofs of the classical Erdős–Rankin bound.

Let $\F_q$ be the field of order $q = p^f$ for prime $p$. If $G(\chi)$ is the Gauss sum attached to a multiplicative character $\chi$ on $\F_q^\times$, then $G(\chi) = G(\chi^p)$. We investigate the converse question: when does the equality of Gauss sums $G(\chi_2) = G(\chi_1)$ imply that $\chi_2 = \chi_1^{p^j}$ for some integer $j$. If $\chi_2 = \chi_1^{p^j}$, we say that $\chi_1$ and $\chi_2$ are Frobenius-conjugate. We use the Stickelberger factorization of ideals in cyclotomic fields to give an easily testable criterion for equality of Gauss sums, based on $p$-adic digit expansion. As an application, we develop several conditions under which Gauss sum equalities between characters on $\F_q^\times$ are explained by Frobenius-conjugacy. For example, if $\chi_1$ has order $q-1$ or $\frac{1}{2}(q-1)$ and $G(\chi_2) = G(\chi_1)$, then $\chi_1$ and $\chi_2$ are Frobenius-conjugate. We also include several examples, based on the explicit evaluation of certain pure Gauss sums by R.J. Evans.

Friedlander and Iwaniec proved that the number of points of the orbit $\{\gamma i \colon \gamma\in\SL_2(\Z)\}$ lying at a distance $p-2$ from the origin $i$ of the upper half-plane, with $p\le x$ prime, is of order $x/\log x$; the upper bound is unconditional, while the lower one rests on a strong hypothesis concerning the distribution of primes in arithmetic progressions. We consider instead square-free distances and prove unconditionally an asymptotic formula, whose main term is of order $x$. Employing the weighted linear sieve, we also show unconditionally that the square-free distances $n$ with at most $7$ prime factors contribute $\gg x/\log x$. Finally, assuming that the sequence $r(n-2)r(n+2)$ has level of distribution $x^{\theta}$ in arithmetic progressions, where $r(n)$ is the number of ways to write $n$ as a sum of two squares, we obtain the same lower bound for the distances with at most $N$ prime factors (the value $7$ corresponds to $\theta=1/6$).

Let $G$ be a $p$-adic classical group (orthogonal, symplectic, or unitary) and $\pi$ be an epipelagic representation of $G$ in the sense of Reeder-Yu. Using Mœglin's theory of extended cuspidal supports and Bushnell-Kutzko's theory of covering types, we determine explicitly the endoscopic lift of $\pi$ to the general linear group, whose Langlands dual expresses the dual group of $G$ as a complex matrix group, in terms of the inducing type of $\pi$ that extends the character of the first Moy-Prasad filtration subgroup defined by a stable functional. We interpret the inducing type of $\pi$ via Stevens' construction of supercuspidal representations by skew semisimple strata and introduce what we will call epipelagic strata, requiring only that the residual characteristic $p$ be odd. As an application, we reprove M. Oi's results on the endoscopic lifts of simple supercuspidal representations, in the sense of Gross-Reeder, of quasi-split classical groups. Finally, on the Galois side, we show that the epipelagic Langlands parameters of $G$ constructed by Reeder-Yu can be recovered via the self-duality of Bushnell-Henniart's admissible triples and endoscopic embeddings of L-groups. We also establish some related results concerning epipelagic parameters that were previously proved under restrictions on $p$, including a parity result on rectifying characters and an adjoint Swan conductor result related to the Hiraga-Ichino-Ikeda conjecture.

For any $p$-adic Lie group $G$, we construct a group stack $G^\mathrm{la}$ over the $p$-adic branch of Clausen–Scholze's stack of norms such that quasicoherent sheaves on its classifying stack $*/G^\mathrm{la}$ recover locally analytic representations of $G$ after base change to any suitable Tate Huber pair, even in positive or mixed characteristic. We also show that smooth $\mathbb{F}_p$-representations of $G$ embed fully faithfully into sheaves on the mod $p$ fibre of $*/G^\mathrm{la}$ and establish Poincaré duality for cohomology of locally analytic representations in mixed characteristic.

We establish weighted bilinear summation identities for two families of Apéry-like polynomials $g_n(x)$ and $v_n(x)$. The identities express weighted sums in terms of consecutive endpoint values and, when necessary, lower moments. For $g_n(x)^2$ we obtain identities with weights $(2n+1)^r$ for $1\le r\le4$; for $v_n(x)^2$ we treat the cubic and quintic weights. Combining these formulas with congruences for the endpoint values gives supercongruences modulo $p^3$ and $p^4$, together with special evaluations modulo $p^5$ and $p^7$, where $p$ is a prime greater than $3$. In particular, $$\sum_{n=0}^{p-1}(2n+1)^3v_n\!\left(\frac52\right)^2 \equiv 6p^4-\frac{143}{3}p^6\pmod {p^7},$$ confirming a congruence conjectured by Sun. The proofs use explicit quadratic telescoping identities and $p$-adic endpoint expansions.

This text is a self-contained expository survey of the Lagrange and Markov spectra, centered on a comprehensive exposition of Markov's theorem and its generalizations. Its purpose is to provide a systematic text for learning the theory, with detailed proofs and explanations of the connections among its arithmetic, combinatorial, and geometric descriptions. The necessary background in continued fractions, quadratic irrationals, binary quadratic forms, and bi-infinite sequences is developed step by step, followed by an exposition of generalized Markov numbers, fence posets, curve lengths, and generalized Cohn matrices. One goal of this exposition is to explain the formula connecting generalized Markov numbers with the two spectra. For nonnegative integer parameters $(k_1,k_2,k_3)$, a permutation $\sigma\in\mathfrak S_3$, and a fraction label $t\in\mathbb Q_{\geq0}\cup\{\infty\}$, let $m_t$ be the associated generalized Markov number and let $k_t=k_{i_t}$ be the parameter assigned to its position $i_t\in\{1,2,3\}$. The text explains the construction of an associated finite sequence $S(t)$ of positive integers and the identity \[ \mathcal L(\alpha_{S(t)}) =\mathcal M(Q_{S(t)}) =\frac{\sqrt{((3+k_1+k_2+k_3)m_t-k_t)^2-4}}{m_t}, \] where $\alpha_{S(t)}=[\overline{S(t)}]$ and $Q_{S(t)}=(x-\alpha_{S(t)}y)(x-\alpha'_{S(t)}y)$, with the prime denoting quadratic conjugation. Here $\mathcal L$ and $\mathcal M$ denote the Lagrange and Markov constants, respectively. The survey explains how this identity relates generalized discrete Markov spectra to the classical theory and how Markov's theorem is recovered when the parameters vanish. The account also includes boundary values arising from irrational slopes and generalizations of Frobenius's uniqueness conjecture, providing a unified perspective on the classical theorem and its extensions.

We study the arithmetic of the realizations for hypergeometric motives attached to ${}_3F_2(1)$ hypergeometric series defined over number fields, focusing on the case where the étale realization is reducible and irregular. We then use motivic techniques to prove new evaluation formulas for certain ${}_3F_2(1)$ series, understood as a period of the hypergeometric motive. In addition, we provide examples which are closely related to the exact values of $L$-functions for modular forms in special cases.

We study pairs consisting of a rational isosceles triangle and a rational triangle having a prescribed angle $\theta$, with equal perimeter and equal area. Writing $\rho=\cos(\theta)\in\mathbb{Q}$, we obtain a two-parameter description of the solutions and, for fixed $\rho$, a family of curves $C_\rho$ which are generically of genus $2$. We determine the two singular specializations: there is no genuine pair for $\rho=\frac{1}{2}$, while $\rho=\frac{47}{49}$ admits infinitely many genuine pairs. We completely parametrize two natural one-parameter families in which the $\theta$-triangle is isosceles, and prove several density results for values of $\rho$ admitting one or more genuine pairs. For thirty-five explicit values of $\rho$, Chabauty–Coleman computations determine $C_\rho(\mathbb{Q})$ completely and show that the corresponding pair is unique up to homothety. We also study multiplicity when one triangle is fixed; the condition for a scalene $\theta$-triangle to have two isosceles partners leads to an elliptic $K3$ surface and yields a dense set of such values of $\rho$. Finally, we construct infinite families with at least four genuine pairs and exhibit examples carrying at least five, and in two cases at least six, genuine pairs.

We classify all $D(u^2)$-triples of triangular numbers whose smallest element is $T_{4u-1}$ or $T_{4u}$. We study the regularity of $D(N)$-triples of triangular numbers. We also show that infinitely many positive integers $u$ admit an all-triangular $D(u^2)$-quadruple. For every positive integer $k$, triangular number $T_{16k^2-1}$ belongs to infinitely many $D(u^2)$-quadruples of triangular numbers with distinct positive integer parameters $u$.

We prove zero density estimates for $L$-functions of cuspidal automorphic representations $\pi$ of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$. We show that $N_\pi(\sigma, T) \ll T^{\frac{5}{2}(1 - \sigma) + o(1)}$, where $N_\pi(\sigma, T)$ denotes the number of zeros $\rho = \beta + i\gamma$ of $L(s,\pi)$ with $\beta \geq \sigma$ and $|\gamma| \leq T$. The key input is an extension of the Guth$\unicode{x2013}$Maynard argument to Dirichlet polynomials whose coefficients satisfy a weaker $\ell^q$ bound.

We construct an adelic counterpart of Scheithauer's lifting map from scalar-valued modular forms to vector-valued modular forms transforming with the Weil representation. More precisely, we introduce suitable spaces of vector-valued adelic automorphic forms and define an adelic lifting map between the corresponding scalar-valued and vector-valued automorphic forms. We then prove that this adelic construction is compatible with Scheithauer's classical lifting map under adelization. In this way, the relation between scalar-valued and vector-valued modular forms is transferred naturally to the adelic setting.

To each modular knot, we attach a weight $2$ modular integral for $\mathrm{SL}_2(\mathbb{Z})$ whose homogenized cycle integrals recover its linking numbers with other modular knots in $S^3$. This construction answers Choie-Zagier's question of explicitly constructing modular integrals with prescribed rational period functions in weight $2$. Our formula applies to individual modular knots without symmetrization, thereby realizing an approach discussed but not pursued by Duke-Imamoḡlu-Tóth. This gives an answer to Ghys' question on pairwise linking numbers in terms of modular integrals, as sought by Simon.

We give a unified treatment of definite Albert algebras over $\mathbb{Z}$, with exceptional Siegel–Weil identities as the main arithmetic tool. Starting from the standard octonion order alone, its first scalar theta coefficient saturates the genus average and forces every remaining class to be rootless. A cubic-ring coefficient of the $G_2$ theta lift then gives the mass of the remaining scale-$2$ framed orders, also recovered from shell geometry. A uniform Niemeier–Golay stabilizer bound saturates this mass, proving that there is exactly one further class and determining both automorphism orders, frame transitivity, and the $2457$ frames on $819$ rank-one elements. The associated generalized hexagon identifies the nonstandard group as ${}^3D_4(2):3$ only after its order has been proved. An independent weighted-theta argument gives the trace-zero root-system alternative $\varnothing$ or $A_2E_8^3$. The construction relates positive masses of framed objects to exceptional Eisenstein coefficients.

For any integers \(N \ge 2\) and \(m \ge 1\), we prove that there exist infinitely many pairs of real quadratic fields \(\QQ(\sqrt{D}), \QQ(\sqrt{D + m})\), with \(D \in \ZZ\) and \(D > 0\), such that the class numbers of both fields are divisible by \(N\). Write \(N = 2^{e}n\) with \(n\) odd. If \(n > 1\), then the class groups of both fields in each such pair contain an element of order \(n\).

Andrews and Dastidar introduced the statistic $\SOME(n)$, defined as the difference between the sum of all odd parts and the sum of all even parts occurring in the partitions of $n$, and investigated its arithmetic properties. Motivated by their work, Baruah and Gogoi defined two analogous statistics for overpartitions, namely $\OSOMEo(n)$ and $\OSOMEe(n)$, which record the sums of all odd and all even overlined parts, respectively. They established several congruences for these statistics and their difference, and proposed a conjecture on congruences modulo $7$. In this paper, we prove their conjecture by elementary $q$-series methods. Our proof combines two specializations of Watson's quintuple product identity, a $(p,k)$-parameterization of divisor-sum series, and termwise logarithmic differentiation

We develop a geometric lifting method for inverse sumset problems in compact connected abelian groups. The first result is an analogue of Freiman's $3k-4$ theorem, that is every compact set $A\subseteq G$ of sufficiently small Haar measure satisfying $\mu_G(A+A)<3\mu_G(A)$ is contained in a one dimensional Bohr set of measure at most $\mu_G(A+A)-\mu_G(A)$. This resolves a question of Christ and Iliopoulou. The proof combines Bilu's theorem with a geometric refinement of the spillover argument. We also establish a sharp projection theorem. Under a continuous surjective homomorphism with connected kernel, a compact set of sufficiently small positive measure and doubling at most $K$, where $2\le K<3$, has image of doubling at most $2K-2$, and this factor is best possible. Further consequences include variants of the $3k-4$ theorem for popular sumsets and an inverse theorem for Tao's convolution inequality.

We study the locally analytic vectors of completed cohomology of Hilbert modular varieties below middle degree. Our main result says that the universal enveloping algebra action has to factor through certain quotients. Application includes bounds on the Gelfand-Kirillov dimension and a density result. The method is a continuation of our joint work with Kai-Wen Lan.