Papers

We construct an infinite family of solutions to the Mondrian art problem with absolute defect bounded above by $280$, of side lengths $n_0 = 315$, $n_1 = 106785$, and $n_{j+2} = 340n_{j+1} - n_j$, where $n_j$ is congruent to $315 \pmod{630}$. This absolute bound shows that the minimum defect does not necessarily increase with $n$, providing a counterexample to a conjecture which has been open since the problem's original proposal. Our design is derived from a well-known seven-rectangle dissection over the real quadratic field $\mathbb{Q}(\sqrt{19})$, with non-integer side lengths; we tailor the side lengths to be integers via an infinite Pell family. Using the same method, we devise a second family with defect $560$ that contains infinitely many side lengths which are divisible by any given positive integer, implying that for any $k$ there are infinitely many side lengths $n \equiv 0 \pmod{k}$ with defect bounded above by $560$. This constant bound offers a dramatic improvement on the previous quadratic bounds.

We carefully construct the profinite iterated monodromy group of a rational map as a quotient of the étale fundamental group of the punctured projective line. In doing so, we prove a number of folklore results regarding profinite IMGs; in particular, their topological generation by inertia concentrated in the post-critical orbit, and their relationship with the discrete iterated monodromy group over $\mathbb{C}$.

Let $\pi$ be a genuine irreducible cuspidal automorphic representation of the $n$-fold cover $SL_2^{(n)}(\mathbb{A})$ of $SL_2(\mathbb{A})$, $n \geq 3$ odd, and let $\tau$ be the genuine representation of $GL_1^{(n)}(\mathbb{A})$ obtained from a unitary Hecke character of $\mathbb{A}^\times$. We prove that, for a certain choice of data, a new way integral for $SL_2^{(n)}(\mathbb{A})$ represents the partial $L$-function $L^S(ns+\tfrac12,\pi\times\tau)$. The integral is derived from the generalized doubling integral of Cai, Friedberg, Ginzburg and Kaplan, following the idea of Ginzburg and Soudry.

Let M be an imaginary quadratic field and A be an abelian variety of odd dimension g defined over Q for which there is a unitary algebra embedding of M in End^0(A_M) of signature ((g+1)/2,(g-1)/2). Extending a result by Fité and Goodman, we show the existence of an elliptic curve E over Q with complex multiplication by M such that H_et^1(E_Qbar, Q_l) is a G_Q-subrepresentation of H_et^g ( A_Qbar, Q_l((g-1)/2) ). The main results of the article consist on explicit descriptions of E for A in two different families. First, when A is the Jacobian of the curve y^2 = xh(x^2), where h in Z[x] is a degree g monic separable polynomial with nonzero h(0), we show that E is given by y^2 = x^3 + h(0)x. Second, when A is the Jacobian of y^3=f(x), where h in Z[x] is a degree 4 monic separable polynomial, we show that E is given by y^2 = x^3 + 16 disc(f). This latter result builds on work by Laga and Shnidman in the bielliptic case, of which we give an alternative proof by means of a new p-adic method. As an application, we derive a practical average polynomial time algorithm to compute the Zeta functions of the reductions of a Picard curve.

In this paper, the relationship is studied between the Lehmer index and the reciprocal sum of the prime factors of a Lehmer number. Some upper bounds and a lower bound are proposed for the Lehmer index. Another constraint on Lehmer index is also proposed if a Lehmer number has 3 as its prime factor.

We study the distribution and oscillatory properties of prime-supported Fourier coefficients attached to a Hecke–Maass form for $\mathrm{SL}(3, \mathbb{Z})$ twisted by additive characters. For linear phases involving irrationals of finite type, we construct an automorphic variant of Vaughan's identity and exploit bilinear cancellation to obtain an unconditional power-saving estimate. For non-linear phases $e(\alpha n^\theta)$ with $0 < \theta < 1$, we refine the Type~II bilinear forms via Weyl differencing and modern exponent pair methods. Combined with the analysis of untwisted $\mathrm{GL}(3)$ sums, this framework breaks the $2/3$-barrier of Jiang and Lü and widens the admissible range of non-trivial cancellation.

We prove a case of a conjecture of Gundlach on the number of Galois extensions over $\QQ$ with bounded multi-invariants for the group $G=Q_8$. We also obtain results when imposing local conditions at the wildly ramified place. As an application, we prove asymptotics for the number of $Q_8$-extensions with bounded product of ramified primes.

We establish bounds on the norm of signed indecomposable integers in totally real number fields, via a geometric argument. We also improve the bound on the norm of totally positive indecomposable integers in real quadratic fields by Tinková and Voutier.

We prove four conjectures of Takigiku and Tsuchioka consisting of eight Rogers–Ramanujan type identities. Two conjectures involve triple sums and a third involves quadruple sums, and their product sides are certain level $2$ principal characters of $A_{13}^{(2)}$. The fourth conjecture gives multisum representations of certain level $2$ principal characters of $A_{2\ell-1}^{(2)}$. The proofs rely on the Bailey pair method, finite sum identities involving Gaussian polynomials, and the quintuple product identity.

We develop a new approach to the Joint Intersectivity Conjecture of Bergelson, Leibman, and Lesigne for commuting transformations. The conjecture asserts that the necessary divisibility condition is the only obstruction to polynomial multiple recurrence and, consequently, to the multidimensional polynomial Szemerédi theorem. We establish the conjecture for pairwise independent polynomials, obtaining the first general result for sequences of arbitrary length beyond the classical one-transformation setting. Our argument is inspired by the Furstenberg-Katznelson approach and its polynomial enhancement due to Bergelson-Leibman. Its main novelty, however, lies in the development of new techniques that allow us to bypass the use of the polynomial van der Waerden theorem, on which these methods traditionally rely, and for which no suitable analogue is currently available in our setting. In particular, we prove a Nil-Bohr returns theorem for compact extensions in systems with nilpotent structure, as well as a nil-plus-null decomposition for polynomial multicorrelation sequences that preserves joint intersectivity and positivity.

For $f(X)=X^r(X^s+a)\in\mathbb F_{p^m}[X]$, with $m,r,s\geqslant1$ and $a\in\mathbb F_{p^m}^\times$, put $g=\gcd(s,p^m-1)$. We prove that, for each fixed $g$, only finitely many prime characteristics $p$ admit such permutation binomials, even as $m,r,s,a$ vary. If $6\nmid g$, every such $p$ divides Wendt's determinant $W_g=\operatorname{Res}_X(X^g-1,(X+1)^g-1)$, which is nonzero. When $6\mid g$, this determinant vanishes; inverse power sums and iterated resultants yield an explicit nonzero integer divisible by every such $p>2g+1$. We determine the exact admissible characteristics for $1\leqslant g\leqslant10$, obtain a further divisibility condition over prime fields, and show that in prime fields $g=11$ occurs precisely when $p=23$.

We construct non-holomorphic Siegel modular forms whose coefficients encode linking numbers in arithmetic hyperbolic manifolds of simplest type. Our proofs rely on an improved understanding of the transgression forms appearing in Kudla–Millson theory, and along the way we establish some structural properties including in particular their expected current equations.

We prove an effective version of the Chebotarev density theorem, conditional on the quasi-Riemann hypothesis for the Dedekind zeta function. Drawing on recent breakthroughs of OpenAI and Cureton–Thorner, we make unconditional the Ellenberg–Venkatesh bounds on $\ell$-torsion in class groups of number fields and extend an averaged result of Lemke Oliver–Smith to arbitrary families of degree-$d$-fields. Further applications are given to the least prime splitting and to small generators in number fields.

In this paper, we give a constructive proof that for any integer n>7 (resp. n>6), the numbers 1,2,...,n can be arranged in a circle (resp. in a line) such that the sum of the reciprocals of all adjacent sums equals 1. The above conjecture was originally posed by Zhi-Wei Sun. Partitioning the integers according to n mod 12 into nine residue classes, we prove a recursive proposition and inductively construct new examples. Main lemmas and propositions in introduction are proved in Section 2. The rest of the cases are handled by explicit constructions in appendix.

In 1994, Friedman, Joichi and Stanton conjectured that coefficients of the power-series $q+(1-q)\prod_{i=n}^{2n-1}\frac{1}{1-q^{i}}$ are nonnegative for $n\geq3$ and $n$ being odd. Let $\qbinom nm$ denote the Gaussian polynomial. In 2003, Prellberg and Stanton conjectured that coefficients of the power-series $(1-q)\qbinom nm\prod_{i=0}^{m-1}\frac{1}{1-q^{i+m}}$ are nonnegative for $0<m<n$, except when $n$ is even and $m\in\{2,n-2\}$. The latter conjecture implies the former. In this paper, we prove a stronger result concerning coefficientwise nonnegativity of another power series, which implies the conjecture of Prellberg and Stanton. The proof uses a Möbius correction from Gaussian polynomials, weight-preserving injections on partitions, explicit linearly independent vectors in the kernel of a derivation operator and the KOH identity.

In 1984, Bressoud and Subbarao established an interesting weighted partition identity connecting the partition function and the divisor function. Recently, the first author, jointly with Bhoria, Eyyunni, and Maji, extended the work of Bressoud and Subbarao and obtained the following analogous weighted partition identity. For any $n \in \mathbb{N}$, they obtained \begin{equation*} p^{(2)}(n) = \sum_{\pi \in \mathcal{D}(n)} (-1)^{ \# (\pi) }s(\pi) \left( \ell(\pi) - s(\pi) \right). \end{equation*} In the present paper, we provide a combinatorial proof of the above identity and extend the result for $p^{(k)}(n)$, where $p^{(k)}(n)$ denotes the number of partitions of $n$ into exactly $k$ distinct part sizes.

In this paper, we study a partition function with parts separated by parity from a family of partition functions introduced by Andrews in 2019. We first show that its generating function admit representation in terms of mock modular forms of weight $3/2$. We then initiate a study of smallest-part partition function analogues of partitions with parts separated by parity, which, to the best of our knowledge, has not previously been investigated in the literature. We show that the generating functions of these smallest-part partition functions are likewise expressible in terms of mock modular forms, mock Maass theta functions and Zagier's $q$-series $H(q)$, which he describes as a “half-derivative of the Dedekind eta-function". Our study also leads to new combinatorial interpretations of the coefficients of the aforementioned mock modular form, function $\sigma^*(q)$, and mock theta functions $\psi(q)$ and $A(q)$ in terms of odd Ferrers graphs. Moreover, we provide a quantum $q$-series identity involving Ramanujan's famous $\sigma(q)$ function. We also derive the leading asymptotic term of one of these partition functions, whose generating function we first express in terms of $\sigma^*(q)$.

A theorem of Landau and Ramanujan states that almost all integers are not sums of two squares (when ordered by their absolute value). Equivalently, almost all integers are not norms from the Gaussian rationals $\mathbb{Q}(i)$. In this paper, we prove a version of this norm result for elliptic curves $E$. Let $E$ be an elliptic curve over a number field $K$ with complex multiplication, such that $K$ contains the endomorphism ring of $E$. For a fixed non-torsion point $P \in E(K)$ and an abelian extension $L/K$, we show that subject to a technical condition, the $y$-coordinate of almost all multiples of $P$ does not belong to the norm group of $L$. The proof is an adaptation of an argument of Bhakta et al. , whose main tools are elliptic divisibility sequences and a sieve technique known as the elliptic sieve. To account for the points in $E(K)$ arising from the additional endomorphisms, we apply the elliptic sieve to elliptic nets, a higher-rank generalisation of elliptic divisibility sequences. This approach demonstrates the possibility of applying the elliptic sieve to $\mathbb{Z}$–linear combination of points on elliptic curves and the usefulness of elliptic nets to counting problems on elliptic curves.

Let $\mathcal O$ be the Lubin–Tate deformation ring of a one-dimensional commutative formal group of finite height $n\geq1$ over a perfect field $k$ of characteristic $p$. We identify Rezk's Witt filtration $\{I_d\}$ with the divisibility filtration of integral trace pairings on subgroup algebras of fixed source lifts. This gives the geometric recursion \[ I_{d+1}=pI_d+J_{d,z}\qquad(d\geq0), \] where $J_{d,z}$ defines the targets of order-$p^d$ quotient isogenies from a $W(k)$-valued lift $z$ of the closed point. The corrected ideal is independent of $z$. We also compute the cokernel of the universal source trace pairing over $\mathcal O$: it is $\bigoplus_{j=1}^d(\mathcal O/p^j)^{r_j-r_{j-1}}$, where $r_j$ counts order-$p^j$ subgroups of $(\mathbf Q_p/\mathbf Z_p)^n$. The same duality extends the generic trace along two-step isogeny flags to an integral surjection, even when the flag-forgetting map is not flat. This computes composition in the subring of weighted trace operations. Applications include sharp projector denominators and congruence fibre products for full-torsion loci.

Let $f$ be a monic polynomial of degree $n$ with integer coefficients. We use Fourier analysis to prove a new upper bound on the number of $n\times n$ matrices with characteristic polynomial $f$ and integer entries in $[-T,T]$, as $T\to \infty$. The implied constant in our bound only depends on $n$ and is otherwise uniform in $f$. Our bound improves on recent work of Habegger, Ostafe and Shparlinski for $n\geq 4$.

We give a directed comparison beginning with the classical zero strip \(\lvert\Ima z\rvert\le1\) for the de Bruijn–Newman heat deformation. Under explicitly stated complete model-to-function source, localisation and global contact premises, translated-probe estimates imply \[ \Lambda\le\frac{148611789321711}{10^{15}} =0.148611789321711<0.148612. \] The construction uses Raymond's finite source data and 57 fixed translated kernels, independently checked by rational Bernstein certificates. An explicit finite Weyl estimate improves the positive first-head tail and its centred logarithmic moment. A centred cone transfer retains the original physical error allowances, exceptional debit and additive cone reserve, and charges the change in existing phase-surplus credits. Directed tail enclosures are regenerated at 192 and 256 bits; a separate exact-rational auditor accepts all 19,023 comparison steps. The controlled decrease from the complete two-source comparison is \(0.000021760452379\). We also prove a compatible matched logarithmic tail transfer and identify the finite requirements for stronger exponent pairs. A supplementary section records Raymond's ceiling below \(0.116398\) under the distinct \(7/8\) zero-free hypothesis.

We settle two conjectures of Berndt and Straub asserting that all nonzero zeros of the generalized Ramanujan polynomials $R_k(x;\chi,\psi)$ and $S_{2n}(x;\chi,\chi)$ lie on the unit circle for real nonprincipal Dirichlet characters. We prove their conjecture for $R_k$. For $S_{2n}$, we prove that at most four nonzero zeros lie off the unit circle, all purely imaginary. If $\chi_{163}=\left(\frac{\cdot}{163}\right)$ is the primitive quadratic character modulo $163$, then $S_6(x;\chi_{163},\chi_{163})$ has four exceptional zeros, showing that this bound is sharp. This disproves the second conjecture and establishes that the `Riemann hypothesis for period polynomials' can fail for Eisenstein newforms. Further, we introduce a transcendental invariant whose sign determines whether exceptional zeros of $S_{2n}(x;\chi,\chi)$ occur in all sufficiently large degrees, and show that these zeros converge to transcendental limits off the unit circle. For primitive odd characters, we relate this invariant and these limits to Fekete polynomials.

We establish a generating function identity for finite and symmetric multiple zeta values using Dougall's bilateral ${}_2H_2$ summation formula. As an application, we prove a restricted sum formula in which the parity of each component of the index is prescribed. We also derive a generating function identity for finite multiple zeta values $\zeta^{}_{\mathcal{A}}(\{1,2\}^n)$ and symmetric multiple zeta values $\zeta^{}_{\mathcal{S}}(\{1,2\}^n)$, thereby proving Kaneko's conjecture for finite multiple zeta values and its symmetric analogue.

We propose a uniform moduli-theoretic description of the v-sheaf local model $\mathbb{M}_{\mathcal G,\mu}$ for an arbitrary parahoric group scheme $\mathcal G$ and an arbitrary dominant cocharacter $\mu$. Our approach is Tannakian and builds on a new theory of parahoric induced modules. When $\mu$ is minuscule, this framework explains the linear-algebraic conditions arising in the theory of local models of Shimura varieties.

For every even Galois representation in a family of prime conductor $\ell$, we prove that if $k \in \mathbb{Z}_{\geq 1}$ is maximal such that \[\ell \equiv 1 \bmod 3^k\] then the even mod $3$ deformation ring $R_S \otimes_{\mathbb{Z}_3} \mathbb{F}_3$ of level $S=\{\ell,3\}$ does not have a quotient isomorphic to $\mathbb{F}_3[T]/(T^{2+3^{k-1}})$. As a corollary, we deduce that the global even deformation rings $R_S$ are finite and flat in the category of $\mathbb{Z}_3$-modules.

Let $\ell$ and $p$ be distinct prime numbers, $K$ a local field with residue characteristic $p$, $C$ a complete algebraically closed extension of $K$, and $X$ a quasi-compact quasi-separated rigid space over $K$. When $\mathop{\mathrm{char}} K = 0$, we prove that the alternating sum of traces of elements in the positive part of the Weil group acting on $R\Gamma(X_C,\mathbb{Q}_{\ell})$ is independent of $\ell$ and is an integer, extending a result of Mieda to the singular case. We also give optimal bounds for the $p$-adic valuations of the eigenvalues of geometric Frobenius liftings acting on $H^i(X_C,\mathbb{Q}_{\ell})$, proving a conjecture of Lu–Zheng. In the general case, we give optimal bounds for the $p$-adic valuations of the eigenvalues of geometric Frobenius liftings acting on $H^i_c(X_C,\mathbb{Q}_{\ell})$, and we show that these eigenvalues are algebraic integers and Weil numbers, thereby extending results of Lu–Zheng and Mieda to the positive characteristic case.

We derive a recurrence for logarithmically weighted Möbius sums by combining Möbius inversion with the Euler–Maclaurin summation formula. Consequently, starting from estimates of orders zero, one and two, we obtain polynomial approximations for these sums for every nonnegative integer exponent of $\log$. We then extend these estimates to sums with coprimality conditions via a convolution identity and to sums weighted by the reciprocal of the sum-of-divisors function. We also establish the nonnegativity of the latter sums for every positive integer logarithmic order, proving that they never change sign.

We prove Yau's number-theoretic conjecture on positive integral points in real right simplices in every dimension $n\ge3$, and apply the resulting sharp polynomial bound to quantum harmonic oscillators. The lattice-point bound holds for every choice of positive real intercepts for which the count is nonzero, with equality precisely when all intercepts equal the same integer. For oscillators in dimensions $n\ge3$, we obtain explicit upper bounds for eigenvalue counts and spectral remainders for arbitrary positive frequencies at every threshold at or above the lowest eigenvalue. For frequencies with rational ratios in dimensions $n\ge2$, an additional lattice coordinate and interpolation give bounds for energy sums, or first Riesz means, at every threshold. When all frequencies are equal (the isotropic case), the oscillator attains these bounds: the counting bound is exact at its energy levels, and interpolation on its natural energy grid recovers the energy sum at every threshold. In three dimensions, the exact formula displays the periodic contribution associated with the resonance discussed by Fefferman and Seco. The proof builds on the slicing method of Yau and Zhang. Arithmetic and geometric consequences include explicit bounds for smooth numbers and lattice-point discrepancies.

We establish a matching lower bound for the sextic large sieve over the Eisenstein integers in an unbalanced range, with both indices restricted to squarefree ideals. Let $\Xi_6(A,B)$ denote the squared operator norm of the sextic residue-character matrix with row and column norms at most $A$ and $B$, respectively, outside a fixed finite set of primes containing those above $6$. For every fixed $0<\delta<3/11$ and every $\varepsilon>0$, we prove $\Xi_6(A,B)\gg_{\delta,\varepsilon}(AB)^{-\varepsilon}A^{5/6}B^{1/3}$ for $B^{16/11+\delta}\le A\le B^{2-\delta}$. Consequently, the full-cutoff norm satisfies $\Xi_6(X,X)\gg_\varepsilon X^{17/16-\varepsilon}$. The main difficulty is to preserve the large second moment supplied by Gauss sums when the column moduli are restricted to be squarefree. We prove a second-moment formula uniform in the prime exclusions and character twists arising from Möbius inversion, and use it to show that the squarefree projection retains a positive proportion of the original moment. A separate argument combining cubic and sextic large-sieve bounds then transfers the lower bound to squarefree rows.

Let $k=\overline{\mathbb{F}}_p$ and let $f\in k[X]$ be a polynomial of degree at least $2$ such that $f-f(0)$ is non-additive. For $\alpha,\beta$ in an algebraically closed extension $L$ of $k$, we prove that there are infinitely many $\lambda\in L$ for which both points are preperiodic under $f(X)+\lambda$ if and only if $\alpha,\beta\in k$ or $f(\beta)-f(\alpha)\in E_f$. Here $E_f=\{c\in G_f:(f-f(0))^n(c)\in G_f\text{ for every }n\ge0\}$, where $G_f=\{c\in k : f(X+c)-f(X)\in k\}$. We also provide a necessary and sufficient criterion for the corresponding colliding orbits set to be infinite.