Papers

We compute the stability degree of curves of genus g and show that it is equal to the classical Minkowsky bound. We also compute the stability degree of semi-abelian varieties with prescribed toric and abelian ranks. The proof uses a specialization result for finite monodromy groups in degenerating families, obtained from a variant of Krasner's lemma, together with an explicit construction at the prime 2.

We study root numbers in the isotrivial families $y^2=x^3+a$ and $y^2=x^3+ax$. For a broad class of fixed binary forms, we prove that the average root number over primitive pairs exists. Assuming finiteness of the relevant Tate–Shafarevich groups, we deduce Zariski density for certain del Pezzo surfaces of degree $1$ arising from separable binary sextics. We also establish explicit averages of root numbers for almost all polynomials of each fixed degree $d\geq2$, ordered by coefficient height. The proof develops a new quantitative transference principle for polynomial values.

M. Kaneko, T. Murakami and A. Yoshihara introduced finite multiple zeta values of level two and conjectured a weighted sum formula for indices whose components belong to $\{1,2\}$. In this paper, we use generating functions and linear recurrence relations to prove the conjecture. More precisely, we reduce the problem to a polynomial identity and solve the resulting second-order difference equation by identifying its specialized solutions with $_4F_3$ hypergeometric polynomials of Racah-type.

We introduce mutation-preserving generalized cluster algebras, for which the generalized cluster mutation in each direction is independent of the seed in the mutation equivalence class. We classify all irreducible generalized cluster algebras with this property. Then, a Markov-type Diophantine equation $x^2+y^2+z^2+2yz=kxyz$ is studied, which has a structure of the mutation-preserving generalized cluster algebra. We prove that positive integer solutions exist if and only if $1\leq k\leq 5$ and determine all their orbits under the associated generalized cluster mutation groups. In particular, multiple orbits occur for each $k=1,3$, whereas the solutions form a single orbit for each $ k=2,4,5$. We then classify all generalized Markov Laurent mutation invariants. As an application, a conjecture proposed by Chen-Li is proved, showing that every Laurent mutation invariant of irreducible sign-equivalent cluster algebras is essentially a polynomial in the corresponding basic invariant.

Recently, Lamzouri proved a lower bound for the least positive integer $n_f$ for which the Hecke eigenvalue $\lambda_f(n_f)<0$, showing it satisfies $n_f \ge (\log k)^{1-o(1)}$ for many holomorphic Hecke cusp forms of even integral weight $k$. In this paper, we extend Lamzouri's result to Hecke-Maass forms without assuming the Generalized Ramanujan Conjecture, and further prove that similar results also hold for the coefficients of symmetric power $L$-functions of Hecke-Maass forms.

In this work, we show that if $h(x)$ is an irreducible monic integer polynomial of degree $2q$ (with $q$ prime), whose defining field extension of $\mathbb{Q}$ contains a Galois extension of degree $q$, then there is a positive density of integers $n$ such that $h(n)$ is squarefree; in particular, $h(n)$ is squarefree for infinitely many integers $n$. As an application, we prove that the family of exceptional cubic fields contains an infinite subfamily whose unit shapes converge to the hexagonal lattice. To the best of our knowledge, this is the first example of a family of non-Galois totally real cubic fields whose unit shapes converge to the hexagonal lattice.

This paper studies the modularity problem for elliptic curves from the elementary geometry of peels of decorated equilateral triangulations. A peel is a finite developing domain together with the boundary identifications that reconstruct the triangulated surface. In genus one it recovers the universal cover and its rank-two deck lattice. For a fixed prime $\ell$, the finite unramified coverings $[\ell^n]:E\to E$ form a compatible tower whose deck groups are $E[\ell^n]$; their inverse limit is the Tate module $T_\ell(E)$, and over $\Q$ the compatible Galois action gives the usual $\ell$-adic representation. Cyclic isogenies, by contrast, give the index-$q$ lattice neighbors that form the local model for Hecke and Brandt correspondences. For a semistable elliptic curve $E/\Q$ of prime conductor $p$, the missing reciprocity is the construction, without using modularity, of a nonzero class $c_E$ in the degree-zero supersingular Brandt module such that $B_qc_E=a_q(E)c_E$ at every good prime. I make this obstruction explicit and study oriented CM packets attached to the Frobenius discriminants $D_q=a_q(E)^2-4q$. New elementary results show that $D_q$ is a nonsquare modulo $p$ exactly when the residual Frobenius polynomial is irreducible, that any residual image containing such an element gives a positive-density set of primes for which $p$ is inert in the quadratic field determined by $D_q$, and that the corresponding discriminants are necessarily unbounded.

Recently, we announced a proof of a characteristic-free resolution of singularities. This is a survey article on the proof. We motivate the approach, describe the new ideas, and introduce the proof in the work.

We classify Fourier summation identities in which the measure on the Fourier side is supported in a horizontal strip of $\mathbb{C}$. Let $\mu=\nu+\eta$, where $\nu$ is a strongly tempered measure on $\mathbb{R}$, $\eta=\sum_{m\geq1} b(\gamma_m)\delta_{\gamma_m}$ is a strongly tempered pure point measure supported off the real line, and $a:\Lambda\rightarrow\mathbb{C}$, with $\Lambda=\{\lambda_n\}_{n\geq1}\subset\mathbb{R}$, has finite exponential growth. Under natural real-antipodal and conjugation-symmetry assumptions, we characterize summation identities of the form \begin{align} \sum_{n\geq1} a(\lambda_n)\varphi(\lambda_n)=\int_{\mathbb{R}} \widehat{\varphi}(t)\mathrm{d}\nu(t)+\sum_{m\geq1} b(\gamma_m)\widehat{\varphi}(\gamma_m), \end{align} valid for every $\varphi\in C^\infty_c(\mathbb{R})$, where $\widehat{\varphi}$ denotes the Fourier transform. We prove that each such identity determines a unique generating function $F$ that is holomorphic and almost periodic in the half-plane above the strip and admits a meromorphic continuation to $\mathbb{C}^+$. The measure $\eta$ encodes the poles and residues of $F$, while $\nu$ describes the boundary behavior of its regular part through a generalized Nevanlinna representation, and $a$ determines its Fourier coefficients. Conversely, every function in the corresponding meromorphic class whose Fourier coefficients satisfy a local summability condition determines a unique summation identity of this form. The proof combines a strip version of the Bridge Lemma with a Cauchy-transform argument that accounts for the off-real poles. As an application, we show that the Guinand-Weil explicit formula for every member of the Selberg class, including non-self-dual members, fits into our framework, and we identify its associated generating function.

We study the $\beta=2$ partition function $\int_{|h| \leq \log^{\theta}(q)/2}|L(1/2+ih,\chi)|^2dh$ for typical Dirichlet characters modulo a large prime $q$ and $\theta \in (-1/2,0]$ motivated by a $q$-analogue of the Saksman–Webb conjectures. When $\theta <0$, we use Harper's randomisation argument to introduce explicit conditioning to recover moment upper bounds consistent with critical normalisation predicted there. As an application, we prove that for $q(1-o(1))$ Dirichlet characters modulo $q$, $\max_{|h| \leq 1/2}|L(1/2+ih,\chi)|\ll \frac{\log(q)}{(\log\log(q))^{3/4+o(1)}}$, establishing an upper bound matching the predictions of the $q$-analogue of the Fyodorov–Hiary–Keating conjectures up to second order.

By a metacyclic field $K$ we mean the Galois closure of a pure field $\mathbb{Q}(\sqrt[\ell]{D})$, $D\in\mathbb{Z}$, of odd prime degree $\ell$. Let $\mathcal{M}_\ell$ denote the collection of isomorphism classes of metacyclic fields of degree $\ell(\ell-1)$. Write $N_\ell(X)=\#\{K\in\mathcal{M}_\ell: |\Delta_K|\leq X\}$ for the associated counting function, where $\Delta_K$ denotes the discriminant of $K$. We show $$N_\ell(X)\sim A_\ell X^{\frac{1}{(\ell-1)^2}}(\log X)^{\ell-2}\,,$$ for an explicit constant $A_\ell$. We express $A_\ell$ as a rational number times a product over primes of a degree $\ell$ polynomial in $1/p$.

Recently, Hirose proved an analogue of the linear part of Kawashima's relations for refined symmetric multiple zeta values. In this paper, we establish an analogue of Hirose's relation for finite multiple harmonic $q$-series at roots of unity. As an application, we prove the cyclic sum conjecture proposed by Kh. Hessami Pilehrood, T. Hessami Pilehrood, and R. Tauraso.

Let $k$ be a field, $G$ an affine $k$-group scheme, $X$ a connected scheme proper over $k$, $f:(Y,y)\to(X,x)$ a pointed $G$-torsor, $\mathcal{C}_X$ and $\mathcal{C}_Y $ the Tannakian categories of vector bundles on $X$ and $Y$, such that $f^*\mathcal{C}_X\subseteq \mathcal{C}_Y$, we give the necessary and sufficient conditions for the exactness of the natural sequence of $k$-group schemes $1\rightarrow \pi(\mathcal{C}_Y,y) \rightarrow \pi(\mathcal{C}_X,x) \rightarrow G \rightarrow 1.$ This provides a common framework of exact sequences for the $S$-, Nori, $F$-, étale fundamental group schemes and their extended variants. Finally, an isogeny of an elliptic curve shows that the relevant preservation property fails for the local, extended local, and unipotent categories.

In this paper, we use finite fields and linear algebra methods to resolve a conjecture by T. Zhang, L. Zheng, H. Wang, J. Peng and Y. Li (Finite Fields Appl. 110 (2026) 102743) concerning permutation pentanomials.

We prove that for every integer $N\geq2$, there exist positive integers $a$ and $b$ such that $N=a+b$ and $\Omega(ab)\leq33$, where $\Omega(n)$ denotes the number of prime factors of $n$, counted with multiplicity. This improves the previous bound of $40$ obtained by Dudek and Dunn. The proof applies the explicit Friedlander–Iwaniec $\Lambda^-\Lambda^2$ lower-bound sieve to a sequence derived from the products $n(N-n)$. The main new ingredient is pre-sieving at the prime $3$, which eliminates the extremal small-prime case in the dimension condition while keeping the resulting remainder terms under explicit control. We complete the proof using analytic estimates for large $N$, finite verification over an intermediate range, and explicit prime-gap data for small $N$.

We prove a Plancherel formula for a family of affine $\Psi$-bundles. As an application, we construct Fourier transforms and asymptotic Schwartz spaces for the family. We then prove the corresponding Poisson summation formula under suitable assumptions. The choice of affine $\Psi$-bundles we consider is motivated by applications to triple product $L$-functions explored in another paper of the authors and Leslie.

Let $Q = ax^2+bxy+cy^2$ be a primitive positive definite integral binary quadratic form with discriminant $\Delta = b^2-4ac$. It is known that $Q$ admits a local obstruction; that is, there exist $k,l \in \mathbb{Z}$ such that $Q \not\equiv l \pmod k$. We study the minimal obstruction modulus $\kappa_Q := \min \{k \in \mathbb{Z}_{\geq 1} \mid \text{there exists } l \text{ such that } Q \not\equiv l \pmod k \}$, and we determine $\kappa_Q$ completely, treating the cases $\Delta \equiv 0 \pmod 4$ and $\Delta \equiv 1 \pmod 4$ separately. We also determine the analogous invariants for primitive ternary diagonal forms.

Let $Q\geq 1$ be large, and $\delta \in(0,1)$ be small. Denote by $\mathcal C \subset \mathbb R^3$ a sufficiently smooth curve with non-vanishing curvature and torsion. How many rational points $\mathbf{a}/q$ of height $q\in[1, Q]$ are $\delta/q$-near $\mathcal C$? This manuscript provides an essentially optimal answer, thereby addressing a problem stated by Beresnevich and Kleinbock, for space curves. We show that the folklore conjectures are incorrect for certain manifolds with codimension $\ge 2$, including the moment curve $(t,t^2,t^3)$. The reason is a {hitherto} hidden `major arc' type obstruction. We also establish matching upper bounds, up to endpoints. Our argument combines purely Fourier analytical techniques with the planar counting results by Vaughan and Velani.

For $n>2$ and $k>1$, define the polynomial \[F_{n,k}(x) = \Phi_n(1 + x + \cdots + x^k),\] where $\Phi_n$ denotes the $n$-th cyclotomic polynomial. The cyclotomic conjecture proposed by Gimbert (1999) exactly describes the irreducibility of $F_{n,k}(x)$ over $\mathbb{Q}$ in terms of $n$ and $k$. Conde, Gimbert, González, Miller and Miret (2014) established that the cyclotomic conjecture, if true, would imply the non-existence of almost Moore digraphs - a well-known open question concerning the directed degree-diameter problem. In this article, we prove the cyclotomic conjecture and, as a consequence, show that there are no almost Moore digraphs with maximum out-degree $d$ and diameter $k$ for any $d>1$ and $k>2$.

Let $\Pp=\{2,3,5,7,\ldots\}$ denote the set of prime numbers. We prove that if every sufficiently large positive even integer can be represented as a difference of two primes, then there is no Borel probability measure $\mu$ on $\R$ for which \(\left\{e^{2\pi i p x}:p\in\Pp\right\}\) is an orthonormal basis of $L^2(\mu)$. In particular, under the Polignac conjecture, the prime numbers $\Pp$ cannot be a spectrum (i.e., the set of frequencies of an exponential orthonormal basis) of any probability measure on $\R$.

Building on prior results [1], we introduce vector-valued modular resurgent series, whose components exhibit a single infinite tower of singularities in the Borel plane, trivial secondary resurgent series, and Stokes constants given by linear combinations of the coefficients of a vector of $L$-functions. We extend the paradigm of modular resurgence to this setting, emphasizing the role of the Stokes constants and the interplay between the associated vectors of $q$-series and Dirichlet series, and describing the resulting symmetry relating canonical pairs of vector-valued modular resurgent series. Moreover, we conjecture that certain vectors of $q$-series with modular resurgent asymptotics are vector-valued quantum modular forms and can be reconstructed via median resummation. Finally, we show that vectors of $q$-Pochhammer symbols, previously considered in [2], and Eichler integrals of vector-valued modular forms of weight $1/2$ and $3/2$ can be studied within the framework of vector-valued modular resurgence. While the first case amounts to a convenient repackaging of scalar modular resurgence, the second involves a non-trivial representation of the modular group and therefore illustrates the necessity of the vector-valued framework; we establish its modular resurgent structure in general, and work it out in full detail for the unary theta series, whose Eichler integrals are the false theta functions of quantum topology.

Let $d\geq 4$ and let $R>0$. When $d=4$, assume that $R^2\in\mathbb{N}\setminus 4\mathbb{N}$; when $d\geq 5$, let $R^2\in\mathbb{N}$ be arbitrary. We prove the fixed-radius estimate $$\|A_R f\|_{\ell^{p'}(\mathbb{Z}^d)}\leq C_{d,p,\varepsilon}R^{-d(2/p-1)+\varepsilon}\|f\|_{\ell^p(\mathbb{Z}^d)}$$ for $(d+2)/d\leq p\leq 2$, where $p'$ is the Hölder conjugate exponent of $p$ and $A_R$ is the probability average over the lattice sphere of radius $R$. This extends the fixed-radius estimates of Kesler–Lacey and Hughes to the sharp lower endpoint $p=(d+2)/d$.

In a previous work we computed the number of integral forms, over its field of definition, of an irreducible representation of a dihedral group. Here we apply the theory of Bruhat-Tits buildings to give similar formulas for a wider family of metacyclic groups that have representations of arbitrarily large dimension. Occasionally, these formulas can be extended to groups containing a subgroup in that family.

For $k\geq2$, let $S_r(k;\ell)$ be the smallest $n$, if exists, such that every $r$-coloring of $\{1,2,\ldots,n\}$ has a monochromatic solution $\mathcal{S}$ to the equation \[ x_1+x_2+\cdots+x_k=x_{k+1} \] such that the number of distinct integers in $\mathcal{S}$ is exactly $\ell+1$. We prove that, if $\ell\geq2$ is fixed, then \[ S_2(k;\ell)= k^2+\left[\frac{(\ell+1)(\ell-2)}{2}+2\right]k+\ell(\ell-2) \] for all large enough $k$. In particular, we have $S_2(k;2)=k^2+2k$ for all $k\geq3$.

We investigate the asymptotic behavior of the rectangle partition functions $p(n,n)$ and $p(3,n)$. The function $p(n,n)$ counts partitions of the square $n\times n$ into rectangular blocks with integer sides, while $p(3,n)$ counts such partitions of the rectangle $3\times n$. Two rectangle partitions are identified when they contain the same multiset of rectangle types, and a block $a\times b$ is identified with a block $b\times a$. Our main results are $$ p(n,n) = \exp\left( \left(\tfrac{\pi}{\sqrt3}+o(1)\right)n\sqrt{\log n} \right) $$ and $$ p(3,n) = \exp\left( \pi\sqrt{\tfrac{11n}{3}}+O(\log n) \right). $$ We also present simpler upper and lower bounds for $p(n,n)$ and an independent upper bound for $p(3,n)$.

The divided generalized Bernoulli values $b_{\chi,j} = fB_{1,\chi\omega^{-j}} \bmod p$, for $\chi$ an odd primitive Dirichlet character of conductor $f$ and order $d$ with $d \mid p-1$, control (for $p \nmid \varphi(f)$) the odd isotypic components of the $p$-class group of $Q(\zeta_{fp})$ through the characterwise abelian Main Conjecture; a zero is a twisted irregular pair, a branch with positive Iwasawa lambda-invariant, in the tradition studied by Ernvall, Holden, Delbourgo-Knospe and Knospe. We survey these zeros in the regime complementary to existing tabulations: fixed small conductor and large $p$ ($f = 3, 5$ to $p < 10^5$; all odd primitive characters of conductor at most 20 to $p < 2 \cdot 10^4$), computing each spectrum by a residue-class weight formula and one Bluestein convolution over $F_p$, a direct finite-field alternative of the same quasi-linear order as the standard power-series method. The survey records 27,508 zero lines over 55,121 character-prime pairs, each verified by two independent code paths with an exact order of vanishing; the counts and digits are consistent with the random model, and eight lines are non-simple, including one of depth three at $(f,p,j) = (19,37,16)$, giving class components of order exactly $p^2$ and $37^3$. The main contribution converts zeros into explicit certified generators: the conductor-three catalogue of arXiv:2607.23177 is extended from $p < 500$ to $p < 10^5$, all 2,441 zero lines simple, each projected circular unit proven to generate its complete order-$p$ Hilbert class field component by a fresh split-prime Artin certificate – the largest in the degree-199,980 field $Q(\zeta_{299973})$. Ancillary files contain all tables, certificates, and a verification program.

We study interlacing between real polynomial quotients obtained from centered binomial samples of a fixed even or odd real entire function at consecutive odd degrees. For sources of order at most one, we determine the optimal vertical-strip half-widths guaranteeing weak interlacing for every adjacent pair: $\sqrt{15/28}$ for even sources and the explicit algebraic constant $h_*\approx 1.0656$ for odd sources. These constants are forced by the first nontrivial pair and suffice at every degree. Under the same sharp bounds, nonpolynomial sources have simple zeros in $(-2,2)$ and strict interlacing at every positive index. We also prove an eventual result: if a nonpolynomial source has growth order less than two and only finitely many zeros off the imaginary axis, then all sufficiently high adjacent quotients have simple interior zeros and strictly interlace; the starting index depends only on the finite factor formed from those zeros. The proof turns the sampling problem into an interval real-rootedness problem for a Jacobi differential operator. For fixed degree, we also determine the maximal range of scalar shifts preserving real-rooted polynomials with at most one zero outside the interval. Applications give strict interlacing for quotients attached to completed Dedekind zeta derivatives and to successive centered blocks of critical values of self-dual newforms satisfying explicit spacing and parity conditions.

An $(\alpha,\beta)$ Somos-$4$ sequence $S_n$ is defined by the recurrence $S_nS_{n-4}=\alpha S_{n-1}S_{n-3}+\beta S_{n-2}^2$ ($n\geq 4$), with suitable initial values, where $\alpha$ and $\beta$ are constant parameters. A widely studied question is the following: When does the Hankel transform of a generating function become an $(\alpha,\beta)$ Somos-4 sequence? In particular, how can $\alpha$ and $\beta$ be derived for such a function? A sufficient condition for this problem has been established by Wang and Zhang. In this paper, we obtain the following three main results. (i): We extend the Wang–Zhang sufficient condition by working over the rational function field. Then we combine this result with the Sulanke–Xin quadratic transformation to resolve all of Barry's currently unsolved $(\alpha,\beta)$ Somos-4 conjectures, which arise in diverse contexts, including generalized Catalan recurrences, Riordan arrays, generalized Bernstein arrays, and elliptic curves. (ii): We show that the odd and even subsequences of an $(\alpha,\beta)$ Somos-4 sequence are again $(\alpha,\beta)$ Somos-4 sequences with transformed parameters. This is employed to establish Barry's Hurwitz transform conjecture. (iii): Using the theory of orthogonal polynomials, we prove a Hankel determinant formula and thereby prove a conjecture related to the $(\alpha,\beta)$ Somos-4 sequence. In addition, we prove some conjectures on formulas for periodic Hankel determinants.

Let \[ \xi\!\left(\frac12+z\right) =\sum_{n\geq 0}\frac{\gamma(n)}{n!}z^{2n}, \qquad J^{d,n}(X) =\sum_{j=0}^{d}\binom dj\gamma(n+j)X^j . \] The Riemann hypothesis is equivalent to the hyperbolicity of $J^{d,n}$ for every $d,n\geq0$. We prove that there is an absolute constant $K>0$ such that \[ n^3\log^2(n+2)\geq Kd^5 \quad\Longrightarrow\quad J^{d,n}\ \text{is hyperbolic}. \] Along every sequence with $n,d\to\infty$ in this region, the empirical measure of the naturally centered and scaled zeros also converges to Wigner's semicircle law. This gives a simultaneous degree–derivative version of the global semicircle consequence of the fixed-degree Hermite limit of Griffin, Ono, Rolen, and Zagier.

Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over $\mathbb{Q}$ during 1984-1989. We complete this program by proving that the last remaining sporadic group, the Mathieu group $M_{23}$, occurs as a Galois group over $\mathbb{Q}$. In fact, we produce an explicit degree $23$ polynomial with rational coefficients whose splitting field has Galois group $M_{23}$ over $\mathbb{Q}$. To accomplish this, we discover an unanticipated splitting of the Nielsen class associated to a non-rigid triple of conjugacy classes of $M_{23}$. We compute the Belyi maps associated with this Nielsen class to construct an explicit regular Galois extension of $\mathbb{Q}(t)$ with Galois group $M_{23}$. Essential for our computation is the numerical Belyi map algorithm developed and implemented by Costa, Klug, Musty, Schiavone, Sijsling, and Voight, inspired by ideas of Hejhal and Stark. As a bonus, one specialization of the regular extension yields an $M_{22}$-extension of $\mathbb{Q}$.