Papers

Let $G$ be a quasi-split reductive group over a finite field and let $\^G$ be the Langlands dual group. Assuming the derived subgroup of $G$ is almost simple, we use techniques from the geometric Langlands program to relate special values of Bessel functions for generic principal series representations of $G$ to the trace of Frobenius acting on Kloosterman sheaves of Heinloth-Ngô-Yun for $\^G$. We also give explicit examples in essentially all possible cases, including exceptional groups.

Let $k=\mathbb F_q$. We reconstruct every $k$-indecomposable $g\in k(X)$ of degree greater than one whose normal closure has genus one. Such a map is automatically separable and, after independent degree-one changes of the source and target coordinates over $k$, arises from a separable equivariant isogeny between elliptic curves equipped with compatible finite group actions stable under Frobenius conjugation. In particular, $\deg g=\ell$ or $\ell^2$ for a prime $\ell$, with $\ell\ne\operatorname{char}k$ in the latter case. More generally, every separable rational function with genus-one Galois closure admits a canonical factorization class, modulo degree-one changes of the intermediate coordinates over $k$, determined by the intrinsic translation subgroup of its geometric monodromy group; every $k$-indecomposable factor of the remaining map has Galois closure of genus zero. For every equivariant-isogeny quotient and every finite extension $k_r/k$, the same exact finite-kernel condition characterizes both permutation of $\mathbf P^1(k_r)$ and exceptionality over $k_r$. The finite kernel and its induced Frobenius and linear symmetry actions also determine the arithmetic and geometric monodromy permutation groups, decomposition classes, and the exact periodic set of permutation extension degrees, including its least period and limiting proportion. Together with the corresponding genus-zero theorem, these results yield permutation if and only if exceptionality over every finite extension for every separable rational function whose Galois closure has genus at most one; under $k$-decomposition, the common extension-degree set is the intersection of the corresponding sets for the factors.

We define a new theory of logarithmic Gromov–Witten invariants allowing negative contact orders using pierced logarithmic curves. We prove that these classes satisfy logarithmic birational invariance and satisfy loop and splitting theorems generalising those for classical Gromov–Witten invariants.

For an elliptically fibered 4-manifold $\pi\colon M\to B$, we introduce a subgroup of the smooth mapping class group of $M$ with explicit geometric representatives: the smooth narrow Mordell-Weil group $MW_0(\pi)$. Leveraging Kodaira's classification of singular fibers, we construct a natural isomorphism between $MW_0(\pi)$ and the orthogonal complement of the trivial lattice in $H^2(M)$. This identification parallels its holomorphic counterpart. This paper generalizes work of Farb and Looijenga, who defined the smooth Mordell-Weil group and computed it for fibrations over a sphere with only nodal fibers. We give an explicit formula for the rank of $MW_0(\pi)$ depending only on the genus of the base curve and the types of singular fibers of $\pi$. As a corollary, we show that rational elliptic surfaces are the only elliptic surfaces with holomorphic and smooth Mordell-Weil groups of the same rank.

We generalize and compare local-global obstructions for algebraic stacks over number fields. For smooth separated Deligne-Mumford stacks of finite type with a quasi-projective coarse moduli space, we prove that the descent obstruction is contained in the Brauer–Manin obstruction. By lifting $\mathbb{G}_m$-gerbes, we obtain an inclusion between the corresponding composite obstructions. We also show that in this setting the descent obstruction coincides with both the étale Brauer-Manin obstruction and the iterated descent obstruction. The Brauer-Manin obstruction also coincides with the iterated Brauer-Manin obstruction.

Let $k=\mathbb F_q$ be a finite field of characteristic $p$, and let \[ U_n(k)=\{F\in\operatorname{TA}_n(k):F(0)=0,\ JF(0)=I_n\}. \] We prove that every special tame polynomial automorphism is repeatedly Cartier-admissible for the de Rham flux class associated with \[ \lambda=x_1\,dx_2\wedge\cdots\wedge dx_n. \] Lowest-weight projection of the iterated Cartier descents gives additive characters \[ \chi_{r,j}:U_n(k)\longrightarrow(k,+), \qquad r\ge0,\quad 1\le j\le n, \] and for every fixed group element all but finitely many of these characters vanish. Put \[ S_{n,p}=\begin{cases} \mathbf N_{\ge1},&(n,p)=(2,2),\\ \mathbf N_0,&\text{otherwise}. \end{cases} \] The joint character map is surjective onto \[ \bigoplus_{s\in S_{n,p}}(k^n,+), \] and every fixed-coordinate submap onto \[ \bigoplus_{s\in S_{n,p}}(k,+) \] admits an explicit group-theoretic section. Consequently, \[ \dim_{\mathbb F_p} \frac{U_n(k)}{[U_n(k),U_n(k)]\,U_n(k)^{[p]}} =\aleph_0. \] The character $\chi_{r,j}$ factors through the jet of order \[ N_r=(n-1)(p^{r+1}-1), \] and this order is optimal for every $r\in S_{n,p}$. Since $U_n(k)$ has finite index in $\operatorname{TA}_n(k)$, the tame polynomial automorphism group over a finite field is finitely generated exactly in dimension one. In particular, this proves the Maubach–Willems finite-generation conjecture for $n\ge3$.

We provide an explicit bound for the image of the Zilber–Pink locus of a curve in an abelian variety under reduction modulo a suitably large prime $p$. We also provide a new proof of finiteness of the Zilber–Pink locus. Under further assumptions on the abelian variety, we prove a bound on the size of the locus.

We prove that every affine terminal Gorenstein toric variety of dimension at most four is unobstructed. In dimension four, the obstruction space need not vanish; instead, we determine the possible homogeneous obstruction degrees and construct simultaneous two-parameter deformations which eliminate the remaining potential obstructions.

We give the first known examples of Oka Calabi–Yau manifolds in any dimension: every smooth hypersurface of multidegree $(2,\ldots,2)$ in a product of projective lines is Oka. Building on this example, we prove that in every local Kuranishi family of K3 surfaces the points with Oka fibers form a dense $G_\delta$ subset of the base, and the sublocus of Oka fibers with vanishing Néron–Severi group is likewise a dense $G_\delta$ subset. We also obtain projective and nonprojective Oka Kummer surfaces and prove an analogous density statement in the Kummer period domain. We also prove the existence of Oka Enriques surfaces and Oka irreducible hyperkähler fourfolds.

We present a conjectural construction of Arthur packets within Fargues-Scholze's framework for the categorical local Langlands correspondence (CLLC). This construction consists of three parts. We first provide an overview the main statement of the CLLC, the underlying moduli stacks – $\mathrm{Par}_G$ of parameters and $\mathrm{Bun}_G$ of $G$-bundles – on the two sides of the correspondence, and its relation to representations of reductive p-adic groups. We then review the geometric Satake correspondence in order to define Hecke operators and the spectral action of sheaves on $\mathrm{Par}_G$ on sheaves on $\mathrm{Bun}_G$, and to construct semisimple parameters using excursion data. Finally, we generalize the geometric construction of Arthur packets from pushing-forward skyscraper sheaves on the regular conormal bundle of $\mathrm{Par}_G$ over $\mathbb C$ to a conjectural analogous operation on the stack of singularities on $\mathrm{Par}_G$ over $\overline{\mathbb Q}_\ell$.

We study the (canonical) 1st derivatives of proper submersions represented as height functions and belonging to a certain class: a proper map means a map the preimage of a compact set by which is always compact. We investigate their Reeb spaces. They are the quotient spaces defined by the equivalence relations on the manifolds of the domains where we identify two points in a same connected component of a same level set of them. They have been important since the establishment of theory of Morse functions, in the 20th century, They are in certain tame situations $0$- or $1$-dimensional and graphs naturally. Related facts have been shown by Gelbukh and Saeki in the 2020s for certain proper smooth real-valued functions. In non-proper cases, even related explicit theory has been difficult, except some previously given case of the author. Our study is on a new related case.

Let $(X,\Delta)\to C$ be a family of pairs over a smooth curve in positive characteristic $p>0$. We study the discrepancies of divisors over the base-change $(X,\Delta)\to C$ along Artin–Schreier and Frobenius covers of $C$, through ramification data for Artin–Schreier and degree $p$ purely inseparable extensions of DVRs. As an application, we recover a result of Hu and Zong about permanence of local stability for such base-change.

For $X$ a conical affine symplectic singularity with $\mathbb{T}=T \times \mathbb{G}_m$-action, the fixed scheme $X^T$ and the map $X^T \rightarrow X$ carry much information about the geometry of $X$. In general, $X^T \rightarrow X$ fails to be a complete intersection. Thus, we study a derived intersection whose classical locus is the $T$-fixed subscheme $X^T$. We show that the structure of the symplectic singularity on $X$ produces a duality theorem for the structure sheaf of the derived intersection. The duality theorem allows us to study the structure of such derived intersections; in particular we describe their cohomological amplitude. An important source of symplectic singularities with $\mathbb{T}$-action are affine Grassmannian slices $\overline{W}^{\lambda}_{\mu}$. We pay particular attention to these slices when $G=\mathrm{SL}_{n+1}$, and we use the previously developed theory to characterize when $(\overline{W}^{\lambda}_{\mu})^T \rightarrow \overline{W}^{\lambda}_{\mu}$ is a complete intersection.

Let $C/K$ be a curve of genus 2 over an algebraically closed field $K$. Every such curve comes equipped with a canonical quadratic form $q_C$ called its refined Humbert invariant. In the case that the Jacobian $J_C$ of $C$ is isogenous to the self-product $E \times E$ for an elliptic curve $E/K$ with complex multiplication (CM), we provide an explicit formula for the finite number $N_C$ of isomorphism classes of genus $2$ curves $C'/K$ whose refined Humbert invariant is equivalent to $q_C$. This formula implies that $N_C$ is unbounded for such curves, and that there are only finitely many isomorphism classes of such curves $C/K$ with a given value of $N_C$. A key step in our approach is a characterization of when $J_C$ is isogenous to a self product of a CM elliptic curve, formulated purely in terms of properties of the refined Humbert invariant $q_C$. We establish that an analogous characterization also holds for superspecial curves of genus 2. The paper concludes with explicit examples illustrating cases where a genus 2 curve is uniquely determined by the invariant $q_C$.

For a non-smooth toric flop of the simplest type, we provide the $\textit{quantum product formulas}$ and construct the $\textit{degree-preserving}$ quantum correspondence via $\textit{generalized}$ analytic continuation which is achieved through a $\textit{regularization}$ map. Moreover, the generating functions of Gromov–Witten invariants with ancestors are invariant for $\textit{all genera}$.

In this paper, after reviewing recent progress on the cohomology of $\overline{{\cal M}}_{0,n}$, we further our investigation on the cohomology of moduli spaces of pointed curves in continuation of [2,4,5,6,7,8]. In particular, we prove that the Betti number distribution of the Fulton-MacPherson compactification $C[n]$ of the space of $n$ ordered distinct points on any smooth projective curve $C$ is asymptotically Gaussian as $n$ goes to infinity.

For a smooth projective curve over a finite field with a fixed point at infinity, we establish a universal correspondence between finite residue duality and geometric kernel functions. We prove that the finite residue Casimir tensor is realized geometrically as the intrinsic principal part of the normalized Szegő kernel for any acyclic line bundle, and equivalently that this kernel acts as a reproducing kernel for the finite residue pairing, in exact analogy with the classical Cauchy integral formula. In the polynomial case these equivalent descriptions yield a closed formula involving the rank-two Weil operator, identified with the classical divided difference, recovering a remainder identity of Hu–Ou. These results provide a geometric foundation for the study of Anderson generating functions and the Weil pairing for Drinfeld modules.

We observe that standard derived equivalences give a counterexample to a conjecture of Kalck on global dimension for weighted projective lines. For the root stack $X=\mathbb{P}^1\langle \infty,0,1;2,3,3\rangle$ we exhibit a $13$-dimensional radical-square-zero algebra $A$ such that $$ D^b(\mathrm{coh}X)\simeq D^b(\mathrm{mod}A), \qquad \mathrm{gldim}A=4>3. $$

We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial \[f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], \] let $U$ be its augmented support matrix whose columns are $(1,u_j)$ with rank $\rho$ and $N(f) := \# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}$ be its torus point count. Our first main result is a quantum algorithm that outputs $\widehat{N}(f)$ satisfying \[ |\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-\rho} \] with success probability $1-\delta$. Provided that $\rho$ and $\|U\|_\infty$ are bounded, the algorithm runs in both classical bit and quantum gate complexity $\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/\delta))$. It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in $\log q$. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes $\#$P-hard under randomized polynomial-time Turing reductions when the support matrix $U$ varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.

Finiteness and injectivity of the moduli morphisms $\mu_n\colon|nH|_{\rm{sm}}\to\mathcal{M}_{g_n}$ associated with linear systems on K3 surfaces are understood in many cases. In this paper we study their ramification. Existing vanishing and stability results already imply unramifiedness when the polarisation or the multiple is sufficiently positive, so our focus is on low-degree phenomena. In particular, for a general K3 surface of degree $2$, we prove that the primitive moduli map is quasi-finite but ramified at exactly $171$ points, identified with the jumping lines of a logarithmic tangent bundle associated with the branch sextic. On the other hand, we prove that if $X\subset\mathbb{P}^n$ is a general hypersurface of degree at least $4$, then $\rm{H}^0(Y,T_X|_Y)=0$ for every smooth hyperplane section $Y$ of $X$. In particular, the moduli map for smooth hyperplane sections of a general quartic K3 surface is unramified. We moreover show that ramification does occur for higher multiples of the primitive polarisation on both degree-$2$ and degree-$4$ K3 surfaces, and relate the question to the stability of the tangent bundle.

By a result of Backelin, Veronese subrings of a standard $\mathbb{Z}$-graded algebra are eventually Koszul. Davis, Erman, and Martinova recently conjectured that the analogous statement holds for Veronese subrings of polynomial rings with positive nonstandard $\mathbb{Z}$-gradings. We disprove this conjecture. For the weighted polynomial ring $\mathbb{K}[x_1,x_2,x_3,x_4]$ with weights $(1,4,7,9)$, we prove that the $(9k+12)$-th Veronese subring is not nonstandard Koszul for every $k\geq1$. Our counterexamples arise from certain arrangements of lattice points, which we call cubic obstruction configurations. Each such configuration produces a minimal cubic generator in the defining ideal of the corresponding associated graded ring. This construction yields counterexamples in $n$ variables for all $n\geq4$. Moreover, we prove that the set of primitive four-variable weight vectors for which eventual nonstandard Koszulness fails has positive density. For a fixed three-variable grading, our cubic obstruction can occur at only finitely many Veronese indices, leaving that case open.

It was proven by Yoshioka that given a complex abelian surface $A$ of Picard rank $1$ whose primitive polarization is non-principal of type $(1,d)$, there is an isomorphism $\Psi:\mathrm{Hilb}^d_A\times\hat{A}\to M_{\hat{A}}(0,\hat{l},-1)$ where $\mathrm{Hilb}^d_A$ is the Hilbert scheme of lenth-$d$ subschemes of $A$ and $M_{\hat{A}}(0,\hat{l},-1)$ is a moduli space of Gieseker-stable sheaves on the dual abelian surface. Specifically, $M_{\hat{A}}(0,\hat{l},-1)$ parametrizes rank $1$ torsion-free sheaves with Euler characteristic $-1$ that are supported on a curve whose Néron–Severi class is dual to that of the polarization on $A$. The Fourier–Mukai transform is a crucial component of $\Psi$. In this paper, we use the isomorphism $\Psi$ to deduce information about curves on abelian surfaces. In the case that $Z\in \mathrm{Hilb}^d_A$ is symmetric, i.e., fixed by the inverse map $\iota$ on $A$, we use quadratic forms to compute information about the sheaf $\Psi(Z)$ and its supporting curve. It was recently shown by Knutsen and Lelli-Chiesa that any singularity on a curve of geometric genus $2$ contained in a general $(d_1,d_2)$-polarized abelian surface must have multiplicity at most $6$, among other constraints. In contrast, we demonstrate there are curves with singularities of arbitrarily high multiplicity contained in general $(1,d)$-polarized abelian surfaces for sufficiently large $d$. Furthermore, we identify the isolated fixed points of $\iota$ in acting on the variety of Kummer type $K_{\hat{A}}(0,\hat{l},-1)$ when $d=4$. Along the way, we prove some structural results on symmetric line bundles, showing that if $d$ is even, the dual of an odd line bundle is odd.

A sixer is a configuration of six pairwise skew lines on a smooth cubic surface, equivalently a choice of six exceptional curves defining a blow-down to $\mathbb{P}^2$. We study the $72$ sixers on the Fermat cubic surface $x_0^3+x_1^3+x_2^3+x_3^3=0$. We show that these sixers split into two orbits under the automorphism group of the Fermat cubic, of sizes $18$ and $54$. We give a geometric interpretation of this decomposition through the corresponding plane blow-up models: representatives of the two orbit types determine six-point configurations in $\mathbb{P}^2$ whose projective automorphism groups have orders $36$ and $12$, respectively. These groups identify with the stabilizers of the corresponding sixers and recover the two orbit sizes. We then compute the projective groups associated with representatives of the two orbits over $K=\mathbb Q(\omega)$, where $\omega^2+\omega+1=0$, and distinguish them arithmetically by the determinant square-class character $\delta_K:\mathrm{PGL}_2(K)\longrightarrow K^*/(K^*)^2$. Its images have $\mathbb F_2$-dimensions $1$ and $2$ for the orbits of sizes $18$ and $54$, respectively. Modulo $13$, the corresponding finite images are $\mathrm{PSL}_2(\mathbb F_{13})$ and $\mathrm{PGL}_2(\mathbb F_{13})$, respectively, and the determinant-character distinction persists for all choices of normalization triple.

We prove Dennin's conjecture (Conjecture 8.9 of arXiv:2506.21052) that his variant of dual RSK correspondence is symmetric when restricted to biGrassmannian permutations. For a binary matrix $A$, let $A^\dagger$ denote its transpose-complement, and let $\operatorname{ins}(A)$ and $\operatorname{rec}(A)$ denote its insertion and recording tableaux. We prove that $\operatorname{ins}(A^\dagger) = \overline{\operatorname{rec}(A)}$, where the bar denotes the natural complement of the recording tableau. Our proof uses rectification of super pipe dreams and a downward induction on suffixes of $A$.

We prove the wall-crossing formula conjectured by Gross–Joyce–Tanaka for equivariant enumerative invariants of CY4 categories equipped with framing functors. We also establish a version for stable pairs with fixed-determinant obstruction theories, as used in earlier applications by the first-named author. The main technical ingredient, which we develop in this work, is a construction of well-behaved CY4 pullback virtual classes using Jouanolou devices.

We derive a closed formula for the Masur-Veech volumes of spin-parity components of the stratum of abelian differentials with a single zero of maximal order. Our approach is based on the intersection theory of the virtual subcone of spin-parity squares inside the Hodge bundle, recently developed by Holmes-Politopoulos-Sauvaget. This is a spin refinement of a classical result of Sauvaget and agrees with the lattice-point counting technique of Chen-Möller-Sauvaget-Zagier. We further apply this theory to compute spin counterparts of virtual volumes, defined recently by Sauvaget. These virtual volumes are related to area Siegel-Veech constants and we are able to compute these invariants component-wise. In addition, by comparing our method for non-parity squares with the classical result of Sauvaget, we compute specific Chiodo integrals.

Let $k$ be an algebraically closed field of characteristic zero. Degree-four rational maps on $\mathbb{P}^1$, up to conjugation, correspond to pairs of binary forms $(F,G)\in V_5\oplus V_3$. The associated invariant ring is the joint invariant ring $\mathcal{R}_{5,3}=k[V_5\oplus V_3]^{\mathrm{SL}_2}$. We determine a minimal generating set for $\mathcal{R}_{5,3}$, consisting of fifty explicit joint transvectants of degrees at most $18$. As consequences we describe the null cone of $V_5\oplus V_3$, construct six explicit absolute invariants that generate the function field of the moduli space $\mathcal{M}_4^1$, and give an effective criterion for determining conjugacy of degree-four rational maps.

We prove equidistribution of small points and subvarieties on quasi-projective varieties over function fields, confirming \cite[Conjecture~5.4.1]{yuan2021adelic} for compactified relatively nef metrized line bundles in the function field case. The argument relies on the generic curves to reduce higher-dimensional bases to curves. We also establish the equidistribution theorem for arithmetically big line bundles using positive intersection.

Let $C\subset X(1)\times X(1)$ be a smooth curve defined over $\mathbb{C}_p$ with irreducible reduction. We study the intersection loci of $C$ with the modular subvarieties $Y_0(n)$. If the curve $C$ avoids points where both coordinates have supersingular reduction, or if a sequence $Y_0(a_n)$ is taken where the numbers $a_n$ have increasing divisibility by $p$, then these loci equidistribute to the unique canonical point of the analytification, $C^\text{Berk}$. If neither condition is satisfied, we expect equidistribution to fail and give a family of examples whose behaviour is believed to be typical. We also study the accumulation points of the set $C\cap\bigcup_nY_0(n)$ and show that this set is always non-discrete.

We prove that if a smooth affine complex surface has the same rational mixed Hodge structure as a smooth projective curve of positive genus, then it admits a smooth morphism to the curve with fibers isomorphic to affine line. In particular, every such surface is the total space of a torsor for a line bundle on the curve. The motivation comes from Jouanolou's trick, to which this result gives a kind of converse in dimension 2.