arXiv subjects

Ruofan Jiang

Publications and source records attributed to Ruofan Jiang.

9 recordsLinked to original sources

arXiv2026

Quot scheme of points on torus knot singularities

For $\gcd(a,b)=1$, we show that the moduli space of $m$-codimensional $\Bbbk[\![T^a,T^b]\!]$-submodules of $\Bbbk[\![T]\!]^n$ is paved by affine cells, by proving that each Bia\l ynicki-Birula stratum of a closed related moduli space with respect to the natural $\mathbb{G}_m$-action is an affine bundle over the fixed point locus and that the fixed point locus is an iterated Grassmannian bundle. As an application, we determine the motive of this moduli space in the Grothendieck ring of varieties in terms of an explicit two-variable series $N_{a,b;n}(q,t)$, and use it to explicit compute the groupoid volume of the category of finite modules over $\mathbb{F}_q[\![T^a,T^b]\!]$. The series $N_{a,b;n}$ carries the conjectures we then formulate. At $n=\infty$ we conjecture a bi-infinite family of Rogers--Ramanujan type identities by specializing the $t$-variable; we identify their product side with the normalized character of a module over the $\mathcal{W}$-algebra minimal model $\mathcal{W}_a(a,a+b)$, and observe a connetion to colored Jones tails. At $n<\infty$ we conjecture that $N_{a,b;n}$ is computed by the bottom $\alpha$-row of the trigraded $S^n$-colored HOMFLY homology of the torus knot $T(a,b)$, and that this same bottom row also computes the Quot schemes of finite codimensional $\Bbbk[\![T^a,T^b]\!]$-submoudles of $\Bbbk[\![T^a,T^b]\!]^n$ and the punctual Hilbert schemes of the non-reduced curve $(Y^a-X^b)^n=0$; the three quantities are special values at three points of the trigrading, and when $n=1$ they recover both the conjectures of Oblomkov--Rasmussen--Shende and of Kivinen--Trinh. Finally we conjecture that the one direction of the trigrading these three points do not see is a perverse filtration on the moduli spaces themselves, and we verify its prediction for a smooth germ at $n=2$ by computing the decomposition theorem for the $\mathrm{GL}_2$ spectral-curve family.

math.AG↗

arXiv2026

Picard rank jumps for families of K3 surfaces in positive characteristic

Let X/C be a non iso-trivial family of K3 surfaces over a curve C defined over characteristic p > 2 field. We show that if X avoids a necessary and structural obstruction coming from Frobenius, and satisfies a big monodromy condition, then there are infinitely may geometric fibers that have larger Picard rank than the geometric generic fiber.

math.AG↗

arXiv2025

$p$-adic monodromy and mod $p$ unlikely intersections, II

We study ordinary abelian schemes in characteristic $p$ and their moduli spaces from the perspective of char $p$ Mumford--Tate, log Ax--Lindemann, and geometric Andr\'e--Oort conjectures (abbreviated as $\MTT_p$, $\mathrm{logAL}_p$ and geoAO$_p$). In this paper, we achieve multiple goals: (\textbf{A}) establish the implication $\mathrm{MT}_p\Leftrightarrow \mathrm{logAL}_p \Rightarrow \mathrm{geoAO_p}$, and show that they all follow from the Tate conjecture for abelian varieties. The equivalence $\mathrm{MT}_p\Leftrightarrow \mathrm{logAL}_p$ is exploited from both sides, which enables us to \noindent(\textbf{B}) develop a representation theory approach to $\mathrm{logAL}_p$ and $\mathrm{geoAO_p}$ by first establishing many cases of MT$_p$ via classical techniques, and (\textbf{C}) develop an algebraization approach to $\MTT_p$ that transcends the limitation of classical methods. In particular, we introduce ``crystalline Hodge loci'', a rigid analytic geometric object that encodes the essential information needed for proving $\mathrm{logAL}_p$, while being very approachable via (integral and relative) $p$-adic Hodge theory. This enables us to prove $\mathrm{logAL}_p$ for compact Tate-linear curves with unramified $p$-adic monodromy. As an application, we establish $\MTT_p$ for many abelian fourfolds of $p$-adic Mumford type.

math.NT↗

arXiv2025

Matrix Points on Varieties

We study the cohomology of $C_n(X)$, the moduli space of commuting $n$-by-$n$ matrices satisfying the equations defining a quasi-projective scheme $X$. This space can be viewed as a non-commutative Weil restriction from the algebra of $n$-by-$n$ matrices to the ground field. We introduce a semi-simple counterpart $S_n(X)$, defined as the quotient of $X^n \times \mathrm{GL}_n/\mathrm{T}_n$ by the diagonal $S_n$ action. We show that there exists a natural map $\sigma \colon S_n(X) \to C_n(X)$ inducing isomorphism on $\ell$-adic cohomology under mild restrictions on $X$ or the characteristic of the field. This confirms a heuristic derived from Weil restrictions. Furthermore, we provide explicit combinatorial formulae for the Betti numbers of $C_n(X)$ and prove a Macdonald-type generating series. A version for Hermitian matrix point is also proved in the last section.

math.AG↗

arXiv2023

Motivic Coh and Quot zeta functions of singular curves

We present a general and effective algebraic framework for enumerating finite-length quotients of a torsion-free sheaf of arbitrary rank (the Quot zeta function) and finite-length coherent sheaves (the Coh zeta function) over reduced singular curves. We prove that Quot zeta functions are motivically rational, using a novel parametrization and the geometry of affine Grassmannians, and that they satisfy an arbitrary-rank reflection principle, via harmonic analysis. We show that the a normalized high-rank limit of Quot zeta functions converges to the Coh zeta function. As a first application, we compute explicit formulas for these zeta functions for all $y^2 = x^n$ singularities, revealing a surprising and previously unknown connection to Rogers--Ramanujan type $q$-series. Further applications to affine Springer fibers and commuting varieties are also discussed.

math.AG↗

arXiv2023

$p$-adic monodromy and mod $p$ unlikely intersections, I

We formulate characteristic $p$ analogues of the Mumford--Tate and the Andr\'e--Oort conjectures for ordinary mod $p$ Shimura varieties of Hodge type, and set up general frameworks for studying them. We prove the two conjectures for (subvarieties of) arbitrary products of GSpin Shimura varieties, by reducing them, via a notion of linearity for mod $p$ Shimura varieties, to a third conjecture of Ax--Schanuel type. Along the way, we solve Chai's Tate-linear conjecture for products of GSpin Shimura varieties, and reveal an intimate relation among the four conjectures mentioned above. Our proof uses Crew's parabolicity conjecture which is recently proven by D'Addezio.

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arXiv2023

Punctual Quot schemes and Cohen--Lenstra series of the cusp singularity

The Quot scheme of points $\mathrm{Quot}_{d,n}(X)$ on a variety $X$ over a field $k$ parametrizes quotient sheaves of $\mathcal{O}_X^{\oplus d}$ of zero-dimensional support and length $n$. It is a rank-$d$ generalization of the Hilbert scheme of $n$ points. When $X$ is a reduced curve with only the cusp singularity $\{x^2=y^3\}$ and $d\geq 0$ is fixed, the generating series for the motives of $\mathrm{Quot}_{d,n}(X)$ in the Grothendieck ring of varieties is studied via Gröbner bases, and shown to be rational. Moreover, the generating series is computed explicitly when $d\leq 3$. The computational results exhibit surprising patterns (despite the fact that the category of finite length coherent modules over a cusp is wild), which not only enable us to conjecture the exact form of the generating series for all $d$, but also suggest a general functional equation whose $d=1$ case is the classical functional equation of the motivic zeta function known for any Gorenstein curve. As another side of the story, Quot schemes are related to the Cohen--Lenstra series. The Cohen--Lenstra series encodes the count of "commuting matrix points'' (or equivalently, coherent modules of finite length) of a variety over a finite field, about which Huang formuated a "rationality'' conjecture for singular curves. We prove a general formula that expresses the Cohen--Lenstra series in terms of the motives of the (punctual) Quot schemes, which together with our main rationality theorem, provides positive evidence for Huang's conjecture for the cusp.

math.AG↗

arXiv2023

Splitting of almost ordinary abelian surfaces in families and the $S$-integrality conjectures

Let $A$ be a non-isotrivial almost ordinary abelian surface with possibly bad reductions over a global function field of odd characteristic $p$. Suppose $\Delta$ is an infinite set of positive integers, such that $\left(\frac{m}{p}\right)=1$ for $\forall m\in \Delta$. If $A$ does not admit any global real multiplication, we prove the existence of infinitely many places modulo which the reduction of $A$ has endomorphism ring containing $\mathbb{Z}[x]/(x^2-m)$ for some $m\in \Delta$. This implies that there are infinitely many places modulo which $A$ is not simple, generalizing the main result of arXiv:1812.11679 to the non-ordinary case. As an another application, we also generalize the $S$-integrality theorem for elliptic curves over number fields, as proved in arXiv:math/0509485, to the setting of abelian surfaces over global function fields.

math.NT↗

arXiv2022

Lattices in $\mathbb F_q[[T]]^d$ and spiral shifting operators

We investigate the algebra and combinatorics of an analogue of the Hermite normal form that classifies finite-index submodules of $\mathbb F_q[[T]]^d$. We identity both normal forms as instances of Gr\"obner basis theory under different monomial orders, where the Hermite normal form corresponds to the lex order, and the new normal form the hlex order. We note that the hlex normal form recovers the Smith normal form, a feature not enjoyed by the Hermite normal form. We also identify the combinatorial structure underlying the cell decomposition induced by the hlex normal form, which appears to be of independent interest. Notably, the statistics tracking the cell dimensions is compatible, in a certain way, with a collection of $d$ ``spiral shifting operators'' on $\mathbb N^d$, which pairwise commute and collectively act freely and transitively. Using these operators, we give direct proofs of some new combinatorial identities obtained by translating the results of Solomon and Petrogradsky in terms of the hlex normal form.

math.CO↗