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We consider two motivic generating functions defined on a variety, and reveal their tight connection. They essentially count torsion-free bundles and zero-dimensional sheaves. On a reduced singular curve, we show that the ``Quot zeta function'' encoding the motives of Quot schemes of 0-dimension quotients of a rank $d$ torsion-free bundle satisfies rationality and a Serre-duality-type functional equation. This is a high-rank generalization of known results on the motivic Hilbert zeta function (the $d=1$ case) due to Bejleri, Ranganathan, and Vakil (rationality) and Yun (functional equation). Moreover, a stronger rationality result holds in the relative Grothendieck ring of varieties. Our methods involve a novel sublattice parametrization, a certain generalization of affine Grassmannians, and harmonic analysis over local fields. On a general variety, we prove that the ``Cohen--Lenstra zeta function" encoding the motives of stacks of zero-dimensional sheaves can be obtained as a ``rank $\to\infty$'' limit of the Quot zeta functions. Combining these techniques, we prove explicit results for the planar singularities associated to the $(2,n)$ torus knots/links, namely, $y^2=x^n$. The resulting formulas involve summations of Hall polynomials over partitions bounded by a box or a vertical strip, which are of independent interest from the perspectives of skew Cauchy identities for Hall--Littlewood polynomials, Rogers--Ramanujan identities, and modular forms., Comment: 50 pages, 3 tables. Comments are appreciated!

Let $p$ be an odd prime. We state characteristic $p$ analogues of the Mumford--Tate conjecture and the Andr\'e--Oort conjecture for ordinary strata of mod $p$ Shimura varieties. We prove the conjectures for arbitrary products of GSpin Shimura varieties (and their subvarieties). Important subvarieties of GSpin Shimura varieties include modular and Shimura curves, Hilbert modular surfaces, $\mathrm{U}(1,n)$ unitary Shimura varieties, and moduli spaces of principally polarized Abelian and K3 surfaces. The two conjectures are both related to a notion of linearity for mod $p$ Shimura varieties, about which Chai has formulated the Tate-linear conjecture. Though seemingly different, the three conjectures are intricately entangled. We will first solve the Tate-linear conjecture for single GSpin Shimura varieties, above which we build the proof of the Tate-linear conjecture and the characteristic $p$ analogue of the Mumford--Tate conjecture for products of GSpin Shimura varieties. We then use the Tate-linear and the characteristic $p$ analogue of the Mumford--Tate conjectures to prove the characteristic $p$ analogue of the Andr\'e--Oort conjecture. Our proof uses Chai's results on monodromy of $p$-divisible groups and rigidity theorems for formal tori, as well as Crew's parabolicity conjecture which is recently proven by D'Addezio., Comment: 49 pages. Included the proof for the product case. Simplified the other parts. Comments welcome!

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 Gr\"obner 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., Comment: 64 pages, 14 figures, 4 tables; minor changes

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 $A$ doesn't 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 generalizes the $S$-integrality conjecture for elliptic curves over number fields, as proved in arXiv:math/0509485, to the setting of Abelian surfaces over global function fields. As a corollary, we show 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.

Fix an integer $d\geq 1$. Solomon counts index-$n$ submodules of the free module $R^d$ for any Dedekind domain $R$ that admits a Dedekind zeta function. The proof involves the Hermite normal form for matrices. We propose another normal form in the case where $R=\mathbb{F}_q[[T]]$, which unlike the Hermite normal form, recovers the Smith normal form. We use our normal form to give a uniform approach to Solomon's result and a refinement of it due to Petrogradsky, which counts finite-index submodules $M$ of $R^d$ such that $R^d/M$ has a given module structure. Our normal form and Hermite's can be interpreted as reduced Gr\"obner bases with respect to two different monomial orders. Counting with our normal form gives rise to a ``total distance'' statistics $W$ and certain ``spiral shifting'' operators $g_1,\dots,g_d$ on the set $X=\mathbb{N}^d$ of $d$-tuples of nonnegative integers. These combinatorial structures are of independent interest. We show that these operators commute, and collectively act on $X$ freely and transitively, which in particular gives a nontrivial self-bijection on $\mathbb{N}^d$. We give a formula on how $g_j$ interacts with the $W$-statistics on $X$, leading to a rational formula for the generating function for $W$-statistics over any forward orbit in $X$ under these operators., Comment: 25 pages