arXiv subjects

Zhao Yu Ma

Publications and source records attributed to Zhao Yu Ma.

5 recordsLinked to original sources

arXiv2026

Optimal homological vanishing: cancellation of character sums and Patterson's conjecture over $\mathbb{F}_q[t]$

Many arithmetic sums over function fields can be expressed in terms of $H_i(B_n, V^{\otimes n})$ for some braided vector space $V$, and a vanishing line for these homology groups gives power-savings cancellation for the arithmetic sum. We prove an explicit vanishing line for $H_i(B_n,V^{\otimes n})$ depending only on the homology up to some finite $n$. Moreover, as the range of $n$ increases, the slope of the resulting vanishing line converges to the optimal slope. We also apply our methods to two different families of arithmetic sums. Firstly, we prove an upper bound for the bias of higher order Gauss sums over function fields, extending Patterson's conjecture beyond the cubic and quartic cases over number fields, and we conjecture this bound is sharp for orders that are prime powers. Secondly, we show that over Galois $G$-extensions, almost all character sums exhibit near square-root cancellation.

math.NT↗

arXiv2026

On the pointwise convergence of the number of abelian varieties over $\mathbb{F}_p$ with fixed trace

Extending Katz-Sarnak heuristics, Ballini-Lombardo-Verzobio [BLV25] conjectures a limiting distribution as $p \to \infty$ for $\# A_g(\mathbb F_p,t)$, the number of $g$-dimensional PPAVs over $\mathbb F_p$ with trace $t$, as a product of natural local factors $v_\ell(t)$ for non-archimedean places $\ell$ and the Sato-Tate measure $\text{ST}_g$ corresponding to $\infty$. We prove that their conjecture is true for all $g$. As a consequence, we obtain analogous results on the distribution of curves of genus $2$ and $3$, answering questions of Bergstr\"om-Howe-Garc\'ia-Ritzenthaler [BHLR24] and [BLV25].

math.NT↗

arXiv2026

Weight filtration of Hurwitz spaces and quantum shuffle algebras

We prove an equivalence between filtrations of primitive bialgebras and filtrations of factorizable perverse sheaves, generalizing the results obtained by Kapranov-Schechtman. Under this equivalence, we find that the word length filtration of quantum shuffle algebras as defined in Ellenberg-Tran-Westerland corresponds to the codimension filtration of factorizable perverse sheaves. Furthermore, we find that the geometric weight filtration of factorizable perverse sheaves corresponds to a filtration on quantum shuffle algebras which has not been previously defined in the literature, and we call this the algebraic weight filtration. To apply this to Hurwitz spaces, we prove a comparison theorem between the weight filtrations for Hurwitz spaces over $\mathbb F_p$ and $\mathbb C$, generalizing the comparison theorem of Ellenberg-Venkatesh-Westerland. This allows us to determine the cohomological weights for Hurwitz spaces explicitly using the algebraic weight filtration of the corresponding quantum shuffle algebra. As a consequence, we find that most weights of Hurwitz spaces are smaller than expected from cohomological degree, and we prove explicit nontrivial upper bounds for weights in some cases, such as when $G=S_3$ and $c$ is the conjugacy class of transpositions.

math.NT↗

arXiv2023

Refinements on vertical Sato-Tate

Vertical Sato-Tate states that the Frobenius trace of a randomly chosen elliptic curve over $\mathbb F_p$ tends to a semicircular distribution as $p\rightarrow \infty$. We go beyond this statement by considering the number of elliptic curves $N_{t,p}'$ with a given trace $t$ over $\mathbb F_p$ and characterizing the 2-dimensional distribution of $(t,N_{t,p}')$. In particular, this gives the distribution of the size of isogeny classes of elliptic curves over $\mathbb F_p$. Furthermore, we show a notion of stronger convergence for vertical Sato-Tate which states that the number of elliptic curves with Frobenius trace in an interval of length $p^\epsilon$ converges to the expected amount. The key step in the proof is to truncate Gekeler's infinite product formula, which relies crucially on an effective Chebotarev's density theorem that was recently developed by Pierce, Turnage-Butterbaugh and Wood.

math.NT↗

arXiv2022

The extremals of Stanley's inequalities for partially ordered sets

Stanley's inequalities for partially ordered sets establish important log-concavity relations for sequences of linear extensions counts. Their extremals however, i.e., the equality cases of these inequalities, were until now poorly understood with even conjectures lacking. In this work, we solve this problem by providing a complete characterization of the extremals of Stanley's inequalities. Our proof is based on building a new ``dictionary" between the combinatorics of partially ordered sets and the geometry of convex polytopes, which captures their extremal structures.

math.CO↗