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Yigeng Zhao

Publications and source records attributed to Yigeng Zhao.

9 recordsLinked to original sources
arXiv2024

On the multiplicity of the eigenvalues of discrete tori

It is well known that the standard flat torus $\mathbb{T}^2=\mathbb{R}^2/\Z^2$ has arbitrarily large Laplacian-eigenvalue multiplicities. We prove, however, that $24$ is the optimal upper bound for the multiplicities of the nonzero eigenvalues of a $2$-dimensional discrete torus. For general higher dimension discrete tori, we characterize the eigenvalues with large multiplicities. As consequences, we get uniform boundedness results of the multiplicity for a long range and an optimal global bound for the multiplicity. Our main tool of proof is the theory of vanishing sums of roots of unity.

math.SP↗
arXiv2024

Special values of spectral zeta functions and combinatorics: Sturm-Liouville problems

In this paper, we apply the combinatorial results on counting permutations with fixed pinnacle and vale sets to evaluate the special values of the spectral zeta functions of Sturm-Liouville differential operators. As applications, we get a combinatorial formula for the special values of spectral zeta functions and give a new explicit formula for Bernoulli numbers.

math.CO↗
arXiv2023

Special values of spectral zeta functions of graphs and Dirichlet L-functions

In this paper, we establish relations between special values of Dirichlet $L$-functions and that of spectral zeta functions or $L$-functions of cycle graphs. In fact, they determine each other in a natural way. These two kinds of special values were bridged together by a combinatorial derivative formula obtained from studying spectral zeta functions of the first order self-adjoint differential operators on the unit circle.

math.NT↗
arXiv2022

Cohomological Milnor formula and Saito's conjecture on characteristic classes

We confirm the quasi-projective case of Saito's conjecture, namely that the cohomological characteristic classes defined by Abbes and Saito can be computed in terms of the characteristic cycles. We construct a cohomological characteristic class supported on the non-acyclicity locus of a separated morphism relatively to a constructible sheaf. As applications of the functorial properties of this class, we prove cohomological analogs of the Milnor formula and the conductor formula for constructible sheaves on (not necessarily smooth) varieties.

math.AG↗
arXiv2018

Duality for relative logarithmic de Rham-Witt sheaves on semistable schemes over $\mathbb{F}_q[[t]]$

We study duality theorems for the relative logarithmic de Rham-Witt sheaves on semi-stable schemes $X$ over a local ring $\mathbb{F}_q[[t]]$, where $\mathbb{F}_q$ is a finite field. As an application, we obtain a new filtration on the maximal abelian quotient $π^{\text{ab}}_1(U)$ of the étale fundamental groups $π_1(U)$ of an open subscheme $U \subseteq X$, which gives a measure of ramification along a divisor $D$ with normal crossing and $\text{Supp}(D) \subseteq X-U$. This filtration coincides with the Brylinski-Kato-Matsuda filtration in the relative dimension zero case.

math.AG↗
arXiv2018

On the relative twist formula of $\ell$-adic sheaves

We propose a conjecture on the relative twist formula of $\ell$-adic sheaves, which can be viewed as a generalization of Kato-Saito's conjecture. We verify this conjecture under some transversal assumptions. We also define a relative cohomological characteristic class and prove that its formation is compatible with proper push-forward. A conjectural relation is also given between the relative twist formula and the relative cohomological characteristic class.

math.AG↗
arXiv2018

Characteristic class and the epsilon factor of an étale sheaf

We prove a twist formula for the epsilon factor of a constructible sheaf on a projective smooth variety over a finite field in terms of characteristic class of the sheaf. This formula is a modified version of the formula conjectured by Kato and Saito in [Ann. Math., 168 (2008):33-96, Conjecture 4.3.11]. We give two applications of the twist formula. Firstly, we prove that the characteristic classes of constructible étale sheaves on projective smooth varieties over a finite field are compatible with proper push-forward. Secondly, we show that the two Swan classes in the literature are the same on proper smooth surfaces over a finite field.

math.AG↗
arXiv2017

Duality for relative logarithmic de Rham-Witt sheaves and wildly ramified class field theory over finite fields

In order to study $p$-adic étale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect field of characteristic $p>0$, we introduce new $p$-primary torsion sheaves. It is a modification of the logarithmic de Rham-Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$. Then we establish a perfect duality between cohomology groups of the logarithmic de Rham-Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite field, the duality can be used to study wild ramification class field theory for the open subvariety $U$.

math.AG↗