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Yichao Tian

Publications and source records attributed to Yichao Tian.

17 recordsLinked to original sources
arXiv2026

Adaptively Point-weighting Curriculum Learning

Curriculum learning (CL) mimics human learning, in which easy samples are learned first, followed by harder samples, and has become an effective method for training deep networks. However, many existing automatic CL methods maintain a preference for easy samples during the entire training process regardless of the constantly evolving training state. This is just like a human curriculum that fails to provide individualized instruction, which can delay learning progress. To address this issue, we propose an adaptively point-weighting (APW) curriculum learning method that assigns a weight to each training sample based on its training loss. The weighting strategy of APW follows the easy-to-hard training paradigm, guided by the current training state of the network. We present a theoretical analysis of APW, including training effectiveness, training stability, and generalization performance. Experimental results validate these theoretical findings and demonstrate the superiority of the proposed APW method.

cs.LG↗
arXiv2025

A prismatic-etale comparison theorem in the semistable case

Let $K|\mathbb{Q}_p$ be a complete discrete valuation field with perfect residue field, $O_K$ be its ring of integers. Consider a semistable $p$-adic formal scheme $X$ over $\mathrm{Spf}(O_K)$ with smooth generic fiber $X_η$. Du--Liu--Moon--Shimizu showed recently that the category of analytic prismatic $F$-crystals on the absolute log prismatic site of $X$ is equivalent to the category of semistable étale $\mathbb{Z}_p$-local systems on the adic generic fiber $X_η$. In this article, we prove a comparison between the Breuil--Kisin cohomology of an analytic log prismatic $F$-crystal on $X$ and the étale cohomology of its corresponding étale $\mathbb{Z}_p$-local system. This generalizes Guo--Reneicke's prismatic--étale comparison for crystalline $\mathbb{Z}_p$-local systems to the semi-stable case

math.AG↗
arXiv2024

Iwasawa's main conjecture for Rankin-Selberg motives in the anticyclotomic case

In this article, we study the Iwasawa theory for cuspidal automorphic representations of $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ over CM fields along anticyclotomic directions, in the framework of the Gan--Gross--Prasad conjecture for unitary groups. We prove one-side divisibility of the corresponding Iwasawa main conjecture: when the global root number is $1$, the $p$-adic $L$-function belongs to the characteristic ideal of the Iwasawa Bloch--Kato Selmer group; when the global root number is $-1$, the square of the characteristic ideal of a certain Iwasawa module is contained in the characteristic ideal of the torsion part of the Iwasawa Bloch--Kato Selmer group (analogous to Perrin-Riou's Heegner point main conjecture).

math.NT↗
arXiv2023

Finiteness and Duality for the cohomology of prismatic crystals

Let $(A, I)$ be a bounded prism, and $X$ be a smooth $p$-adic formal scheme over $\Spf(A/I)$. We consider the notion of crystals on Bhatt--Scholze's prismatic site $(X/A)_{\prism}$ of $X$ relative to $A$. We prove that if $X$ is proper over $\Spf(A/I)$ of relative dimension $n$, then the cohomology of a prismatic crystal is a perfect complex of $A$-modules with tor-amplitude in degrees $[0,2n]$. We also establish a Poincaré duality for the reduced prismatic crystals, i.e. the crystals over the reduced structural sheaf of $(X/A)_{\prism}$. The key ingredient is an explicit local description of reduced prismatic crystals in terms of Higgs modules.

math.AG↗
arXiv2021

On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives

In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg $L$-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin-Selberg $L$-function, then the Bloch-Kato Selmer group is of rank one.

math.NT↗
arXiv2021

Deformation of rigid conjugate self-dual Galois representations

In this article, we study deformations of conjugate self-dual Galois representations. The study has two folds. First, we prove an R=T type theorem for a conjugate self-dual Galois representation with coefficients in a finite field, satisfying a certain property called rigid. Second, we study the rigidity property for the family of residue Galois representations attached to a symmetric power of an elliptic curve, as well as to a regular algebraic conjugate self-dual cuspidal representation.

math.NT↗
arXiv2018

Tate cycles on some quaternionic Shimura varieties mod p

Let $F$ be a totally real field in which a prime number $p>2$ is inert. We continue the study of the (generalized) Goren--Oort strata on quaternionic Shimura varieties over finite extensions of $\mathbb F_p$. We prove that, when the dimension of the quaternionic Shimura variety is even, the Tate conjecture for the special fiber of the quaternionic Shimura variety holds for the cuspidal $π$-isotypical component, as long as the two unramified Satake parameters at $p$ are not differed by a root of unity.

math.NT↗
arXiv2018

Supersingular locus of Hilbert modular varieties, arithmetic level raising and Selmer groups

This article has three goals. First, we generalize the result of Deuring and Serre on the characterization of supersingular locus of modular curves to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use the previous results to study the Selmer group of certain triple product motive of an elliptic curve, in the context of the Bloch--Kato conjecture.

math.NT↗
arXiv2017

On Tate conjecture for the special fibers of some unitary Shimura varieties

Let $F$ be a totally real field in which a fixed prime $p$ is inert, and let $E$ be a CM extension of $F$ in which $p$ splits. We fix two positive integers $r,s \in \mathbb N$. We investigate the Tate conjecture on the special fiber of $G(U(r,s) \times U(s,r))$-Shimura variety. We construct cycles which we conjecture to generate the Tate classes and verify our conjecture in the case of $G(U(1,s) \times U(s,1))$. We also discuss the general conjecture regarding special cycles on the special fibers of unitary Shimura varieties.

math.NT↗
arXiv2016

On Goren-Oort stratification for quaternionic Shimura varieties

Let $F$ be a totally real field in which $p$ is unramified. We study the Goren-Oort stratification of the special fibers of quaternionic Shimura varieties over a place above $p$. We show that each stratum is a $(\mathbb{P}^1)^N$-bundle over other quaternionic Shimura varieties (for some appropriate $N$).

math.AG↗
arXiv2016

p-adic Cohomology and classicality of overconvergent Hilbert modular forms

Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method follows the original cohomological approach of Coleman. The key ingredient of the proof is giving an explicit description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety. A byproduct of the proof is to show that, at least when $p$ is inert, of the rigid cohomology of the ordinary locus has the same image as the classical forms in the Grothendieck group of Hecke modules.

math.NT↗
arXiv2013

Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case

We extend the modularity lifting result of the arXiv:1111.2804 to allow Galois representations with some ramification at p. We also prove modularity mod 2 and 5 of certain Galois representations. We use these results to prove many new cases of the strong Artin conjecture over totally real fields in which 5 is unramified. As an ingredient of the proof, we provide a general result on the automatic analytic continuation of overconvergent p-adic Hilbert modular forms of finite slope which substantially generalizes a similar result in arXiv:1111.2804.

math.NT↗
arXiv2010

An upper bound on the Abbes-Saito filtration for finite flat group schemes and applications

Let $\cO_K$ be a complete discrete valuation ring of residue characteristic $p>0$, and $G$ be a finite flat group scheme over $\cO_K$ of order a power of $p$. We prove in this paper that the Abbes-Saito filtration of $G$ is bounded by a simple linear function of the degree of $G$. Assume $\cO_K$ has generic characteristic 0 and the residue field of $\cO_K$ is perfect. Fargues constructed the higher level canonical subgroups for a Barsotti-Tate group $\cG$ over $\cO_K$ which is "not too supersingular". As an application of our bound, we prove that the canonical subgroup of $\cG$ of level $n\geq 2$ constructed by Fargues appears in the Abbes-Saito filtration of the $p^n$-torsion subgroup of $\cG$.

math.NT↗
arXiv2008

p-adic Monodromy of the Universal Deformation of a HW-cyclic Barsotti-Tate Group

Let k be an algebraically closed field of characteristic $p>0$, and $G_0$ be a Barsotti-Tate group (or $p$-divisible group) over k. We denote by $S$ the "algebraic" local moduli in characteristic p of $G_0$, by $G$ the universal deformation of $G_0$ over $S$, and by $U\subset S$ the ordinary locus of $G$. The etale part of $G$ over $U$ gives rise to a monodromy representation $ρ$ of the fundamental group of $U$ on the Tate module of $G$. Motivated by a famous theorem of Igusa, we prove in this article that $ρ$ is surjective if $G_0$ is connected and HW-cyclic. This latter condition is equivalent to that Oort's $a$-number of $G_0$ equals 1, and it is satisfied by all connected one-dimensional Barsotti-Tate groups over $k$.

math.AG↗
arXiv2008

Canonical subgroups of Barsotti-Tate groups

Let $S$ be the spectrum of a complete discrete valuation ring with fraction field of characteristic 0 and perfect residue field of characteristic $p\geq 3$. Let $G$ be a truncated Barsotti-Tate group of level 1 over $S$. If ``$G$ is not too supersingular'', a condition that will be explicitly expressed in terms of the valuation of a certain determinant, we prove that we can canonically lift the kernel of the Frobenius endomorphism of its special fibre to a subgroup scheme of $G$, finite and flat over $S$. We call it the canonical subgroup of $G$.

math.NT↗