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Yujie Xu

Publications and source records attributed to Yujie Xu.

27 records · Page 2Linked to original sources
arXiv2022

Surjectivity of Galois Representations in Rational Families of Abelian Varieties

In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-$1$ subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension $g \geq 3$, there are infinitely many abelian varieties over $\mathbb Q$ with adelic Galois representation having image equal to all of $\operatorname{GSp}_{2g}(\widehat{\mathbb Z})$.

math.NT↗
arXiv2022

Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians

In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations.

math.NT↗
arXiv2022

An Unsupervised Masking Objective for Abstractive Multi-Document News Summarization

We show that a simple unsupervised masking objective can approach near supervised performance on abstractive multi-document news summarization. Our method trains a state-of-the-art neural summarization model to predict the masked out source document with highest lexical centrality relative to the multi-document group. In experiments on the Multi-News dataset, our masked training objective yields a system that outperforms past unsupervised methods and, in human evaluation, surpasses the best supervised method without requiring access to any ground-truth summaries. Further, we evaluate how different measures of lexical centrality, inspired by past work on extractive summarization, affect final performance.

cs.CL↗
arXiv2021

Normalization in integral models of Shimura varieties of Hodge type

Let $(G,X)$ be a Shimura datum of Hodge type, and $\mathscr{S}_K(G,X)$ its integral model with hyperspecial (resp. parahoric, assuming the group is unramified) level structure. We prove that $\mathscr{S}_K(G,X)$ admits a closed embedding, which is compatible with moduli interpretations, into the integral model $\mathscr{S}_{K'}(\mathrm{GSp},S^{\pm})$ for a Siegel modular variety. In particular, the normalization step in the construction of $\mathscr{S}_K(G,X)$ is redundant. In particular, our results apply to the earlier integral models constructed by Rapoport, Kottwitz etc. (resp. Rapoport-Zink etc.), as those models agree with the Hodge type integral models for appropriately chosen Shimura data. Moreover, combined with a result of Lan's on the boundary components of toroidal compactifications of integral models, our result also implies that there exist closed embeddings of toroidal compactifications of integral models of Hodge type into toroidal compactifications of Siegel integral models, for suitable choices of cone decompositions.

math.NT↗
arXiv2021

Finiteness of fppf cohomology

Let $R$ be a Henselian DVR with finite residue field. Let $G$ be a finite type, flat $R$-group scheme (not necessarily commutative) with smooth generic fiber. We show that $H^1_{\mathrm{fppf}}(\mathrm{Spec} R,G)$ is finite. We then give an application of the global analogue of this finiteness result to PEL type integral models of Shimura varieties.

math.NT↗
arXiv2020

Bounding Selmer groups for the Rankin--Selberg convolution of Coleman families

Let $f$ and $g$ be two cuspidal modular forms and let $\mathcal{F}$ be a Coleman family passing through $f$, defined over an open affinoid subdomain $V$ of weight space $\mathcal{W}$. Using ideas of Pottharst, under certain hypotheses on $f$ and $g$ we construct a coherent sheaf over $V \times \mathcal{W}$ which interpolates the Bloch-Kato Selmer group of the Rankin-Selberg convolution of two modular forms in the critical range (i.e. the range where the $p$-adic $L$-function $L_p$ interpolates critical values of the global $L$-function). We show that the support of this sheaf is contained in the vanishing locus of $L_p$.

math.NT↗
arXiv2017

Lifting Subgroups of Symplectic Groups over $\mathbb{Z} / \ell \mathbb{Z}$

For a positive integer $g$, let $\mathrm{Sp}_{2g}(R)$ denote the group of $2g \times 2g$ symplectic matrices over a ring $R$. Assume $g \ge 2$. For a prime number $\ell$, we give a self-contained proof that any closed subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z}_\ell)$ which surjects onto $\mathrm{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$ must in fact equal all of $\mathrm{Sp}_{2g}(\mathbb{Z}_\ell)$. The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties.

math.GR↗
arXiv2016

Quantum statistical mechanics in arithmetic topology

This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes system, with knots replacing primes, and cyclic branched coverings of the 3-sphere replacing abelian extensions of the field of rational numbers. The operator algebraic properties of this system differ significantly from the Bost-Connes case, due to the properties of the action of the semigroup of knots on a direct limit of knot groups. The resulting algebra of observables is a noncommutative Bernoulli crossed product. We describe the main properties of the associated quantum statistical mechanical system and of the relevant partition functions, which are obtained from simple knot invariants like genus and crossing number.

math-ph↗