arXiv subjects

Frank Lu

Publications and source records attributed to Frank Lu.

4 recordsLinked to original sources

arXiv2026

Finiteness for Étale Fundamental Groups of Néron Models

In this paper, we prove that the étale fundamental group of the Néron model of an abelian variety over a number field $K$ is the semidirect product of a finite group with the étale fundamental group of the ring of integers of $K.$ We prove this by studying how the Faltings height of an abelian variety changes under covers that spread out to finite étale covers of its Néron model. We then strengthen this result for elliptic curves. Using Merel's torsion theorem, we show the size of this finite group can be uniformly bounded for a fixed number field. We conclude by giving the list of all possible étale fundamental groups for the Néron model of an elliptic curve over $\mathbb{Q}.$

math.NT↗

arXiv2025

Special Cases of the Shafarevich Conjecture for Complete Intersections in Abelian Varieties

In this paper, we prove the Shafarevich conjecture for certain complete intersections of hypersurfaces in abelian varieties defined over a number field $K$ using the Lawrence-Venkatesh method. The main new inputs we need are computation of certain Euler characteristics of these complete intersections and a big monodromy statement for the variation of Hodge structure arising from the middle cohomology of a family of such complete intersections. Following \cite{ls25}, we prove the latter by relating this monodromy statement to a statement about Tannaka groups, which we then convert into a combinatorial statement.

math.NT↗

arXiv2023

%-Immanants and Temperley-Lieb Immanants

In this paper, we investigate the relationship between Temperley-Lieb immanants, which were introduced by Rhoades and Skandera, and %-immanants, an immanant based on a concept introduced by Chepuri and Sherman-Bennett. Our main result is a classification of when a Temperley-Lieb immanant can be written as a linear combination of %-immanants. This result uses a formula by Rhoades and Skandera to compute Temperley-Lieb immanants in terms of complementary minors. Using this formula, we also derive an explicit expression for the coefficients of a Temperley-Lieb immanant coming from a $321$-, $1324$-avoiding permutation $w$ containing the pattern $2143,$ which we use to derive our main result.

math.CO↗

arXiv2014

Numerical and Experimental Studies on the Separation Topology of the MVG Controlled Flow at M=2.5

In this paper, the implicitly implemented LES method and fifth order bandwidth optimized WENO scheme are used to make comprehensive studies on the separation topology of the MVG controlled flow at M=2.5 and Reθ=5760. Experiments are also made to verify the prediction of the computation. Analyses are conducted on three categories of the topology: the surface separation, cross-section separation and the three dimensional structure of the vortices. A complete description about the separation topology and a series of new findings are obtained. Among them, a pair of spiral point is first predicted by the computation and verified by the experiment. A corresponding new vortex model with 7 vortex tubes is presented also.

physics.flu-dyn↗