### Jiuya Wang's Research

• Pointwise Bound for $$\ell$$-torsion in Class Groups II: Nilpotent Extensions, submitted for publication, [arxiv] [pdf].

• This is a sequel paper of my previous paper on bounding $$\ell$$-torsion in class groups for elementary abelian groups. For every finite $$p$$-group $$G_p$$ that is non-cyclic and non-quaternion and every positive integer $$\ell\neq p$$ that is greater than $$2$$, we prove the first non-trivial bound on $$\ell$$-torsion in class group of every $$G_p$$-extension. More generally, for every nilpotent group $$G$$ where every Sylow-$$p$$ subgroup $$G_p\subset G$$ is non-cyclic and non-quaternion, we prove a non-trivial bound on $$\ell$$-torsion in class group of every $$G$$-extension for every integer $$\ell>1$$. All results are unconditional and pointwise.

• Generalized Bockstein Maps and Massey products, with Yeuk Hay Joshua Lam, Yuan Liu, Romyar Sharifi and Preston Wake, submitted for publication, [arxiv] [pdf].

• Given a profinite group $$G$$ of finite $$p$$-cohomological dimension and a pro-$$p$$ quotient $$H$$ of $$G$$ by a closed normal subgroup $$N$$, we study the filtration on the cohomology of $$N$$ by powers of the augmentation ideal in the group algebra of $$H$$. We show that the graded pieces are related to the cohomology of $$G$$ via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups $$H$$, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on $$H$$. We apply our study to give a new proof of the vanishing of triple Massey products in Galois cohomology.

• Malle's Conjecture for $$G\times A$$ with $$G = S_3,S_4,S_5$$, with Riad Masri, Frank Thorne and Wei-Lun Tsai, submitted for publication, [arxiv] [pdf].

• We prove Malle's conjecture for $$G \times A$$, with $$G=S_3, S_4, S_5$$ and $$A$$ an abelian group. This builds upon my previous work, which proved this result with restrictions on the primes dividing $$A$$.

• $$\ell$$-torsion Bounds for the Class Group of Number Fields with an $$\ell$$-group as Galois group, with Jürgen Klüners, submitted for publication, [arxiv] [pdf].

• We describe the relations among the $$\ell$$-torsion conjecture for $$\ell$$-extensions, the discriminant multiplicity conjecture for nilpotent extensions and a conjecture of Malle giving an upper bound for the number of nilpotent extensions. We then prove all of these conjectures in these cases.

• Pointwise Bound for $$\ell$$-torsion in Class Groups: Elementary Abelian Extensions, Journal für die reine und angewandte Mathematik (Crelle) , Published online 13 Oct 2020, DOI: https://doi.org/10.1515/crelle-2020-0034 [arxiv] [pdf].

• Elementary abelian groups are finite groups in the form of $$A = (Z/pZ)^r$$ for a prime number $$p$$. For every integer $$\ell> 1$$ and $$r > 1$$, we prove a non-trivial upper bound on the $$\ell$$-torsion in class groups of every $$A$$-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the $$\ell$$-torsion in class groups are bounded non-trivially for every G-extension and every integer $$\ell> 1$$. When $$r$$ is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg-Venkatesh under GRH.

• Secondary Term of Asymptotic Distribution of $$S_3\times A$$ Extensions over $$\mathbb{Q}$$ , submitted for publication, [arxiv] [pdf].

• We combine a sieve method together with good uniformity estimates to prove a secondary term for the asymptotic estimate of $$S_3\times A$$ extensions over $$\mathbb{Q}$$ when $$A$$ is an odd abelian group with minimal prime divisor greater than 5. At the same time, we prove the existence of a power saving error when $$A$$ is any odd abelian group.

• Malle's conjecture for $$S_n\times A$$ for $$n = 3,4,5$$, Compositio Mathematica, Published online Jan 2021, DOI: https://doi.org/10.1112/S0010437X20007587 [arxiv] [pdf].

• We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method we can prove Malle's conjecture for $$S_n\times A$$ over any number field $$k$$ for $$n=3$$ with $$A$$ an abelian group of order relatively prime to 2, for $$n= 4$$ with $$A$$ an abelian group of order relatively prime to 6 and for $$n= 5$$ with $$A$$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $$C_3\wr C_2$$ in its $$S_9$$ representation, whereas its $$S_6$$ representation is the first counter example of Malle's conjecture given Klüners.

• The 2-Class Tower of $$\mathbb{Q}(\sqrt{-5460})$$ , with Nigel Boston, Geometry, Algebra, Number Theory, and Their Information Technology, Toronto, Canada, June 2016 and Kozhikode, India, August 2016 , [arxiv] [pdf].

• The seminal papers in the field of root-discriminant bounds are those of Odlyzko and Martinet. Both papers include the question of whether the field $$\mathbb{Q}(\sqrt{-5460})$$ has finite or infinite 2-class tower. This is a critical case that will either substantially lower the best known upper bound for lim inf of root-discriminants (if infinite) or else give a counterexample to what is often termed Martinet's conjecture or question (if finite). Using extensive computation and introducing some new techniques, we give strong evidence that the tower is in fact finite, establishing other properties of its Galois group en route.

#### PhD Thesis

• Malle's Conjecture for Compositum of Number Fields [pdf].

#### Others

• Oberwolfach Report: Inductive Methods for Proving Malle's Conjecture, with Robert J. Lemke Oliver and Melanie Matchett Wood, [Abstract].

• We propose a general framework to inductively count number fields. By using this method, we prove the asymptotic distribution for extensions with Galois groups in the form of $$T\wr B$$ where $$T = S_3$$ or every abelian groups and $$B$$ is an arbitrary group with the associated counting function not growing too fast.