Jiuya Wang's Research

• Counterexamples for Tünrkelli's Modification on Malle's Conjecture, [pdf].

  • We give counterexamples for the modification on Malle's Conjecture given by Tünrkelli. Tünrkelli's modification on Malle's conjecture is inspired by an analogue of Malle's conjecture on function field. As a byproduct, our counterexamples show that the \(b\) constant can be different between function field and number field. Along the same line, we also show that Klünners' counterexamples give counterexamples for a natural extension of Malle's conjecture on counting number fields by product of ramified primes.

• Inductive Methods for Counting Number Fields, with Brandon Alberts, Robert J.Lemke Oliver and Melanie Matchett Wood, [pdf].

  • We give a new method for counting extensions of a number field asymptotically by discriminant, which we employ to prove many new cases of Malle’s Conjecture and counterexamples to Malle’s Conjecture. We consider families of extensions whose Galois closure is a fixed permutation group G. Our method relies on having asymptotic counts for \(T\)-extensions for some normal subgroup \(T\) of \(G\), and possibly weak, but uniform bounds on the asymptotic number of \(G/T\)-extensions. Our new results use \(T\) either abelian or \(S_3^m\), though our framework is general.

• The Residually Indistinguishable Case of Ribet's Method for \(GL_2\), with Samit Dasgupta, Mahesh Kakde and Jesse Silliman, [arxiv].

  • Ribet's method provides a strategy for constructing a nontrivial extension of a \(p\)-adic Galois representation \(\rho1\) by another such representation \(\rho2\). Suppose we are working over a local ring. An important assumption that occurs throughout literature is that the representations \(\rho_i\) are residually distinguishable i.e. are residually non-isomorphic. The main theorem of this paper is a general version of Ribet's Lemma for \(GL_2\) where we do not impose the assumption that the associated characters are residually distinguished.

• The Brumer-Stark Conjecture over \(\mathbb{Z}\), with Samit Dasgupta, Mahesh Kakde and Jesse Silliman, [arxiv].

  • In this paper we give a complete proof of the Brumer-Stark conjecture over \(\mathbb{Z}\).

• The Average Size of \(3\)-torsion in Class Groups of \(2\)-extensions, with Robert J.Lemke Oliver and Melanie Matchett Wood, [arxiv] [pdf].

  • We determine the average size of the \(3\)-torsion in class groups of \(G\)-extensions of a number field when \(G\) is any transitive \(2\)-group containing a transposition, for example \(D_4\). It follows from the Cohen--Lenstra--Martinet heuristics that the average size of the \(p\)-torsion in class groups of \(G\)-extensions of a number field is conjecturally finite for any \(G\) and most \(p\) (including \(p\nmid|G|\)). Previously this conjecture had only been proven in the cases of \(G=S_2\) with \(p=3\) and \(G=S_3\) with \(p=2\). We also show that the average \(3\)-torsion in a certain relative class group for these \(G\)-extensions is as predicted by Cohen and Martinet, proving new cases of the Cohen--Lenstra--Martinet heuristics. Our new method also works for many other permutation groups \(G\) that are not \(2\)-groups.

• Improved bounds on the number of number fields of small degree, with Theresa C. Anderson, Ayla Gafni, Kevin Hughes, Robert J. Lemke Oliver, David Lowry-Duda, Frank Thorne, and Ruixiang Zhang, [arxiv].

  • We study the number of degree n number fields with discriminant bounded by \(X\). In this article, we improve an upper bound due to Schmidt on the number of such fields that was previously the best known upper bound for \(6 \le n \le 94\).

• Generalized Erdős-Turán Inequalities and Stability of Energy Minimizers, with Ruiwen Shu, [arxiv] [pdf].

  • The classical Erdős-Turán inequality on the distribution of roots for complex polynomials can be equivalently stated in a potential theoretic formulation, that is, if the logarithmic potential generated by a probability measure on the unit circle is close to 0, then this probability measure is close to the uniform distribution. We generalize this classical inequality from d=1 to higher dimensions d>1 with the class of Riesz potentials which includes the logarithmic potential as a special case. In order to quantify how close a probability measure is to the uniform distribution in a general space, we use Wasserstein-infinity distance as a canonical extension of the concept of discrepancy. Then we give a compact description of this distance. Then for every dimension d, we prove inequalities bounding the Wasserstein-infinity distance between a probability measure ρ and the uniform distribution by the \(L^p\)-norm of the Riesz potentials generated by ρ. Our inequalities are proven to be sharp up to the constants for singular Riesz potentials. Our results indicate that the phenomenon discovered by Erdős and Turán about polynomials is much more universal than it seems. Finally we apply these inequalities to prove stability theorems for energy minimizers, which provides a complementary perspective on the recent construction of energy minimizers with clustering behavior.

• The Sharp Erdős-Turán Inequality, with Ruiwen Shu, submitted, [arxiv] [pdf].

  • Erdős and Turán proved a classical inequality on the distribution of roots for a complex polynomial in 1950, depicting the fundamental interplay between the size of the coefficients of a polynomial and the distribution of its roots on the complex plane. Various results have been dedicated to improving the constant in this inequality, while the optimal constant remains open. In this paper, we give the optimal constant, i.e., prove the sharp Erdős-Turán inequality. To achieve this goal, we reformulate the inequality into an optimization problem, whose equilibriums coincide with a class of energy minimizers with the logarithmic interaction and external potentials. This allows us to study their properties by taking advantage of the recent development of energy minimization and potential theory, and to give explicit constructions via complex analysis. Finally the sharp Erdős-Turán inequality is obtained based on a thorough understanding of these equilibrium distributions.

• Hilbert Transforms and the Equidistribution of Zeros of Polynomials, with Emanuel Carneiro, Mithun Das, Alexandra Florea, Angel V. Kumchev, Amita Malik, Micah B. Milinovich and Caroline Turnage-Butterbaugh, To appear in Journal of Functional Analysis [arxiv] [pdf].

  • We improve the current bounds for an inequality of Erdős and Turán from 1950 related to the discrepancy of angular equidistribution of the zeros of a given polynomial. Building upon a recent work of Soundararajan, we establish a novel connection between this inequality and an extremal problem in Fourier analysis involving the maxima of Hilbert transforms, for which we provide a complete solution. Prior to Soundararajan (2019), refinements of the discrepancy inequality of Erdős and Turán had been obtained by Ganelius (1954) and Mignotte (1992).

• 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, To appear in Proceedings of the American Mathematical Society , [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.