Peikai Qi

My current mathematical interests are algebraic geometry and number theory, more specifically, Iwasawa theory, modular forms, elliptic curves, Massey products. Here is my arXiv author page and ORCID .

Papers and Writings

PDF versions available here are more up to date than those posted on arXiv.

• On the Quotient of a Pseudo-null Module (2026) PDF
  Abstract. Motivated by questions arising in noncommutative Iwasawa theory, let \(R\) be a (not necessarily commutative) ring, \(T \in R\) a regular central element, and \(M\) a pseudo-null \(R\)-module. We investigate necessary and sufficient conditions under which the quotient \(M/TM\) is pseudo-null as an \(R/TR\)-module. We first give necessary and sufficient conditions in terms of associated prime ideals when \(R\) is commutative. Then we apply this to the case when \(R\) is a Krull domain and obtain a precise relationship between the characteristic ideals of \(M/TM\) and the \(T\)-torsion submodule \(M[T]\). We give necessary and sufficient conditions in terms of \(\operatorname{Ext}\) groups when \(R\) is a noncommutative ring and then compare them with the criterion when \(R\) is commutative. Lastly, we give an application to noncommutative Iwasawa theory. Roughly speaking, if 'big' dual fine Selmer group is pseudo-null, then 'most' specialized dual fine Selmer group is pseudo-null.
• Pseudonullity for fine Selmer groups over multiple \(\mathbb{Z}_p^d\)-extensions (Joint with Ruichen Xu, submitted, 2026)
  Abstract. In this article, we develop vertical control theorems for fine Selmer groups associated with Galois representations and their deformations in \(p\)-adic families. We apply them to provide new evidence for Conjectures A and B of Coates–Sujatha and to obtain pseudonullity of the fine Selmer groups for cyclotomic characters, elliptic curves, classical cuspidal newforms, and Hida families of \(p\)-ordinary cuspidal newforms.
• Machine Learning and LLM-Boost Symbolic Regression for Predicting \(\mathbb{Q}\)-Gonality of Modular Curves (Joint with Xu Zhuang, Yuxiang Yao, Po-Chu Hsu, and Xiaokang Wang; accepted, 2025)
  Abstract. We investigate the prediction of \(\mathbb{Q}\)-gonality of modular curves using other arithmetic invariants recorded in the LMFDB. Gradient-boosted trees, feedforward neural networks, and transformer-based models attain over 90% exact-match accuracy on curves with known gonality, while more than 89% of their predictions satisfy the available theoretical bounds. To make the predictions more interpretable, we develop LLM-Boost, an LLM-assisted symbolic-regression approach that proposes nonlinear combinations of arithmetic features and produces concise analytic expressions. The resulting formulas perform competitively with symbolic-regression baselines and suggest relationships among the arithmetic invariants of modular curves.
• An Analogue of Greenberg's Pseudo-null Conjecture for CM Fields (Joint with Matt Stokes, accepted, Acta Arithmetica, 2026) PDF
  Abstract. Let \(K\) be a CM field, and let \(K^+\) be its maximal totally real subfield. Assume that the primes above \(p\) in \(K^+\) split in \(K\). Let \(S\) be a set containing exactly half of the prime ideals of \(K\) lying above \(p\). We show that, assuming Leopoldt's conjecture holds for \(K\) and \(p\), there exists a unique \(\mathbb{Z}_p\)-extension of \(K\) that is unramified outside \(S\) (the \(S\)-ramified \(\mathbb{Z}_p\)-extension of \(K\)). Such \(\mathbb{Z}_p\)-extensions for CM fields exhibit properties similar to the cyclotomic \(\mathbb{Z}_p\)-extensions of totally real fields. For example, Greenberg established a criterion ensuring that the Iwasawa invariants satisfy \(\mu=\lambda=0\) for the cyclotomic \(\mathbb{Z}_p\)-extension of a totally real field. We prove analogous results for the \(S\)-ramified \(\mathbb{Z}_p\)-extension of a CM field. Additionally, we provide a numerical criterion for \(\mu=\lambda=0\) in the case of an imaginary biquadratic field, which parallels the criterion given by Fukuda and Komatsu for real quadratic fields. We also give a generalization of de Shalit's result regarding the index of unit groups.
• On the Non \(p\)-Rationality and Iwasawa Invariants of Certain Real Quadratic Fields (Joint with Matt Stokes, submitted, 2024) PDF
  Abstract. Let \(p\) be an odd prime, and \(m,r\in\mathbb{Z}^{+}\) with \(m\) coprime to \(p\). In this paper we investigate the real quadratic fields \(K=\mathbb{Q}(\sqrt{m^2p^{2r}+1})\). We first show that for \(m < C\), where the constant \(C\) depends on \(p,r\), the fundamental unit \(\varepsilon\) of \(K\) satisfies the congruence \(\varepsilon^{p-1}\equiv 1\pmod{p^2}\), which implies that \(K\) is a non-\(p\)-rational field. Varying \(r\) then gives an infinite family of non-\(p\)-rational fields. When \(m=1\) and \(p\) is a non-Wieferich prime, we use a criterion of Fukuda and Komatsu to show that if \(p\) does not divide the class number of \(K\), then the Iwasawa invariants for a cyclotomic \(\mathbb{Z}_p\)-extension of \(K\) vanish. We conjecture that there are infinitely many \(r\) such that \(p\) does not divide the class number of \(K\).
• Iwasawa \(\lambda\) invariant and Massey product (Submitted, 2024) PDF
  Abstract. We compute the Iwasawa \(\lambda\)-invariant in terms of Massey products in Galois cohomology with restricted ramification. Applied to imaginary quadratic fields and cyclotomic fields, this approach yields a new proof and a generalization of results by Gold and McCallum–Sharifi. The main tool is the generalized Bockstein map introduced by Lam, Liu, Sharifi, Wake, and Wang.

Papers with undergraduate students

• Combinatorial and Analytical Properties of Higher-Order Recursive Polynomial Sequences (2023) (Joint with Luke Bridges, Max Budnick, Zachary Martin, Janine Wang, and Aklilu Zeleke; draft in progress.)
• Multivariate Fibonacci-like Polynomials and their Applications (2022) (Joint with Sejin Park, Etienne Phillips, Zhan Zhan, and Ilir Ziba)
• Universally uniformly continuous metric spaces (2017) (Joint with Katrina Gensterblum and Willie Wong)

Papers with assistance of AI

• Partial results on common roots of positive derivatives of geometric sums (2026)

Talks

Notes