January 20 – 22, 2026
Harbin Engineering University, Harbin, China
The scope of the Ice City Number Theory conference is Algebraic Number Theory, specifically focusing on Iwasawa theory, elliptic curves and modular forms, as well as algebraic groups.
Our goal is to bring everyone together to talk about math, share ideas, and foster new connections in a collaborative environment amidst the unique winter atmosphere of Harbin.
| Time | Speaker | Title |
|---|---|---|
| 09:00 - 10:00 | Yi Ouyang | Isogeny-based cryptosystems and structure of isogeny graphs |
| 10:00 - 10:30 | Tea Break | |
| 10:30 - 11:30 | Andrei Jorza | $p$-adic $L$-functions for $GL(2n)$ |
| 11:30 - 12:30 | Daniel Delbourgo Zoom | Selmer groups of abelian varieties and the non-vanishing of the $\mu$-invariant |
| 12:30 - 14:00 | Lunch Break | |
| 14:00 - 15:00 | Yang Zhang | Using computation algebraic geometry for Feynman integrals |
| 15:00 - 16:00 | Katherine Stange Zoom | The arithmetic of thin orbits |
| 16:00 - 16:30 | Tea Break | |
| 16:30 - 17:30 | Daniel Kriz | A canonical splitting of the $p$-adic Hodge filtration and applications |
| Time | Speaker | Title |
|---|---|---|
| 09:00 - 10:00 | Chan-Ho Kim | Applications of Kolyvagin's conjecture to elliptic curves of arbitrary rank |
| 10:00 - 10:30 | Tea Break | |
| 10:30 - 11:30 | Andrei Rapinchuk Zoom | On almost strong approximation in reductive groups |
| 11:30 - 12:30 | Yu Kuang | An Integral Approach to Iwasawa Theory |
| 12:30 - 14:00 | Lunch Break | |
| Afternoon: Excursion to Harbin Ice and Snow World (冰雪大世界) | ||
| Time | Speaker | Title |
|---|---|---|
| 09:00 - 10:00 | Dinesh Thakur Zoom | Expected, average ranks and rank stability for elliptic curves over number fields |
| 10:00 - 10:30 | Tea Break | |
| 10:30 - 11:30 | Dong Quan Nguyen Zoom | An analogue of the Kronecker-Weber theorem for rational function fields over ultra-finite fields |
| 11:30 - 12:30 | Mulun Yin | Iwasawa theory of elliptic curves at Eisenstein primes |
| 12:30 - 14:00 | Lunch Break | |
| 14:00 - 15:00 | Peikai Qi | Iwasawa Invariants And Massey Products |
| 15:00 - 16:00 | Weitong Wang | Distribution of $K$-groups and reflection principle |
| 16:00 - 16:30 | Tea Break | |
| 16:30 - 17:30 | Di Zhang | From Gauss to Bianchi: An Algorithm for Computing Class Representatives of Binary Hermitian Forms |
Abstract: Let $p$ be an odd prime, and suppose that $f_1$ and $f_2$ newforms of weight two sharing the same irreducible Galois representation modulo $p$. We establish a transition formula relating the Iwasawa invariants $\lambda(f_1)$ and $\lambda(f_2)$ in the case where their underlying modular abelian varieties have the same dimension. For abelian extensions $F=\mathbb{Q}$, this essentially removes the $\mu(f_i)=0$ condition present in earlier work of Greenberg-Vatsal and Emerton-Pollack-Weston.
Abstract: In this talk, I will talk about my recent work that establishes a correspondence between Galois extensions of rational function fields over arbitrary fields $F_s$ and Galois extensions of the rational function field over the ultraproduct of the fields $F_s$. As an application, I will discuss an analogue of the Kronecker-Weber theorem for rational function fields over ultraproducts of finite fields. I will also describe an analogue of cyclotomic fields for these rational function fields that generalizes the works of Carlitz from the 1930s, and Hayes in the 1970s. If time permits, I will talk about how to use the correspondence established in my work to study the inverse Galois problem for rational function fields over finite fields.
Abstract: Lying between arithmetic and analytic aspects of modular forms and Galois representations, $p$-adic $L$-functions have been crucial in many of the incredible progress towards the Birch and Swinnerton-Dyer of the last decade, particularly in the work of Skinner, Loeffler, Zerbes, et al. Two main difficulties arise in their construction: an algebraic geometric input for controlling the rate of growth of $p$-adic $L$-functions, which characterizes them uniquely, and an automorphic input for relating them to classical $L$-functions. I will talk about recent work with Mladen Dimitrov on $p$-adic $L$-functions attached to general parahoric representations of $GL(2n)$ of symplectic type, a novel setting. Different constructions of $p$-adic $L$-functions are better suited for different applications, and we are using this construction for the study of trivial zeros.
Abstract: In early 1990's, Kolyvagin formulated the conjecture on the non-triviality of Heegner point Kolyvagin systems motivated by understanding the arithmetic of elliptic curves of arbitrary rank. The conjecture itself is now proved for a large class of elliptic curves thanks to the work of Wei Zhang and its various generalizations. We discuss some interesting applications of Kolyvagin's conjecture as well as a different proof of Kolyvagin's conjecture for semi-stable elliptic curves with supersingular reduction.
Abstract: I will describe a functorial splitting of the $p$-adic Hodge filtration on universal de Rham cohomology over the infinite-level Shimura curve which specializes to the unit root splitting on the ordinary locus. Using the splitting, I define theories of quasi-overconvergent modular forms and $p$-adic Maass-Shimura operators acting on them, which have applications to the construction of anticyclotomic $p$-adic $L$-functions for $p$ inert or ramified in the CM field, generalizing my previous work. I will discuss applications of these $p$-adic $L$-functions, including Sylvester's conjecture on primes expressible as a sum of two rational cubes, showing 100% of positive squarefree integers congruent to 5, 6, 7 mod 8 are congruent numbers, and results toward Goldfeld's conjecture.
Abstract: Classical Iwasawa theory is largely formulated after fixing a prime $p$ and working over the $p$-adic Iwasawa algebra $\mathbb{Z}_p [[ G ]] $, where $G$ is a compact $p$-adic Lie group (for example $\mathbb{Z}_p$). In this talk, we explore an integral approach by working directly over the completed group ring $\mathbb{Z} [[ G ]] $, prior to any $p$-adic specialisation. We discuss algebraic aspects of the resulting module theory over this non-Noetherian ring, and indicate how this perspective provides a more intrinsic algebraic framework accommodating Iwasawa-theoretic structures arising from arithmetic contexts. This talk is based on joint work with David Burns and Dingli Liang.
Abstract: Isogeny-based cryptography is one technical route for post-quantum cryptography. In this talk, we shall explain the mathematical problems about isogeny computation and the cryptosystems based on those problems. We will then explain several results about the local structure of the isogeny graphs of supersingular elliptic curves and of superspecial abelian surfaces. This talk is based on joint works with Zheng Xu and Zijian Zhou.
Abstract: Gold's criterion and the results of McCallum and Sharifi establish a connection between the Iwasawa invariant $\lambda \geq 2$ and the vanishing of cup products in cyclotomic $\mathbb{Z}_p$-extensions. We propose two directions for generalizing these results. First, by replacing cup products with Massey products, we establish a relationship between $\lambda \geq n$ and the vanishing of $n$-fold Massey products. Second, by replacing the cyclotomic $\mathbb{Z}_p$-extension with more general $\mathbb{Z}_p$-extensions, we construct $S$-ramified $\mathbb{Z}_p$-extensions and demonstrate their connection to Greenberg's pseudo-null conjecture.
Abstract: A criterion for strong approximation in algebraic groups was obtained by Platonov in characteristic zero, and by Margulis and Prasad in positive characteristic. It follows from this criterion that strong approximation never holds for non simply connected groups (in particular, algebraic tori) and a finite set of places. We will report on a recent work where we show that a slightly weaker property, which we termed “almost strong approximation” can hold for non simply connected reductive groups and some special infinite sets of places. Applying this fact to maximal tori of an absolutely almost simple simply connected group, we generalize some results on the congruence subgroup problem. Joint work with Wojciech Tralle.
Abstract: We consider the local-to-global question for orbits of thin groups/semigroups. We will discuss Apollonian circle packings, continued fractions, and some related problems. In the Apollonian case, we ask about the integers which occur as curvatures in a packing. We observe that they satisfy certain congruence restrictions, and ask whether all sufficiently large integers otherwise occur. In the case of continued fractions, we consider variants of Zaremba's conjecture on the rationals with bounded continued fractions. Joint work includes work with Haag, Kertzer, and Rickards.
Abstract: The talk will present and discuss a mixture of results, heuristics, conjectures, evidence and counter-examples related to the topics in the title.
Abstract: In this talk, we first introduce the classical Cohen-Lenstra Heuristics, which predict the distribution of class groups. We then discuss this framework for the "distribution problem of subfamilies", where the random fields contain a fixed subfield. Focusing on real quadratic fields $K$, we apply the result of Lim on $K$-groups to relate the $p$-part of the class group of $K(\mu_p)$ to the $p$-part of $K_2(O_K)$. Finally, using the reflection principle for the case $p=3$, we demonstrate that the distribution of these groups is determined by the 3-part of the class groups of imaginary quadratic fields. This is a joint work with Meng Fai Lim and Chao Qin.
Abstract: Recent decades have seen many advances in arithmetics of elliptic curves of analytic rank 0 and 1 at a prime $p$, most of which are restricted to curves whose $p$-torsion is an irreducible $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-module. In 2022, Castella--Grossi--Lee--Skinner first proved so-called anticyclotomic Iwasawa Main Conjectures for elliptic curves in the residually reducible case ('Eisenstein case') at primes of good ordinary reduction. We will discuss generalizations and subsequent works on Iwasawa theory of elliptic curves at Eisenstein primes, leading to new cases of the BSD conjecture.
Abstract: This talk presents a computational framework for determining class representatives of binary Hermitian forms over imaginary quadratic fields, generalizing the classical theory of binary quadratic forms to the context of Bianchi modular forms. Driven by the Shimura-Shintani-Waldspurger correspondence and its implications for central $L$-values, the work establishes a geometric algorithm based on the action of the Bianchi group $PSL_2(\mathcal{O}_K)$ on hyperbolic 3-space $\mathbb{H}^3$. By integrating height minimization, translation reduction, and Swan's method for fundamental domain verification, the algorithm systematically identifies unique representatives for specific discriminants. This is a joint work with Chao Qin.
Abstract: Feynman integrals are the central objects for perturbative quantum fields, for collider physics and formal theories, as well as the effective theory approach for gravitational wave. The linear reduction of Feynman integrals is a major bottleneck for Feynman integral computations. We introduce the computational algebraic geometry method, based on syzygy computations, for Feynman integral reductions. The cutting-edge examples of three-loop five-point Feynman integrals and two-loop Higgs plus b quark pair production for the LHC will be explained to demonstrate the power of this algebraic method.
More titles and abstracts will be posted here as they become available.