数论及相关领域青年学者研讨会

方江学： Galois representation of Drinfeld modules
In this talk, we study the Galois representations attached to products of Drinfeld modules. As an analogue of Serre’s classical result on the images of Galois representations associated with products of elliptic curves, we prove that for any finite set of primes p, the image of the corresponding product representation is sufficiently large, in the sense that it is commensurable with a subgroup defined by a natural determinant condition. Our approach combines Pink’s minimal model theory for compact subgroups of linear groups over local fields with explicit reciprocity laws for global function fields.

王浩然： Homological branching for GL₂
I will talk about the branching problem for mod p representations of GL₂(F), where F is a p-adic field. This is a joint work in progress with Yang Chen.

范洋宇： p-adic Gross-Zagier formulae for supersingular elliptic curves
The p-adic Gross-Zagier formula— the p-adic analogue of the celebrated Gross-Zagier formula— relates the p-adic height of Heegner points to the derivative of p-adic Rankin series. It plays an essential role in the Iwasawa Main Conjecture for elliptic curves and the Birch-Swinnerton-Dyer Formula. In this talk, we report some recent progress regarding the p-adic Gross-Zagier formulae for supersingular elliptic curves. This work is partly based on joint work with Prof. Y. Tian and Dr. J. Pan.

蔡立： Bessel distributions and Kloosterman integrals
In this talk, we shall give the germ expansions of Kloosterman integrals. As an application, we prove that if Kloosterman integrals have nontrivial bounds for all split groups, then Bessel distributions are regular for all split groups.

杨金榜： A Lefschetz Theorem for Crystalline Representations
As a corollary of nonabelian Hodge theory, Simpson proved a strong Lefschetz theorem for complex polarized variations of Hodge structure. In this talk, I will introduce an arithmetic analog. The primary technique is p-adic nonabelian Hodge theory. This is based on joint work with Raju Krishnamoorthy and Kang Zuo.

张神星： 具有最大弯曲分量的二次二项式向量函数
设 \(n=2m\), \(F(x)=x^{d_1}+x^{d_2}\in\mathbb{F}_{2^n}[x]\) 具有最多弯曲分量. 我们证明了: 当 \(2\) 进汉明权重 \(\mathrm{wt}_2(d_1),\mathrm{wt}_2(d_2)\le 2\) 时, 若 \(\ell(n):=\min\limits_{\gamma:\mathbb{F}_2(\gamma)=\mathbb{F}_{2^n}} \dim_{\mathbb{F}_2}\mathbb{F}_2[\sigma]\gamma >m\), 即 \(\mathbb{F}_{2^n}\) 任一生成元的Frobenius轨道的线性复杂度均大于 \(m\), 且 \(\gcd(d_1,d_2,2^m-1)>1\), 则 \(F(x)\) 等价于 \(x^{2^m+1}\) 或 \(x^{2^i}(x+x^{2^m})\). 我们还给出该条件下 \(F\) 的非线性性和差分均匀度的更好的下界. 本文与谢贤红、欧阳毅、程奇合作完成.

李加宁： Class groups of \( \mathbb{Q}(N^{1/p}) \) and related Eisenstein congruences
Let \(K=\mathbb{Q}(N^{1/p})\), where p and N are two primes such that \(p \mid N+1\). Iimura proved in 1980s that \(p\) divides the class number of \(K\) using genus theory. In 2021, Lang-Wake gave a new proof by constructing a special newform of level \(N^2\) which cuts out a unramified cyclic of degree \(p\) extension of \(K\). If \(p^r\) divides \(N+1\) exactly, Lang-Wake further in 2025 shows that there exists exactly \(r\) such newforms. We show that these \(r\) newforms cut out the same unramified cyclic degree \(p\)-extension of \(K\), under the assumption that \(p\) is a regular prime. We will also talk about some results on the Eisenstein congruences.