Research

We study the nonvanishing of central values of ray class \(L\)-functions over a fixed imaginary quadratic field \(K\). We prove that, as \(\mathrm{N}(\mathfrak f)\) tends to infinity, at least a proportion \(1/3-\mathsf{o}_K(1)\) of the primitive ray class characters \(\chi\) modulo \(\mathfrak f\) satisfy \(L(1/2,\chi)\neq 0\). Our proof adapts the classical mollifier method to the ray class setting and establishes asymptotic formulas for the first and second mollified moments.

Let \(q=p^h>7\) be odd, put \(m=(q+1)/2\), and suppose that \(i=(m-2)/2\) satisfies \(\gcd(i,m)=\gcd(i+2,m)=1\). For the \(\mathbf{F}_{q^2}\)-maximal function field \(\mathcal{F}_i=\mathbf{F}_{q^2}(x,y)\) defined by \(y^m=x^i(x^2+1)\), Peter Beelen, Maria Montanucci, Jonathan Niemann, and Luciane Quoos showed that the geometric automorphism group contains a subgroup of order \(4(q+1)\) and conjectured that its order is exactly \(4(q+1)\). We prove this equality by establishing the reverse inequality.

We study certain extremal problems in combinatorial geometry that ask about configurations of points in an \(n \times n\) grid that satisfy strict, global geometric constraints. Classical exact solvers suffer from combinatorial explosion for these types of problems, and standard reinforcement learning and transformer-based models struggle with the sparse reward "validity cliff" and quadratic token-consumption limits. To overcome these bottlenecks, we propose a Geometry-Aware Monte Carlo Tree Search (MCTS) framework. Our approach strictly enforces geometric constraints through incremental updates to the feasible action space. For constraints about collections of collinear points, like those that occur in the classic No-Three-in-Line problem (Max-N3IL), this mechanism reduces the constraint checking complexity from \(O(n^3)\) to \(O(n^2)\). To improve search efficiency, we exploit geometric symmetries in two ways: canonical pruning during node expansion to reduce the branching factor, and symmetric batch transitions to accelerate the discovery of promising configurations. We perform extensive experiments and establish new best-known computational results on five out of six of the problems that we considered. Notably, for Max-N3IL we find configurations of size roughly \(1.8n\) for grids of size \(82 \le n \le 119\). For the Smallest Complete Set problem, we find configurations of size roughly \(0.95n\), providing new upper bounds within the tested grids. This work establishes Geometry-Aware MCTS as a highly adaptable framework for discovering novel configurations in combinatorial geometry.

The Jacobian group (also called the sandpile group, Picard group, or critical group) of a graph or, more generally, of a regular matroid has been well studied. Sixth-root-of-unity matroids, also called complex unimodular matroids, are generalizations of regular matroids. This paper provides a definition, and establishes some basic properties, of the Jacobian group of a sixth-root-of-unity matroid.

We aim to predict the \(\mathbb{Q}\)-gonality of modular curves, an invariant measuring the minimal degree of a nonconstant rational map to \(\mathbb{P}^1\). Three machine-learning architectures—extreme gradient-boosted trees, feedforward neural networks, and transformer-based models—achieve over 90\% exact-match accuracy on existing curves, with more than 89\% of predictions falling within known theoretical bounds. To improve interpretability, we employ an LLM-guided boost symbolic regression pipeline that proposes nonlinear feature combinations and uncovers concise analytic formulas. LLM-Boost outperforms the symbolic-regression baselines while producing compact, human-readable expressions that suggest how core arithmetic invariants interact. Our results highlight the effectiveness of combining data-driven prediction with LLM-enhanced symbolic discovery in arithmetic geometry.