arXiv subjects
Publications and source records attributed to Sun Woo Park.
Let $E$ be a non-isotrivial elliptic curve over a global function field $\mathbb{F}_q(t)$ of characteristic coprime to $2$ and $3$. Under some explicit conditions, we determine the secondary terms for the first moments of prime Selmer groups of cyclic prime twist families of $E$ over $\mathbb{F}_q(t)$.
Fix a prime number $\ell \geq 5$. Let $K = \mathbb{F}_q(t)$ be a global function field of characteristic $p$ coprime to $2,3$, and $q \equiv 1 \text{ mod } \ell$. Let $C:y^\ell = F(x)$ be a non-isotrivial superelliptic curve over $K$ such that $F$ is a degree $3$ polynomial over $\mathbb{F}_q(t)$. Denote by $C_f: fy^\ell = F(x)$ the twist of $C$ by a polynomial $f$ over $\mathbb{F}_q$. Assuming some conditions on $C$, we show that the expected number of $K$-rational points of $C_f$ is bounded, and at least $99\%$ of such curves $C_f$ have at most $(3p)^{4\ell} \cdot \ell!$ many $K$-rational points, as $f$ ranges over the set of polynomials of sufficiently large degree over $\mathbb{F}_q$. To achieve this, we compute the distribution of dimensions of $1-ζ_\ell$ Selmer groups of Jacobians of such superelliptic curves. This is done by generalizing the technique of constructing a governing Markov operator, as developed from previous studies by Swinnerton-Dyer, Klagsbrun--Mazur--Rubin, Yu, and the author.
We find all zero-dimensional spaces of newforms of weight $2$ and squarefree level $N$ with a fixed Atkin--Lehner sign pattern. As an application, we classify unexpected primes of good reduction of Atkin--Lehner quotients of modular and Shimura curves of squarefree levels.
Using modifications to work of Klagsbrun, Mazur, and Rubin, we study (assuming the Extended Riemann Hypothesis) the distribution of Selmer ranks of twist families of some given even-dimensional Galois modules satisfying some mild technical conditions. As a corollary, we study the probability with which a fixed elliptic curve gains (or does not gain) rank in $p$-cyclic extensions, obtaining bounds for this distribution. Likewise, for some superelliptic curves $C$, we bound the average size of $C(L)$ as $L$ ranges over $p$-cyclic extensions over a number field $K$ containing primitive $p$-th roots of unity. Lastly, we study the probability with which a fixed hyperelliptic curve gains (or does not gain) rank in quadratic extensions, also obtaining bounds for this distribution. In all three cases, the extensions under consideration are ordered by the product of ramified primes.
If a self-map $σ\colon \mathcal{X} \rightarrow \mathcal{X}$ has a dynamical zeta function with nonzero radius of convergence $1/Λ$ and the Cesàro mean $B$ of $ \# \mathrm{Fix}(σ^k)/Λ^k$ exists and is positive, we show a large deviation principle for the number of prime orbits occurring in the decomposition of a general orbit of length $\leq X$ (an element of the free abelian monoid generated by the prime orbits or, equivalently, a prime orbit of a finite multiset in $\mathcal{X}$) with speed $B \log X$ and universal rate function equal to that of the Poisson distribution with unit mean. We also show a large deviation principle for more general strongly additive functions. The proof uses asymptotic results on the total number of general orbits, as well as a weak analogue of Mertens's second theorem, that may be of independent interest. The theory applies, for example, to endomorphisms of algebraic groups over finite fields, additive cellular automata, and automorphisms of some solenoids.
We provide a mathematical interpretation of convolutional (or message passing) neural networks by using presheaves and copresheaves of the set of continuous functions over a topological space. Based on this interpretation, we formulate a theoretical heuristic which elaborates a number of empirical limitations of these neural networks by using obstructions on such sets of continuous functions over a topological space to be sheaves or copresheaves.
We prove that for any number field $K$ and any fixed genus $g \geq 2$, there are infinitely many non-isomorphic hyperelliptic curves of genus $g$ over $K$ whose Jacobians have rank over $K$ equal to each of 0, 1, or 2. As an example of our method, over $\mathbb{Q}$, we prove that there exist infinitely many non-isomorphic hyperelliptic curves of genus two, whose Jacobians have rank equal to a fixed number between $1$ and $11$, genus three and four curves with rank between $1$ and $4$, and genus five and six with rank between $1$ and $3$.
Transformers have revolutionized performance in Natural Language Processing and Vision, paving the way for their integration with Graph Neural Networks (GNNs). One key challenge in enhancing graph transformers is strengthening the discriminative power of distinguishing isomorphisms of graphs, which plays a crucial role in boosting their predictive performances. To address this challenge, we introduce 'Topology-Informed Graph Transformer (TIGT)', a novel transformer enhancing both discriminative power in detecting graph isomorphisms and the overall performance of Graph Transformers. TIGT consists of four components: A topological positional embedding layer using non-isomorphic universal covers based on cyclic subgraphs of graphs to ensure unique graph representation: A dual-path message-passing layer to explicitly encode topological characteristics throughout the encoder layers: A global attention mechanism: And a graph information layer to recalibrate channel-wise graph features for better feature representation. TIGT outperforms previous Graph Transformers in classifying synthetic dataset aimed at distinguishing isomorphism classes of graphs. Additionally, mathematical analysis and empirical evaluations highlight our model's competitive edge over state-of-the-art Graph Transformers across various benchmark datasets.
Graph Neural Networks (GNNs) and Transformer-based models have been increasingly adopted to learn the complex vector representations of spatio-temporal graphs, capturing intricate spatio-temporal dependencies crucial for applications such as traffic datasets. Although many existing methods utilize multi-head attention mechanisms and message-passing neural networks (MPNNs) to capture both spatial and temporal relations, these approaches encode temporal and spatial relations independently, and reflect the graph's topological characteristics in a limited manner. In this work, we introduce the Cycle to Mixer (Cy2Mixer), a novel spatio-temporal GNN based on topological non-trivial invariants of spatio-temporal graphs with gated multi-layer perceptrons (gMLP). The Cy2Mixer is composed of three blocks based on MLPs: A temporal block for capturing temporal properties, a message-passing block for encapsulating spatial information, and a cycle message-passing block for enriching topological information through cyclic subgraphs. We bolster the effectiveness of Cy2Mixer with mathematical evidence emphasizing that our cycle message-passing block is capable of offering differentiated information to the deep learning model compared to the message-passing block. Furthermore, empirical evaluations substantiate the efficacy of the Cy2Mixer, demonstrating state-of-the-art performances across various spatio-temporal benchmark datasets. The source code is available at https://github.com/leemingo/cy2mixer.
We apply the Gärtner--Ellis theorem on large deviations to prove a weak version of the Loughran--Smeets conjecture for general fibrations.
Following work of Mehrdad and Zhu and of Liu, we prove a large deviation principle for a broad class of integer-valued additive functions defined over abelian monoids. As a corollary, we obtain a large deviation principle for a generalized form of the Erdős-Kac theorem due to Liu.
We show that the number of $5$-isogenies of elliptic curves defined over $\mathbb{Q}$ with naive height bounded by $H > 0$ is asymptotic to $C_5\cdot H^{1/6} (\log H)^2$ for some explicitly computable constant $C_5 > 0$. This settles the asymptotic count of rational points on the genus zero modular curves $X_0(m)$. We leverage an explicit $\mathbb{Q}$-isomorphism between the stack $\mathscr{X}_0(5)$ and the generalized Fermat equation $x^2 + y^2 = z^4$ with $\mathbb{G}_m$-action of weights $(4, 4, 2)$.
Fix a prime number $p$. Let $\mathbb{F}_q$ be a finite field of characteristic coprime to 2, 3, and $p$, which also contains the primitive $p$-th root of unity $μ_p$. Based on the works by Swinnerton-Dyer and Klagsbrun, Mazur, and Rubin, we prove that the probability distribution of the sizes of prime Selmer groups over a family of cyclic prime twists of non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ satisfying a number of mild constraints conforms to the distribution conjectured by Bhargava, Kane, Lenstra, Poonen, and Rains with explicit error bounds. The key tools used in proving these results are the Riemann hypothesis over global function fields, the Erdös-Kac theorem, and the geometric ergodicity of Markov chains.
We study the probability with which an elliptic curve $E/k$, subject to some technical conditions, gains rank upon base extension to an $S_3$-cubic extension $K/k$ with quadratic resolvent field $F/k$, all three fields of which are subject to some mild technical conditions. To do so, we determine the distribution (under a non-standard ordering) of Selmer ranks of an auxiliary abelian variety associated to $E$ and $S_3$-cubic extensions $K/k$ following ideas of Klagsbrun, Mazur, and Rubin. One corollary of this distribution is that $E$ gains rank by at most one upon base extension to $K$ with probability at least $31.95\%$.
Given a projective smooth curve $X$ over any field $k$, we discuss two notions of global $\mathbb{A}^1$ degree of a finite morphism of smooth curves $f: X \to \mathbb{P}^1_k$ satisfying certain conditions. One originates from computing the Euler number of the pullback of the line bundle $\mathscr{O}_{\mathbb{P}^1}(1)$ as a generalization of Kass and Wickelgren's construction of Euler numbers. The other originates from the construction of global $\mathbb{A}^1$ degree of morphisms of projective curves by Kass, Levine, Solomon, and Wickelgren as a generalization of Morel's construction of $\mathbb{A}^1$-Brouwer degree of a morphism $f: \mathbb{P}^1_k \to \mathbb{P}^1_k$. We prove that under certain conditions on $N$, both notions of global $\mathbb{A}^1$ degrees of covering maps between modular curves $X_0(N) \to X(1)$, $X_1(N) \to X(1)$, and $X(N) \to X(1)$ agree to be equal to sums of hyperbolic elements $\langle 1 \rangle + \langle -1 \rangle$ in the Grothendieck-Witt ring $\mathrm{GW}(k)$ for any field $k$ whose characteristic is coprime to $N$ and the pullback of $\mathscr{O}_{\mathbb{P}^1}(1)$ is relatively oriented.
This paper presents the Persistent Weisfeiler-Lehman Random walk scheme (abbreviated as PWLR) for graph representations, a novel mathematical framework which produces a collection of explainable low-dimensional representations of graphs with discrete and continuous node features. The proposed scheme effectively incorporates normalized Weisfeiler-Lehman procedure, random walks on graphs, and persistent homology. We thereby integrate three distinct properties of graphs, which are local topological features, node degrees, and global topological invariants, while preserving stability from graph perturbations. This generalizes many variants of Weisfeiler-Lehman procedures, which are primarily used to embed graphs with discrete node labels. Empirical results suggest that these representations can be efficiently utilized to produce comparable results to state-of-the-art techniques in classifying graphs with discrete node labels, and enhanced performances in classifying those with continuous node features.
Given a chain of groups $G_0 \le G_1 \le G_2 ... $, we may form the corresponding chain of their representation rings, together with induction and restriction operators. We may let $\textrm{Res}^l$ denote the operator which restricts down $l$ steps, and similarly for $\textrm{Ind}^l$. Observe then that $\textrm{Ind}^l \textrm{Res}^l$ is an operator from any particular representation ring to itself. The central question that this paper addresses is: "What happens if the $\textrm{Ind}^l \textrm{Res}^l$ operator is a polynomial in the $\textrm{Ind} \textrm{Res}$ operator?". We show that chains of wreath products $\{H^n \rtimes S_n\}_{n \in \mathbb{N}}$ have this property, and in particular, the polynomials that appear in the case of symmetric groups are the falling factorial polynomials. An application of this fact gives a remarkable new way to compute characters of wreath products (in particular symmetric groups) using matrix multiplication. We then consider arbitrary chains of groups, and find very rigid constraints that such a chain must satisfy in order for $\textrm{Ind}^l \textrm{Res}^l$ to be a polynomial in $\textrm{Ind} \textrm{Res}$. Our rigid constraints justify the intuition that this property is indeed a very rare and special property.
Previous research on quantum annealing methods focused on effectively modeling systems of linear equations by utilizing quadratic unconstrained binary optimization (QUBO) formulations. These studies take part in enhancing quantum computing algorithms, which extract properties of quantum computers suitable for improving classical computational models. In this paper, we further develop the QUBO formulations of systems of linear equations by applying Sylvester's law of inertia, which explores matrix congruence of any real symmetric matrix to a diagonal matrix. We expect that the proposed algorithm can effectively implement higher dimensional systems of linear equations on a quantum computer. Further experimental verification of the proposed QUBO models as well as their comparisons to classical algorithms are also made.