Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer

This paper presents a novel, interdisciplinary study that leverages a Machine Learning (ML) assisted framework to explore the geometry of affine Deligne-Lusztig varieties (ADLV). The primary objective is to investigate the nonemptiness pattern, dimension and enumeration of irreducible components of ADLV. Our proposed framework demonstrates a recursive pipeline of data generation, model training, pattern analysis, and human examination, presenting an intricate interplay betwee...

As datasets used in scientific applications become more complex, studying the geometry and topology of data has become an increasingly prevalent part of the data analysis process. This can be seen for example with the growing interest in topological tools such as persistent homology. However, on the one hand, topological tools are inherently limited to providing only coarse information about the underlying space of the data. On the other hand, more geometric approaches rely p...

Suppose $E$ is an elliptic curve over $\mathbb{Q}$ and $\chi$ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that $L(E,\chi,1) = 0$. Via the Birch and Swinnerton-Dyer conjecture, this gives a heuristic estimate of the probability that the Mordell-Weil rank grows in abelian extensions of $\mathbb{Q}$. Using this heuristic we find a large class of infinite abelian extensions $F$ where we expect $E(F)$ to be f...

In this paper we give an upper bound for the number of integral points on an elliptic curve E over F_q[T] in terms of its conductor N and q. We proceed by applying the lower bounds for the canonical height that are analogous to those given by Silverman and extend the technique developed by Helfgott-Venkatesh to express the number of integral points on E in terms of its algebraic rank. We also use the sphere packing results to optimize the size of an implied constant. In the e...

Statistical Machine Learning (SML) refers to a body of algorithms and methods by which computers are allowed to discover important features of input data sets which are often very large in size. The very task of feature discovery from data is essentially the meaning of the keyword `learning' in SML. Theoretical justifications for the effectiveness of the SML algorithms are underpinned by sound principles from different disciplines, such as Computer Science and Statistics. The...

It is challenging for humans to enable visual knowledge discovery in data with more than 2-3 dimensions with a naked eye. This chapter explores the efficiency of discovering predictive machine learning models interactively using new Elliptic Paired coordinates (EPC) visualizations. It is shown that EPC are capable to visualize multidimensional data and support visual machine learning with preservation of multidimensional information in 2-D. Relative to parallel and radial coo...

The well-known analogies between number fields and function fields have led to the transposition of many problems from one domain to the other. In this paper, we will discuss traffic of this sort, in both directions, in the theory of elliptic curves. In the first part of the paper, we consider various works on Heegner points and Gross-Zagier formulas in the function field context; these works lead to a complete proof of the conjecture of Birch and Swinnerton-Dyer for elliptic...

In the era of big data, analysts usually explore various statistical models or machine learning methods for observed data in order to facilitate scientific discoveries or gain predictive power. Whatever data and fitting procedures are employed, a crucial step is to select the most appropriate model or method from a set of candidates. Model selection is a key ingredient in data analysis for reliable and reproducible statistical inference or prediction, and thus central to scie...

In recent years, the field of statistics has experienced a surge in interest and application, largely due to significant advances in computer technology. This progress has led to remarkable developments in statistics methods and algorithms, enabling their widespread adoption across various disciplines. Key areas benefiting from these advancements include machine learning, economics, finance, geophysics, molecular modeling, computational systems biology, operations research, a...

Technology is generating a huge and growing availability of observa tions of diverse nature. This big data is placing data learning as a central scientific discipline. It includes collection, storage, preprocessing, visualization and, essentially, statistical analysis of enormous batches of data. In this paper, we discuss the role of statistics regarding some of the issues raised by big data in this new paradigm and also propose the name of data learning to describe all the a...