# Arithmetic Geometry, Number Theory and Computation

## Mathematics and Physical Sciences

Computation and number theory naturally go hand in hand — one of the earliest examples is a Mesopotamian tablet from 1800 BC that lists 15 sets of integers that satisfy the equation *a*^{2} + *b*^{2} = *c*^{2}, now known as Pythagorean triples.

The Simons Collaboration on Arithmetic Geometry, Number Theory and Computation continues the legacy of combining computation with theoretical research by focusing on several central problems in the study of numbers and solutions to polynomial equations.

The collaboration, launched in 2018, germinated from a 2015 meeting at the Institute for Computational and Experimental Research in Mathematics at Brown University, titled “Computational Aspects of the Langlands Program” and organized by collaboration investigator John Voight of Dartmouth College and others.

“Our collaboration grew out of the questions: What does computational number theory look like in the 21st century, and what tools should be developed for use by the arithmetic geometry community?” Voight says.

Voight is one of six principal investigators along with collaboration director Brendan Hassett of Brown University, Jennifer S. Balakrishnan of Boston University, Noam Elkies of Harvard University, and Bjorn Poonen and Andrew Sutherland of the Massachusetts Institute of Technology. The principal investigators meet monthly to discuss their research and trade ideas. The collaboration also includes 20 affiliated scientists from around the world, including graduate students and late-career researchers.

Many of those affiliated scientists support the collaboration by developing tools and databases to create and store examples of mathematical phenomena.

“We’re really motivated in encouraging the growth of mathematical researchers who are equally comfortable in the computer science side and the math theory side,” Hassett says. “That is actually the biggest impact: not just solving particular problems but having a cohort of people who can both do the math and oversee databases of mathematical objects.”

**Pulling rank:**

Collaboration members used computers to upend a broadly held belief in arithmetic geometry concerning the fundamentals of the field. Basic objects of study in arithmetic geometry are elliptic curves, or solutions to equations in the form *y*^{2} = *x*^{3} + *ax* + *b*. Researchers want to find points on elliptic curves that are rational — that is, points that have coordinates that can be written as simple fractions. For example, (25/4, −75/8) is a rational point on the curve defined by the equation *y*^{2} = *x*^{3} − 25*x*.

‘Rank’ is a measure of the complexity of the set of those rational points. If an elliptic curve has only a finite number of rational points, it has rank zero. If it has an infinite number of rational points, then it has some positive rank.

For decades, mathematicians have thought that there is no cap on how high this rank can get. In 2006, Elkies used extensive computer experiments to find an elliptic curve with a rank of at least 28, the highest rank seen so far. More recently, Voight, Poonen and their colleagues Jennifer Park of Ohio State University and Melanie Matchett Wood of the University of Wisconsin-Madison developed a heuristic model suggesting that only a finite number of curves defined over the rational numbers have a rank higher than 21. This conclusion implies there must be a cap on rank — at least 28 because of Elkies’ example.

Numerical evidence will be key to progress. If someone could find an infinite number of curves with higher rank, that would disprove the model. Or, if mathematicians could find infinite sequences of curves with growing rank, they’d prove that the existing paradigm of limitless ranks was right. As it stands, the model by Voight and his colleagues has shaken up the world of arithmetic geometry.

“That is actually the biggest impact: not just solving particular problems but having a cohort of people who can both do the math and oversee databases of mathematical objects.”

**Higher genus:**

Computational power has also allowed collaboration members to extend research from elliptic curves, which are curves with genus 1, to higher-genus curves given by equations with higher-degree polynomials, such as *y*^{2} = *x*^{8} + *ax*^{7} + … + *gx*.

When finding genus 3 curves with a small discriminant, a number that, like rank, measures the complexity of the curve, Sutherland set a record for the largest computation using the Google Cloud Platform. In one afternoon, Sutherland used more than 580,000 Google computing cores around the world — more than 300 years of computer time — to whittle 10 billion candidate curves down to a list of about 80,000 with particularly small discriminants.

After one of the collaboration leaders’ monthly meetings, Balakrishnan and her co-authors found the rational points of 17,000 curves from Sutherland’s list.

**The LMFDB:**

Those rational points will be uploaded to the L-functions and Modular Forms Database (LMFDB), an online repository of information about elliptic curves, modular forms and L-functions. L-functions encode a correspondence between certain kinds of elliptic curves, or equations in the form *y*^{2} = *x*^{3} + *ax* + *b*, and modular forms, a surprisingly useful type of function.

“The L-function encodes the mathematical DNA of these objects,” Sutherland says.

The LMFDB, which is partially supported by the collaboration, is like a modern version of the Mesopotamian tablets. Each encyclopedic page in the LMFDB is dedicated to a mathematical object, such as an elliptic curve, a modular form or an L-function, and its relationships with other objects. Upgrading its infrastructure was one of the first tasks for the collaboration.

“The LMFDB is a big project, bigger than one university or one department could possibly support alone,” Hassett says.

The mathematical community hopes that numerical data in the LMFDB can help establish a correspondence between higher-genus curves and L-functions, just as elliptic curves correspond with modular forms via their L-functions.

**The bigger picture:**

The scope of the collaboration continues beyond these examples to solve computational problems in arithmetic geometry and number theory. Just as we now have the Pythagorean theorem instead of a Mesopotamian catalog of Pythagorean triples, the hope is that more data will lead to more mathematical proofs.

“One of the central premises of our collaboration is to take delight in the wonderful interplay between practical computation and abstract theory,” Voight says. “We’re thinking hard about how computers, algorithmic techniques and big databases can be used to advance number theory. We want to make it easy for researchers to zoom in like a microscope to really dissect a few particular specimens and to zoom out like a telescope to understand large-scale structure of the mathematical universe.”