# Research

## The Arithmetic of Kleinian groups

Some recent work was originally motivated by a connection between Apollonian circle packings and the structure of imaginary quadratic fields (in turn coming out of a connection to abelian sandpiles!). A Schmidt arrangement (named for Asmus Schmidt) is the orbit of the extended real line in the extended complex plane under the Mobius action of the Bianchi group PSL2(OK) (here, OK is the ring of integers of the imaginary quadratic field). The resulting fractal of nested circles is an analogue to the Farey subdivision of the real line into nested intervals. It has interesting connections to the arithmetic of the field: for example, it is connected if and only if OK is Euclidean. It turns out the Schmidt arrangement of the Gaussian integers is a union of all primitive integral Apollonian circle packings. In other fields, there are other new packings — and associated thin groups — that govern the picture. In this age of computer visualization, there are still new things to discover about topics as classical as imaginary quadratic fields, quadratic forms, hermitian forms, etc.

## The arithmetic of Abelian varieties

The Fibonacci numbers are governed by a twisted multiplicative group — for example, primes p appear in the (p2-1)-th term because this is the cardinality of the multiplicative group of Fp2. If you replace the multiplicative group with an elliptic curve, you obtain a recurrence sequence called an elliptic divisibility sequence (this is because the division polynomials satisfy a recurrence coming from the group law). Much of my interest in arithmetic geometry is connected to algebraic divisibility sequences such as these. The sequences carry all sorts of arithmetic information: for example, using a multi-dimensional generalization of elliptic divisibility sequences called elliptic nets, you can compute the Tate-Lichtenbaum pairing as a quotient of certain terms. More recently, my interests are turning toward modular curves.

## Cryptography

I learned from Kristin Lauter that cryptographic applications can give rise to interesting and totally new number theory problems. I’ve worked on elliptic curve cryptography, and more recently, lattice-based cryptography, which provides the promise of secure post-quantum crypto (quantum computers will be able to solve the elliptic curve discrete logarithm problem; NIST is currently running a competition). My interest has often been in the security of hard problems. For example, lattice-based cryptography benefits from the extra structure of the ring of integers of a number field, but I’m interested in how that extra structure may provide needed tools to attack problems such as Ring Learning with Errors (9 of the surviving 26 NIST PQC candidates are LWE or Ring-LWE/LWR based). I find it fascinating to see the interplay between theory and practice, and to work across disciplines.

### Students

• Sarah Arpin (Ph.D. candidate)
• Elizabeth Parsons (M.A. 2016)

## Algebraic Number Theory and Arithmetic Dynamics

The core of all three topics above is, of course, algebraic number theory. I’m interested in quadratic forms, Hermitian forms, class field theory, Euclideanity, continued fractions and elementary number theory. I’m also interested in arithmetic dynamics, particularly in taking a dynamical viewpoint on the structure of number fields.

### Students

• Amy Feaver (Ph.D. 2014)
• Ryan Ibarra (Honors thesis candidate)
• Annie S. Chen (Boulder High School)

### Summer/Lab Projects

• 2018: Ryan Ibarra, Henry Lembeck, Mohammad Ozaslan