My research interests are in arithmetic geometry, computational number theory, and sphere packing. I have also worked on connections between analytic number theory and random matrix theory and in combinatorics.
See also my pages at arXiv and Google Scholar Citations.
On the arithmetic of a family of superelliptic curves (with S. Arpin, R. Griffon, and L. Taylor), submitted (2021). (pdf) (arXiv)
Let \(p\) be a prime, let \(r\) and \(q\) be powers of \(p\), and let \(a\) and \(b\) be relatively prime integers not divisible by \(p\). Let \(C/\mathbb F_{r}(t)\) be the superelliptic curve with affine equation \(y^b+x^a=t^q-t\).
Let \(J\) be the Jacobian of \(C\). By work of Pries--Ulmer, \(J\) satisfies the Birch and Swinnerton-Dyer conjecture (BSD).
Generalizing work of Griffon--Ulmer, we compute the \(L\)-function of \(J\) in terms of certain Gauss sums.
In addition, we estimate several arithmetic invariants of \(J\) appearing in BSD, including the rank of the Mordell--Weil group \(J(\mathbb F_{r}(t))\), the Faltings height of \(J\), and the Tamagawa numbers of \(J\) in terms of the parameters \(a,b,q\).
For any \(p\) and \(r\), we show that for certain \(a\) and \(b\) depending only on \(p\) and \(r\), these Jacobians provide new examples of families of simple abelian varieties of fixed dimension and with unbounded analytic and algebraic rank as \(q\) varies through powers of \(p\).
Under a different set of criteria on \(a\) and \(b\), we prove that the order of the Tate--Shafarevich group ш\((J)\) is
``large'' as \(q \to \infty.\)
Restriction of Scalars Chabauty and the S-unit equation, submitted (2020). (pdf) (arXiv) (Illustrated Slides)
Given a smooth, proper, geometrically integral curve \(X\) of genus \(g\) with Jacobian \(J\) over a number field \(K\), Chabauty's method is a \(p\)-adic technique to bound \(\# X(K)\) when \(\text{rank}\ J(K) < g\). We study limitations of a variant of this approach called `Restriction of Scalars Chabauty' (RoS Chabauty). RoS Chabauty typically bounds \(\# X(K)\) when \(\text{rank}\ J(K)\ \leq [K:\mathbb Q] (g - 1)\), but fails in the presence of a subgroup obstruction, a high-rank subgroup scheme of \(\text{Res}_{K/\mathbb{Q}} J\) which intersects the image of \(\text{Res}_{K/\mathbb{Q}} X\) in higher-than-expected dimension. We define BCP obstructions, which are certain subgroup obstructions arising from the geometry of \(X\). BCP obstructions explain all known examples where RoS Chabauty fails to bound \(\# X(K)\). We also extend RoS Chabauty to compute \(S\)-integral points on affine curves.
Suppose \(K\) does not contain a CM-subfield. We present a \(p\)-adic algorithm which conjecturally computes solutions to the \(S\)-unit equation \(x+y = 1\) for \(x,y \in \mathcal{O}_{K,S}^{\times}\) by using RoS Chabauty to compute \(S\)-integral points on certain genus \(0\) affine curves. As evidence the algorithm succeeds, we prove that all but one of these curves have no subgroup obstructions and that the remaining curve has no BCP obstructions to RoS Chabauty. In contrast, under a generalized Leopoldt conjecture, we prove that analogous methods using classical Chabauty cannot bound solutions to the \(S\)-unit equation when \([K:\mathbb Q] \geq 3\) and \(K\) is not totally real.
There are no exceptional units in number fields of degree prime to 3 where 3 splits completely, to appear in Proceedings of the AMS, Series B (2021). (pdf) (arXiv)
Let \(K\) be a number field with ring of integers \(\mathcal O_{K}\). We prove that if 3 does not divide \([K:\mathbb Q]\) and 3 splits completely in \(K\), then there are no exceptional units in \(K\). In other words,
there are no \(x, y \in \mathcal O_{K}^{\times}\) with \(x + y = 1\).
Our elementary \(p\)-adic proof is inspired by the Skolem-Chabauty-Coleman method applied to the restriction of scalars of the projective line minus three points.
Applying this result to a problem in arithmetic dynamics, we show that if \(f \in \mathcal O_{K}[x]\) has a finite cyclic orbit in \(\mathcal O_{K}\) of length \(n\) then \(n \in \{1, 2, 4\}\).
Two recent p-adic approaches towards the (effective) Mordell conjecture, (with A. J. Best, F. Bianchi, J. Balakrishnan, B. Lawrence, J. S. Müller, and J. Vonk), in Arithmetic L-Functions and Differential Geometric Methods (2021). (pdf) (arXiv) (Numerical Supplement) (journal)
We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usually called the method of Chabauty--Kim or non-abelian Chabauty in the literature, has the advantage that in some cases it has been turned into an effective method to determine the set of rational points on a curve, and we illustrate this by presenting three new examples of modular curves where this set can be determined.
Dual linear programming bounds for sphere packing via modular forms, (with Henry Cohn), Mathematics of Computation (2021). (pdf) (arXiv) (journal preview)
We obtain new restrictions on the linear programming bound for sphere packing, by optimizing over spaces of modular forms to produce feasible points in the dual linear program.
In contrast to the situation in dimensions 8 and 24, where the linear programming bound is sharp, we show that it comes nowhere near the best packing densities known in dimensions 12, 16, 20, 28, and 32.
More generally, we provide a systematic technique for proving separations of this sort.
Restriction of Scalars, the Chabauty-Coleman Method, and \(\mathbb P^1 \smallsetminus \{0, 1, \infty\}\), Ph.D. Thesis (2019). (pdf)
We extend Siksek's development of Chabauty's method for computing the set of \(S\)-integral points on restrictions of scalars of curves over \(\mathcal O_{K,S}\) to curves which are not necessarily complete. When the curve is \(\mathcal C = \mathbb P^1 \smallsetminus \{0, 1, \infty\}\), we show that after replacing the curve with a suitable descent set of covers, the method has no base change obstructions. We also use the method to prove that \( \mathcal C(\mathcal O_{K}) \) is finite for several classes of fields \( K \), including that \( \mathcal C(\mathcal O_{K}) = \emptyset \) when 3 splits completely in \(K\). This represents the first infinite class of cases where Chabauty's method for restrictions of scalars is proved to show that \( \mathcal C(\mathcal O_{K}) \) is finite where the classical Chabauty's method cannot.
Computing zeta Functions of cyclic covers in large characteristic, (with Vishal Arul, Alex J. Best, Edgar Costa, and Richard Magner), Proceedings of ANTS XIII (2018). (pdf) (arXiv)
We describe an algorithm to compute the zeta function of a cyclic cover of the projective line over a finite field of characteristic \(p\) that runs in time \(p^{1/2+o(1)}.\)
We confirm its practicality and effectiveness by reporting on the performance of our SageMath implementation on a range of examples.
The algorithm relies on Gon\c{c}alves's generalization of Kedlaya's algorithm for cyclic covers, and Harvey's work on Kedlaya's algorithm for large characteristic.
Distribution of eigenvalues of weighted, structured matrix ensembles, (with O. Beckwith, V. Luo, S. J. Miller, K. Shen), INTEGERS (2015). (pdf) (arXiv) (journal)
Given a structured random matrix ensemble where each random variable
occurs o(N) times in each row and the limiting rescaled spectral
measure exists, we fix a \(p \in [1/2,1]\) and study the ensemble of signed
structured matrices by multiplying the \((i,j)\)th and \((j,i)\)th entries of
a matrix by a randomly chosen \(\varepsilon(i,j) \in {1,-1}\), with
Prob\((\epsilon(i,j)) = p\) (i.e., the Hadamard product.) For \(p = 1/2\)
the limiting signed rescaled spectral measure is the semi-circle; for
other \(p\) it has bounded (resp., unbounded) support if the limiting
rescaled spectral measure has bounded (resp., unbounded) support, and
converges to the limitin rescaled spectral measure as \(p \to 1\). The
proofs are by Markov's Method of Moments, and involve the pairings of
\(2k\) vertices on a circle. The contribution of each pairing in the
signed case is weighted by a factor depending on p and the number of
vertices involved in at least one crossing. These numbers are of
interest in their own right, appearing in problems in combinatorics
and knot theory. The number of configurations with no vertices
involved in a crossing is well-studied (the Catalan numbes). We
discover and prove similar formulas for other configurations.
Sharp threshold asymptotics for the emergence of additive bases (with A. Godbole,
C. M. Lim, V. Lyzinski), INTEGERS (2013). (pdf) (arXiv) (journal)
A subset \(\mathcal A\) of \(\{0,1,...,n\}\) is said to be a 2-additive basis for \(\{1,2,...,n\}\) if each \(j \in \{1,2,...,n\}\) can be written as \(j=x+y, x,y \in \mathcal A, x \leq y\). If we pick each integer in \(\{0,1,...,n\}\) independently with probability \(p=p_n\) tending to \(0\), thus getting a random set \(\mathcal A\), what is the probability that we have obtained a 2-additive basis? We address this question when the target sum-set is \([(1-\alpha)n, (1+\alpha)n]\) (or equivalently \([\alpha n, (2-\alpha) n])\) for some \(0 < \alpha < 1\). Under either model, the Stein-Chen method of Poisson approximation is used, in conjunction with Janson's inequalities, to tease out a very sharp threshold for the emergence of a 2-additive basis. Generalizations to \(k\)-additive bases are then given.
Omnimosaics, (with J. Banks, A. Godbole) Preprint (2010). (pdf)
(arXiv)
An omnimosaic \(O(n, k, a)\) is defined to be an \(n \times n\) matrix, with
entries from the set \(A ={1, 2,\ldots, a}\), which contains each of the
\(a^{k^ 2}\) different \(k \times k\) matrices over \( A\), as a submatrix. We provide constructions of omnimosaics
and show that for fixed \(a\) the smallest possible size \(\omega (k, a)\) of an \(O(n, k, a)\)
omnimosaic satisfies \(\frac{ka^{k/2}}{e} \leq \omega(k, a) \leq \frac{ka^{k/2}}{e}(1+ o (1))\) for a
well-specified function \(o(1)\) that tends to zero as \(k \to \infty\).
This list is current as of October 2021. My (cv) may contain a more up-to-date list of talks.
Restriction of scalars Chabauty applied to cyclic covers of \(\mathbb P^1\smallsetminus \{0,1, \infty\}\), Oberwolfach workshop ``Explicit methods in number theory'', Oberwolfach, Germany (Zoom), (July 2021). (Slides)
Nonexistence of exceptional units via modified Skolem-Chabauty, Groningen Algebra seminar, Groningen, Netherlands (Zoom), (March 2021).
Computing Isolated Points on Modular Curves, Virtual Seminar on Number Theory and Arithmetic Geometry (VaNtAGe), Zoom, (November 2020). (Video)
Nonexistence of exceptional units via Skolem-Chabauty's method, MSRI Diophantine Problems Seminar, Berkeley, CA (Zoom), (November 2020). (Video)
Restriction of Scalars Chabauty and \(\mathbb P^1\smallsetminus \{0,1, \infty\}\), Chicago Number Theory Days, Zoom (June 2020). (Slides)
Restriction of Scalars, Chabauty's Method, and \(\mathbb P^1\smallsetminus \{0,1, \infty\}\), University of Washington Number Theory Seminar, Seattle, WA (November 2019).
Restriction of Scalars, Chabauty's Method, and \(\mathbb P^1\smallsetminus \{0,1, \infty\}\), Weslyan University Algebra Seminar, Middletown, CT (November 2019).
Restriction of Scalars, Chabauty's Method, and \(\mathbb P^1\smallsetminus \{0,1, \infty\}\), Rice University Number Theory Seminar, Houston, TX (September 2019).
Restriction of Scalars, Chabauty's Method, and \(\mathbb P^1\smallsetminus \{0,1, \infty\}\), University of Georgia Number Theory Seminar, Athens, GA (September 2019).
The Chabauty-Coleman method: variants and computational aspects, University of Georgia AGANT Oberseminar, Athens, GA (August 2019).
Restriction of Scalars, Chabauty's Method, and \(\mathbb P^1\smallsetminus \{0,1, \infty\}\), Boston University/Keio University Workshop at Boston University, Boston, MA (June 2019).
Restriction of Scalars, Chabauty's Method, and \(\mathbb P^1\smallsetminus \{0,1, \infty\}\), Arithmetic of Low-Dimensional Abelian Varieties at ICERM, Providence, RI (June 2019).
Restriction of Scalars, Chabauty's Method, and \(\mathbb P^1\smallsetminus \{0,1, \infty\}\), MIT Thesis Defense, Cambridge, MA (April 2019).
Restriction of Scalars, Chabauty's Method, and \(\mathbb P^1\smallsetminus \{0,1, \infty\}\), AMS Graduate Student Conference in Algebra and Number Theory at Brown University, Providence, RI (April 2019).
Restriction of Scalars, Chabauty's Method, and \(\mathbb P^1\smallsetminus \{0,1, \infty\}\), Emory Algebra and Number Theory Seminar, Atlanta, GA (March 2019).
The Method of Chabauty-Coleman-Skolem for Restrictions of Scalars, Harvard Number Theory Seminar, Cambridge, MA (March 2019).
Variants of Chabauty's method and the thrice-punctured projective line, Joint Meetings of the American Mathematical Society 2019, AMS Contributed Paper Session on Number Theory III, Baltimore, MD, (January 2019).
Computing zeta functions of cyclic covers in large characteristic, (based on joint work with Vishal Arul, Alex J. Best, Edgar Costa, and Richard Magner) AMS Joint Math Meetings 2019, Special Session on Number Theory, Arithmetic Geometry, and Computation, Baltimore, MD (January 2019).
The method of Chabauty-Coleman-Skolem for restrictions of scalars, Junior Number Theory Days, University of Rutgers - Newark (November 2018).
Computing zeta functions of cyclic covers in large characteristic, (based on joint work with Vishal Arul, Alex J. Best, Edgar Costa, and Richard Magner) AMS Fall Central Sectional Meeting 2018 Special Session on From Hyperelliptic to Superelliptic Curves (October 2018).
Computing zeta functions of cyclic covers in large characteristic, (based on joint work with Vishal Arul, Alex J. Best, Edgar Costa, and Richard Magner) ANTS XIII (2018).
Computing Limitations to the LP Method for Sphere Packing
from Non-negative Modular Forms (with H. Cohn) presented at ANTS XIII at University of Wisconsin Madison (2018). (pdf)
In 2016, Vizovska proved that the \(E_8\) lattice gives the densest sphere packing problem in \(8\)-dimesions [V16]. Shortly thereafter, Cohn, Kumar, Miller, Radchenko, and Viazovska [CKM+16] proved that the Leech lattice gives the densest sphere packing in \(24\)-dimensions. Their proofs find feasible solutions to the the Cohn-Elkies LP-method [CE03] to give upper bounds on the density of sphere packings in \(8\) and \(24\) dimensions that match sphere packings centered on the \(E_8\) and Leech lattice respectively.
We prove that in contrast to the situation in dimensions \(8\) and \(24\), the LP-method is insufficient to prove that the densest known sphere packing is indeed the densest sphere packing in \(12, 16, 20, 28, 32\) dimensions. The obstructions comes from modular forms. We describe a general method involving linear programming over spaces of modular forms which appears to be the obstruction making the LP-method insufficient to solve the sphere packing problem in these dimensions.
Limitations from Modular Forms on LP-Bounds for Sphere Packing, (with H. Cohn) presented at Workshop Arithmetic Geometry and Computer Algebra, Carl von Ossietzky Universität (2017). (pdf)
In 2016, Vizovska proved that the \(E_8\) lattice gives the densest sphere packing problem in \(8\)-dimesions [V16]. Shortly thereafter, Cohn, Kumar, Miller, Radchenko, and Viazovska [CKM+16] proved that the Leech lattice gives the densest sphere packing in \(24\)-dimensions. Their proofs find feasible solutions to the the Cohn-Elkies LP-method [CE03] to give upper bounds on the density of sphere packings in \(8\) and \(24\) dimensions that match sphere packings centered on the \(E_8\) and Leech lattice respectively.
We prove that in contrast to the situation in dimensions \(8\) and \(24\), the LP-method is insufficient to prove that the densest known sphere packing is indeed the densest sphere packing in \(12\) and \(16\) dimensions. We also provide evidence that the same is true in dimensions \(20, 28, 32\) and \(36\). The obstructions comes from modular forms. Moreover, we describe a general method involving linear programming over spaces of modular forms which appears to be the obstruction making the LP-method insufficient to solve the sphere packing problem in a wide range of dimensions.