Ryan C. Chen
I am a PhD student in the math department at
MIT. My advisor is
Wei Zhang.
I have been supported by an NSF graduate research fellowship.
I am interested in number theory and arithmetic geometry. I have been working on topics
concerning special cycles on Shimura varieties, such as in Kudla's program.
I am on the job market during the 2024–2025 academic year.
Emimail:
rcchenhrvap@mit.edu
Curriculum vitae:
[CV]
Research
Listed in reverse order of first arXiv appearance (with arXiv year also indicated).
arXiv ID
Comments welcome!
Non-highlighted links such as [abstract] and some collaborator names may also be clicked.
Faltings heights and the subleading terms of adjoint $L$-functions
with
Weixiao Lu and
Wei Zhang.
In preparation.
[abstract]
(From conference, July 2024)
The Kronecker limit formula
may be interpreted as an equality relating the Faltings height of a CM elliptic curve
to the sub-leading term (at $s=0$) of the Dirichlet $L$-function of an
imaginary quadratic character. Colmez conjectured a generalization relating the Faltings height
of any CM abelian variety to the subleading terms of certain Artin $L$-functions.
An averaged (over all abelian varieties with CM by the integer ring of a given CM field)
version was proved by Andreatta–Goren–Howard–Madapusi and Yuan–Zhang.
We formulate a “non-Artinian” generalization of Colmez's conjecture, relating the following two quantities:
(1) the arithmetic intersection numbers from the Hodge bundle and certain cycles on unitary Shimura varieties, and
(2) the sub-leading terms of the adjoint $L$-functions of (cohomological) automorphic representations of $U(n)$.
The case $n=1$ amounts to the averaged Colmez conjecture. We are able to prove our conjecture when $n=2$.
Work in progress with Weixiao Lu and Wei Zhang.
Co-rank 1 Arithmetic Siegel–Weil IV: Analytic local-to-global
Preprint, pp. 1–69.
[abstract]
[arXiv (2024)]
This is the fourth in a sequence of four papers,
where we prove the arithmetic Siegel–Weil formula in co-rank $1$
for Kudla–Rapoport special cycles on exotic smooth integral models
of unitary Shimura varieties of arbitrarily large even arithmetic dimension.
Our arithmetic Siegel–Weil formula implies that degrees of Kudla–Rapoport
arithmetic special $1$-cycles are encoded in the first derivatives of
unitary Eisenstein series Fourier coefficients.
In this paper, we pin down precise normalizations for some $U(m,m)$
Siegel Eisenstein series, give local Siegel–Weil special value
formulas with explicit constants, and record a geometric Siegel–Weil
result for degrees of complex $0$-cycles. Using this, we complete the proof of
our arithmetic Siegel–Weil results by patching together the
local main theorems from our companion papers.
Co-rank 1 Arithmetic Siegel–Weil III: Geometric local-to-global
Preprint, pp. 1–67.
[abstract]
[arXiv (2024)]
This is the third in a sequence of four papers,
where we prove the arithmetic Siegel–Weil formula
in co-rank $1$ for Kudla–Rapoport special cycles
on exotic smooth integral models of unitary Shimura varieties
of arbitrarily large even arithmetic dimension.
Our arithmetic Siegel–Weil formula implies that degrees
of Kudla–Rapoport arithmetic special $1$-cycles are
encoded in the first derivatives of unitary Eisenstein
series Fourier coefficients.
In this paper, we finish the reduction process from
global arithmetic intersection numbers for special cycles
to the local geometric quantities in our companion papers.
Building on our previous companion papers, we also propose
a construction for arithmetic special cycle classes associated
to possibly singular matrices of arbitrary co-rank.
Co-rank 1 Arithmetic Siegel–Weil II: Local Archimedean
Preprint, pp. 1–29.
[abstract]
[arXiv (2024)]
This is the second in a sequence of four papers, where we prove the arithmetic
Siegel–Weil formula in co-rank $1$ for Kudla–Rapoport special cycles
on exotic smooth integral models of unitary Shimura varieties of arbitrarily
large even arithmetic dimension. Our arithmetic Siegel–Weil formula implies
that degrees of Kudla–Rapoport arithmetic special $1$-cycles
are encoded in the first derivatives of unitary Eisenstein series Fourier coefficients.
In this paper, we formulate and prove the key Archimedean local theorem.
In the case of purely Archimedean intersection numbers, we also prove
an Archimedean local arithmetic Siegel–Weil formula, relating Green currents
of arbitrary degree and off-central derivatives of local Whittaker functions.
The crucial input is a new limiting method, which is structurally parallel
to our strategy at non-Archimedean places.
Co-rank 1 Arithmetic Siegel–Weil I: Local non-Archimedean
Preprint, pp. 1–111.
[abstract]
[arXiv (2024)]
[video (2024)]
[combined I–IV]
This is the first in a sequence of four papers, where we prove the
arithmetic Siegel–Weil formula in co-rank $1$ for Kudla–Rapoport
special cycles on exotic smooth integral models of unitary Shimura varieties
of arbitrarily large even arithmetic dimension. Our arithmetic Siegel–Weil
formula implies that degrees of Kudla–Rapoport arithmetic special $1$-cycles
are encoded in the first derivatives of unitary Eisenstein series Fourier coefficients.
The crucial input is a new local limiting method at all places.
In this paper, we formulate and prove the key local theorems at all
non-Archimedean places. On the analytic side, the limit relates
local Whittaker functions on different groups. On the geometric side
at nonsplit non-Archimedean places, the limit relates degrees of $0$-cycles
on Rapoport–Zink spaces and local contributions to heights of
$1$-cycles in mixed characteristic.
This video
is a recording of my talk at the workshop
Arithmetic intersection theory on Shimura varieties
at the American Institute of Mathematics (AIM) in January 2024.
The talk surveys my four-paper sequence "Co-rank $1$ Arithmetic Siegel–Weil I–IV".
Some notation from the talk differs slightly from my papers.
Talk errata
At 17:00 in the video, I should only have claimed that the expression $\Lambda$ is
given by the formula I wrote on the upper left board when $m = n$.
In general, see Section 6 (or Remark 1.3.2) in
Co-rank 1 Arithmetic Siegel–Weil IV.
At 55:00 in the video, I thought I forgot a sign when defining the constant $h^{\mathrm{CM}}_{\widehat{\mathcal{E}}^{\vee}}$
at the beginning of the talk, so I added an extra sign at this point in the talk. But I did not actually
forget the extra sign at the beginning, so the sign here is extraneous. See the formulas in
Section 9 of Co-rank 1 Arithmetic Siegel–Weil IV.
A refined conjecture for the variance of Gaussian primes across sectors
with Yujin H. Kim,
Jared D. Lichtman,
Steven J. Miller,
Alina Shubina,
Shannon Sweitzer,
Ezra Waxman,
Eric Winsor,
and Jianing Yang.
Experimental Mathematics
, vol. 32 no. 1 (2023), pp. 33–53.
[abstract]
[arXiv (2019)]
We derive a refined conjecture for the variance of Gaussian primes across sectors,
with a power saving error term, by applying the $L$-functions Ratios Conjecture.
We observe a bifurcation point in the main term,
consistent with the Random Matrix Theory (RMT) heuristic previously proposed
by Rudnick and Waxman. Our model also identifies a second bifurcation point,
undetected by the RMT model, that emerges upon taking into account lower order terms.
For sufficiently small sectors, we moreover prove an unconditional result that
is consistent with our conjecture down to lower order terms.
$p$-adic Properties of Hauptmoduln with Applications to Moonshine
with Samuel Marks
and Matt Tyler.
Symmetry, Integrability, and Geometry: Methods and Applications (SIGMA)
, vol. 15 (2019), pp. 1–35.
[abstract]
[arXiv (2018)]
The theory of monstrous moonshine asserts that the coefficients of Hauptmoduln,
including the $j$-function, coincide precisely with the graded characters of the monster module,
an infinite-dimensional graded representation of the monster group.
On the other hand, Lehner and Atkin proved that the coefficients of the $j$-function
satisfy congruences modulo $p^n$ for $p\in\{2,3,5,7,11\}$,
which led to the theory of $p$-adic modular forms.
We combine these two aspects of the $j$-function
to give a general theory of congruences modulo powers of primes
satisfied by the Hauptmoduln appearing in monstrous moonshine.
We prove that many of these Hauptmoduln satisfy such congruences,
and we exhibit a relationship between these congruences and the
group structure of the monster. We also find a distinguished class
of subgroups of the monster with graded characters satisfying such congruences.
Lower-Order Biases in the Second Moment of Dirichlet Coefficients in Families of $L$-functions
with Megumi Asada,
Eva Fourakis,
Yujin Hong Kim,
Andrew Kwon,
Jared Duker Lichtman,
Blake Mackall,
Steven J. Miller,
Eric Winsor,
Karl Winsor,
Jianing Yang,
and Kevin Yang.
Experimental Mathematics
, vol. 32 no. 3 (2023), pp. 431–456.
[abstract]
[arXiv (2018)]
Let $\mathcal E: y^2 = x^3 + A(T)x + B(T)$ be a nontrivial one-parameter family of elliptic
curves over $\mathbb{Q}(T)$, with $A(T), B(T) \in \mathbb Z(T)$, and consider the $k$th
moments $A_{k,\mathcal{E}}(p) := \sum_{t (p)} a_{\mathcal{E}_t}(p)^k$ of the Dirichlet coefficients
$a_{\mathcal{E}_t}(p) := p + 1 - |\mathcal{E}_t (\mathbb{F}_p)|$. Rosen and Silverman proved a conjecture
of Nagao relating the first moment $A_{1,\mathcal{E}}(p)$ to the rank of the family over $\mathbb{Q}(T)$,
and Michel proved that if $j(T)$ is not constant then the second moment is equal to $A_{2,\mathcal{E}}(p) = p^2 + O(p^{3/2})$.
Cohomological arguments show that the lower order terms are of sizes $p^{3/2}, p, p^{1/2},$ and $1$.
In every case we are able to analyze, the largest lower order term in the second moment expansion that
does not average to zero is on average negative. We prove this Bias Conjecture for several large
classes of families, including families with rank, complex multiplication, and constant $j(T)$-invariant.
We also study the analogous Bias Conjecture for families of Dirichlet characters,
holomorphic forms on GL$(2)/\mathbb{Q}$, and their symmetric powers and Rankin-Selberg convolutions.
We identify all lower order terms in large classes of families, shedding light on the arithmetic objects
controlling these terms. The negative bias in these lower order terms has implications toward
the excess rank conjecture and the behavior of zeros near the central point in families of $L$-functions.
Spectral statistics of non-Hermitian random matrix ensembles
with Yujin H. Kim,
Jared D. Lichtman,
Steven J. Miller,
Shannon Sweitzer,
and Eric Winsor.
Random Matrices: Theory and Applications
, vol. 8 no. 2 (2019), pp. 1–40.
[abstract]
[arXiv (2018)]
Recently Burkhardt et. al. introduced the $k$-checkerboard random matrix
ensembles, which have a split limiting behavior of the eigenvalues
(in the limit all but $k$ of the eigenvalues are on the order of $\sqrt{N}$
and converge to semi-circular behavior, with the remaining $k$ of size $N$
and converging to hollow Gaussian ensembles). We generalize their work to
consider non-Hermitian ensembles with complex eigenvalues; instead of a blip new
behavior is seen, ranging from multiple satellites to annular rings.
These results are based on moment method techniques adapted to the
complex plane as well as analysis of singular values.
On Reay's relaxed Tverberg conjecture and generalizations of Conway's thrackle cojecture
with Megumi Asada,
Florian Frick,
Frederick Huang,
Maxwell Polevy,
David Stoner,
Ling Hei Tsang,
and Zoe Wellner.
The Electronic Journal of Combinatorics
, vol. 25 no. 3 (2018), pp. 1–14.
[abstract]
[arXiv (2016)]
Reay’s relaxed Tverberg conjecture and Conway’s thrackle conjecture are open problems
about the geometry of pairwise intersections. Reay asked for the minimum number of points
in Euclidean $d$-space that guarantees any such point set admits a partition into r parts,
any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number,
which Reay conjectured to be independent of $k$. We prove a colored version of Reay’s conjecture
for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Requiring convex hulls
to intersect pairwise severely restricts combinatorics. This is a higher-dimensional analog of
Conway’s thrackle conjecture or its linear special case. We thus study convex-geometric and
higher-dimensional analogs of the thrackle conjecture alongside Reay’s problem and conjecture
(and prove in two special cases) that the number of convex sets in the plane is bounded by the
total number of vertices they involve whenever there exists a transversal set for their pairwise
intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture.
We also establish tight bounds for the number of facets of higher-dimensional analogs of linear
thrackles and conjecture their continuous generalizations.
Other (not on arXiv):
Integer points on complements of dual curves and on genus one modular curves
Undergraduate thesis.
[abstract]
[readme]
Higher dimensional analogues of Siegel's theorem on
finiteness of integer points are known in limited cases
and often come with restrictions on the divisor at infinity,
e.g. that the divisor should have many irreducible components.
In the first half of this thesis, we give a new class of
prime divisors in the plane, namely duals of certain smooth
plane curves, whose complements have finitely many integer points.
This is accomplished using a moduli of curves interpretation.
In the second half of this thesis, we give verify finiteness
of integer points on genus one modular curves.
This result is not new, but the proof we give is based
on the $p$-adic period map and ideas from a recent proof of
the Mordell conjecture (Faltings's theorem) by Lawrence and Venkatesh.
This work would benefit from revision.
An original (unrevised) version from 2019 is available
here.
Alternatively, see Princeton's database here.
Last updated: September 2024.
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