Ryan C. Chen

(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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.