Related papers: Class number relations arising from intersections …

We study arithmetic intersections on twisted (quaternionic) Hilbert modular surfaces and Shimura curves over a real quadratic field. Our first main result is the determination of the degree of the top arithmetic Todd class of an arithmetic…

The aim of this paper is to study class number relations over function fields and the intersections of Hirzebruch-Zagier type divisors on the Drinfeld-Stuhler modular surfaces. The main bridge is a particular "harmonic" theta series with…

In this paper we study relations between intersection numbers on moduli spaces of curves and Hurwitz numbers. First, we prove two formulas expressing Hurwitz numbers of (generalized) polynomials via intersections on moduli spaces of curves.…

This is the third of a series of papers relating intersections of special cycles on the integral model of a Shimura surface to Fourier coefficients of Hilbert modular forms. More precisely, we embed the Shimura curve over Q associated to a…

In this paper, we prove Hurwitz-Eichler type formulas for Hurwitz class numbers with each level $ M $ when the modular curve $ X_0(M) $ has genus zero. A key idea is to calculate intersection numbers of modular correspondences with the…

This paper studies the intersections of Hecke correspondences on the modular varieties of $\mathcal{D}$ -elliptic sheaves in the higher-rank setting, where $\mathcal{D}$ is a "maximal order" in a central division algebra $D$ over a global…

We compute the arithmetic intersection numbers of certain Heegner divisors on integral models of Shimura curves over Q. Our formulas generalize the formulas of Gross-Kohnen-Zagier for intersection numbers of Heegner divisors on integral…

We define the dimension 2g-1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P^1 with given ramification over infinity and sufficiently many fixed ramification points…

We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke…

This article proves that Tate classes on Siegel modular 3-folds are spanned by the images of Hilbert modular surfaces at degree 2 and by the images of Shimura curves at degree 4. The proof involves a careful study of the period pairing…

We find that the first Hurwitz triplet possesses two distinct arithmetic structures. As Shimura curves $X_1$, $X_2$, $X_3$, whose levels are with norm 13. As non-congruence modular curves $Y_1$, $Y_2$, $Y_3$, whose levels are 7. Both of…

We find explicit formulas for the Hilbert series of residual intersections of a scheme in terms of the Hilbert series of its conormal modules. In a previous paper we proved that such formulas should exist. We give applications to the…

Kudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension…

This is a note on calculating intersection numbers on moduli spaces of curves. A codimension 3 relation among tautological classes on the moduli space of genus 4 curves is given.

We compute the arithmetic intersections of Hecke correspondences on the product of integral model of modular curve $\mathcal{X}_0(N)$ and relate it to the derivatives of certain Siegel Eisenstein series when $N$ is odd and squarefree. We…

We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…

Shimura curves are moduli spaces of abelian surfaces with quaternion multiplication. Models of Shimura curves are very important in number theory. Klein's icosahedral invariants $\mathfrak{A},\mathfrak{B}$ and $\mathfrak{C}$ give the…

We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and…

We extend a holomorphic projection argument of our earlier work to prove a novel divisibility result for non-holomorphic congruences of Hurwitz class numbers. This result allows us to establish Ramanujan-type congruences for Hurwitz class…

In this work we study the Humbert-Edge's curves of type 5, defined as a complete intersection of four diagonal quadrics in $\mathbb{P}^5$. We characterize them using Kummer surfaces and using the geometry of these surfaces we construct some…