Related papers: Height pairings on orthogonal Shimura varieties

Our aim is to clarify the relationship between Kudla's and Bruinier's Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type. These functions play a key role in the arithmetic geometry of the special…

We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-K\"uhn. Our theorem is thus suitable for…

We consider refined conjectures of Birch and Swinnerton-Dyer type for the Hasse-Weil-Artin L-series of abelian varieties over general number fields. We shall, in particular, formulate several new such conjectures and establish their precise…

We investigate so-called "higher" Siegel theta lifts on Lorentzian lattices in the spirit of Bruinier-Ehlen-Yang and Bruinier-Schwagenscheidt. We give a series representation of the lift in terms of Gauss hypergeometric functions, and…

Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over Q and formulate conjectures about their rationality properties. Moreover, if K…

We study the modularity of the generating series of special cycles on unitary Shimura varieties over CM-fields of degree $2d$ associated with a Hermitian form in $n+1$ variables whose signature is $(n,1)$ at $e$ real places and $(n+1,0)$ at…

We construct a family of special cycle classes on the regular integral model of an orthogonal Shimura variety, and show that these cycle classes appear as Fourier coefficients of a Siegel modular form. Passing to the generic fiber of the…

We apply the theory of Borcherds products to calculate arithmetic volumes (heights) of Shimura varieties of orthogonal type up to contributions from very bad primes. The approach is analogous to the well-known computation of their geometric…

Let $f$ be a primitive Hilbert modular form of parallel weight $2$ and level $N$ for the totally real field $F$, and let $p$ be a rational prime coprime to $2N$. If $f$ is ordinary at $p$ and $E$ is a CM extension of $F$ of relative…

We give a new proof of Howard's $\Lambda$-adic Gross-Zagier formula, which we extend to the context of indefinite Shimura curves over $\mathbf{Q}$ attached to nonsplit quaternion algebras. This formula relates the cyclotomic derivative of a…

We study the intersections of special cycles on a unitary Shimura variety of signature (n-1,1), and show that the intersection multiplicities of these cycles agree with Fourier coefficients of Eisenstein series. The results are new cases of…

Let s be a special point on a Shimura variety, and x a pre-image of s in a fixed fundamental set of the associated Hermitian symmetric domain. We prove that the height of x is polynomially bounded with respect to the discriminant of the…

We study higher spin (pure and mixed spin) representations of the Yangian of $\mathfrak{sl}_2$. We provide a geometric realization in terms of the critical cohomology of representations of the quiver with potential of Bykov and Zinn-Justin…

The Kawaguchi--Silverman conjecture predicts that if $f\colon X \dashrightarrow X$ is a dominant rational-self map of a projective variety over $\overline{\mathbb{Q}}$, and $P$ is a $\overline{\mathbb{Q}}$-point of $X$ with Zariski-dense…

We state and prove an equivariant version of Lehmer's conjecture on heights, generalizing papers by Zagier (1993) and Dresden (1998) which are special cases of this theorem. We also extend their three cases to a full classification of all…

This article sketches relations among algebraic cycles for the Shimura varieties defined by arithmetic quotients of symmetric domains for O(n,2), theta functions, values and derivatives of Eisenstein series and values and derivatives of…

Let $f$ be a newform of weight $2k$ and let $\chi$ be an unramified imaginary quadratic Hecke character of infinity type $(2t, 0)$, for some integer $0 < t \leq k-1$. We show that the central derivative of the Rankin-Selberg $L$-function…

In the 1980s B\"ocherer formulated a conjecture relating the central value of the quadratic twists of the spinor L-function attached to a Siegel modular form F to the coefficients of F . He proved the conjecture when F is a Saito-Kurokawa…

Given a Hecke eigenform $f$ of weight $2$ and square-free level $N$, by the work of Kohnen, there is a unique weight $3/2$ modular form of level $4N$ mapping to $f$ under the Shimura correspondence. Furthermore, by the work of Waldspurger…

The main result of this text is a generalization of Perrin-Riou's p-adic Gross-Zagier formula to the case of Shimura curves over totally real fields. Let $F$ be a totally real field. Let $f$ be a Hilbert modular form over $F$ of parallel…