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We show that the values of Borcherds products on Shimura varieties of orthogonal type at certain CM points are given in terms of coefficients of the holomorphic part of weight one harmonic weak Maass forms. Furthermore, we investigate the…

We extend the work of S. Zhang and Yuan-Zhang-Zhang to obtain a Gross-Zagier formula for modular forms of even weight in terms of an arithmetic intersection pairing of CM-cycles on Kuga-Sato varieties over Shimura curves. Combined with a…

We prove a $p$-adic version of the work by Gross and Zagier on the differences between singular moduli by proving a set of conjectures by Giampietro and Darmon, who investigated the factorisation of a rational invariant associated to a pair…

We study the Faltings height pairing of arithmetic Heegner divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedian contribution to the height pairing and derive a conjecture relating the total…

Given two CM elliptic curves over a number field and a natural number $m$, we establish a polynomial lower bound (in terms of $m$) for the number of rational primes $p$ such that the reductions of these elliptic curves modulo a prime above…

Gross and Zagier defined certain `higher Green's functions' on products of modular curves and conjectured that the value of these functions at complex multiplication points should be logarithms of algebraic numbers. This is now a theorem of…

Elliptic modular graph functions and forms (eMGFs) are defined for arbitrary graphs as natural generalizations of modular graph functions and forms obtained by including the character of an Abelian group in their Kronecker--Eisenstein…

We prove an abstract modularity result for classes of Heegner divisors in the generalized Jacobian of a modular curve associated to a cuspidal modulus. Extending the Gross-Kohnen-Zagier theorem, we prove that the generating series of these…

We describe the complex multiplication (CM) values of modular functions for $\Gamma_0(N)$ whose divisor is given by a linear combination of Heegner divisors in terms of special cycles on the stack of CM elliptic curves. In particular, our…

In this paper, we use Borcherds lifting and the big CM value formula of Bruinier, Kudla, and Yang to give an explicit factorization formula for the norm of $\Psi(\frac{d_1+\sqrt{d_1}}2) -\Psi(\frac{d_2+\sqrt{d_2}}2)$, where $\Psi$ is the…

We prove two formulas in the style of the Gross-Zagier theorem, relating derivatives of L-functions to arithmetic intersection pairings on a unitary Shimura variety. We also prove a special case of Colmez's conjecture on the Faltings…

A general theory of vector-valued modular functions, holomorphic in the upper half-plane, is presented for finite dimensional representations of the modular group. This also provides a description of vector-valued modular forms of arbitrary…

We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all N larger than 2, satisfy a central limit theorem in a…

Multiple modular values are a common generalisation of multiple zeta values and periods of modular forms, and are periods of a hypothetical Tannakian category of mixed modular motives. They are given by regularised iterated integrals on the…

In this work, we derive Gross-Zagier type CM value formulas for Hauptmoduls $j_{p}^{*}(\tau)$ on Fricke groups $\Gamma_{0}^{*}(p)$. We also illustrate how to employ these formulas to obtain certain Hilbert class polynomials.

Given a modular form f of even weight larger than two and an imaginary quadratic field K satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga-Sato variety over a suitable Shimura curve which gives rise…

We establish a rationality result for linear combinations of traces of cycle integrals of certain meromorphic Hilbert modular forms. These are meromorphic counterparts to the Hilbert cusp forms $\omega_m(z_1,z_2)$, which Zagier investigated…

In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics to Siegel modular forms of genus 3. We use this connection to show that certain modular…

This paper is a sequel to our previous work, where we proved the ``modularity theorem'' for algebraic Witt vectors over imaginary quadratic fields. This theorem states that, in the case of imaginary quadratic fields $K$, the algebraic Witt…

The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of…