Yingkun Li

Asymptotics for the number of row-Fishburn matrices

In this paper, we provide an asymptotic for the number of row-Fishburn matrices of size n which settles a conjecture by Vit Jelinek. Additionally, using q-series constructions we provide new identities for the generating functions for the number of such matrices, one of which was conjectured by Peter Bala.

Real-dihedral harmonic Maass forms and CM-values of Hilbert modular functions

In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in [7] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we will give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

Heegner divisors in generalized Jacobians and traces of singular moduli

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 classes is a weakly holomorphic modular form of weight 3/2. Moreover, we show that any harmonic Maass forms of weight 0 defines a functional on the generalized Jacobian. Combining these results, we obtain a unifying framework and new proofs for the Gross-Kohnen-Zagier theorem and Zagiers modularity of traces of singular moduli, together with new geometric interpretations of the traces with non-positive index.

Modularity of generating series of winding numbers

The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of half-integral weight. In this article, we generalize this lift to differentials of the third kind. As an application, we obtain a modularity result concerning the generating series of winding numbers of closed geodesics on the modular curve.

Shimura lifts of weakly holomorphic modular forms

We show how to realize the Shimura lift of arbitrary level and character using the vector-valued theta lifts of Borcherds. Using the regularization of Borcherds’ lift we extend the Shimura lift to take weakly holomorphic modular forms of half-integral weight to meromorphic modular forms of even integral weight having poles at CM points.

Harmonic Eisenstein Series of Weight One

In this short note, we will construct a harmonic Eisenstein series of weight one, whose image under the ξ-operator is a weight one Eisenstein series studied by Hecke (Math Ann 97(1):210–242, 1927).

Level lowering for half-integral weight modular forms

In this paper we provide a level lowering result for half-integral weight modular forms. The main ingredients are the Shimura map from half-integral weight modular forms to integral weight modular forms along with a level lowering result for integral weight modular forms due to Ribet. It is necessary to keep track of the parity of the weight as well as the character involved so that one can apply the Shintani lift to go back to a half-integral weight modular form and establish the result.