In this paper, we consider the question of correcting mock modular forms in order to obtain p-adic modular forms. In certain cases we show that a mock modular form M is a p-adic modular form. Furthermore, we prove that otherwise the unique correction of M is intimately related to the shadow of M.

Our goal is to give an introduction to modular forms and to discuss the role of Hecke operators in the theory. We will focus on the parts of the theory that we will need in studying [GZ86]. For more details, one can refer to [DI95]. Recall the following topics from Jeff’s talks: • The Atkin-Lehner involution Wk is an involution on the modular curve Y0(N) = h/Γ0(N) and permute the cusps. • Eisen...

Abstract In this paper, we prove the algebraicity of some L -values attached to quaternionic modular forms. We follow rather well-established path doubling method. Our main contribution is that include case where corresponding symmetric space non-tube type. make various aspects very explicit, such as embedding, coset decomposition, and definition forms via CM-points.

For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ2−k (resp. D) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms have transcendental coefficients, we show that those...

For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ2−k (resp. D) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we...

We give a new construction of p-adic overconvergent Hilbert modular forms by using Scholze’s perfectoid Shimura varieties at infinite level and the Hodge–Tate period map. The definition is analytic, closely resembling that complex as holomorphic functions satisfying transformation property under congruence subgroups. As special case, we first revisit case elliptic forms, extending recent work C...

We revisit the modular flavor symmetry from a more general perspective. The scalar forms of principal congruence subgroups are extended to vector-valued forms, then we have possible finite groups including $\Gamma_N$ and $\Gamma'_N$ as symmetry. theory provide method differential equation construct multiplets, it also reveals simple structure invariant mass models. review give results for lower...

We define Jacobi forms with complex multiplication. Analogous to modular multiplication, they are constructed from Hecke characters of the associated imaginary quadratic field. From this construction we obtain a form which specializes η(τ)26 present highlight an open question Dyson and Serre. give other examples applications multiplication including constructing theta blocks elliptic curves new...

Introduction 2 1. Shimura curves 6 1.1. Modular interpretation 6 1.2. Integral models 13 1.3. Reductions of models 17 1.4. Hecke correspondences 18 1.5. Order R and its level structure 25 2. Heegner points 29 2.1. CM-points 29 2.2. Formal groups 34 2.3. Endomorphisms 37 2.4. Liftings of distinguished points 40 3. Modular forms and L-functions 43 3.1. Modular forms 44 3.2. Newforms on X 49 3.3. ...