Automorphic properties of generating functions for generalized odd rank moments and odd Durfee symbols

We define two-parameter generalizations of Andrews' $(k+1)$-marked odd Durfee symbols and $2k$th symmetrized odd rank moments, and study the automorphic properties of some of their generating functions. When $k=0$ we obtain families of modular forms and mock modular forms. When $k \geq 1$, we find quasimodular forms and quasimock modular forms

Создание и совершенствование автоматизированных систем управления, связи и оповещения в ЧС

Alfes C, Jameson M, Oliver RJL. Proof of the Alder-Andrews conjecture. Proceedings of the American Mathematical Society. 2011;139(01):63-63

Twisted Traces of CM values of Harmonic Weak Maass Forms

We show that the twisted traces of CM values of weak Maass forms of weight 0 are Fourier coefficients of vector valued weak Maass forms of weight 3/2. These results generalize work by Zagier on traces of singular moduli. We utilize a twisted version of the theta lift considered by Bruinier and Funke.Comment: Some minor corrections, mostly typos. Added Journal referenc

Measures, modular forms, and summation formulas of Poisson type

In this article, we show that Fourier eigenmeasures supported on spheres with radii given by a locally finite sequence, which we call $k$-spherical measures, correspond to Fourier series exhibiting a modular-type transformation behaviour with respect to the metaplectic group. A familiar subset of such Fourier series comprises holomorphic modular forms. This allows us to construct $k$-spherical eigenmeasures and derive Poisson-type summation formulas, thereby recovering formulas of a similar nature established by Cohn-Gon\c{c}alves, Lev-Reti, and Meyer, among others. Additionally, we extend our results to higher dimensions, where Hilbert modular forms yield higher-dimensional $k$-spherical measures

Cycle integrals of meromorphic Hilbert modular forms

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 the context of the Doi-Naganuma lift. We give an explicit formula for these cycle integrals, expressed in terms of the Fourier coefficients of harmonic Maass forms. A key element in our proof is the explicit construction of locally harmonic Hilbert-Maass forms on $\mathbb{H}^2$, which are analogous to the elliptic locally harmonic Maass forms examined by Bringmann, Kane, and Kohnen. Additionally, we introduce a regularized theta lift that maps elliptic harmonic Maass forms to locally harmonic Hilbert-Maass forms and is closely related to the Doi-Naganuma lift