Real Zeros of Random Modular Forms

Özkan, Recep
Modular forms have been a highly important area of interest in many fields such as Algebraic Geometry, Number Theory and Applied Cryptography. These special functions possess very important and interesting arithmetic and geometric properties through which several applications occur in modern mathematics and geometry. Calculating the number of zeros of modular forms on a fundamental domain and finding their distribution behaviour are considered as major problems among them. In this study, the main focus will be on attacking this problem with a probabilistic approach by using standard normal variables and the basis elements of cusp forms through which one can define a so-called random modular form. For this purpose we first give basic definitions and fundamental properties of modular forms, then introduce cusp forms, which are defined as modular forms vanishing as Im(z) tends to infinity, which form a very crucial subspace of the finite dimensional vector space of modular forms. Afterwards, by using the basis elements of the vector space of cusp forms of weight k and independently identically distributed (i.i.d) real random variables, we construct random modular forms. Then we adapt the Croftons formula for random modular forms to obtain the expected number of real zeros which are the zeros defined on some specific geodesic segments on a fundamental domain. In the end we obtain a formula for the expected number of real zeros of a random modular form of weight k and give an upper bound for the infimum of this number.
Citation Formats
R. Özkan, “Real Zeros of Random Modular Forms,” Ph.D. - Doctoral Program, Middle East Technical University, 2024.