The study of structure of spaces of modular forms by the method of cutting

The report is based on the articles of the author [1], [2].

It is well known by the classical theorem that there is the exact cutting of the space of cusp forms of the level $1$ by the function $\Delta(z) = \eta^{24}(z)$: each cusp form of the level 1 and of the even weight $k \geqslant 12$ is the product of $\eta^{24}(z)$ and the modular form of the level 1 and of the weight $k - 12$.

The function $\eta^{24}(z)$ is one of cusp forms of integral weight that are the eta-products with multiplicative coefficients. First they were described in the work [3]. These functions are called multiplicative eta-products (or MacKay's functions).

In the report we discuss structure theorems for spaces of cusp forms with trivial or quadratic characters of higher levels. We show that the exact cutting in such spaces of levels $N \ne 3,~ 17,~ 19$ takes place iff when the cutting function is a multiplicative eta-product. If $N = 3, 17, 19,$ the exact cutting takes place but the cutting function may be a multiplicative eta-product ( it is possible only if $N = 3$) or not.

We also consider several structure theorems in situations when the exact cutting does not take place. In our research we use Biagioli formula for calculating orders of eta-quotients in cusps [4]. The dimensions of spaces are calculated by Cohen-Oesterle formula [5].

[1] G.V. Voskresenskaya, Decomposition of Spaces of Modular Forms, Math. Notes, 99 (2016), № 6, p. 851–860.

[2] G.V. Voskresenskaya, Exact Cutting in Spaces of Cusp Forms with Characters, Math. Notes, 103 (2018), № 6, p. 881–891.

[3] D. Dummit, H. Кisilevsky, J. МасKay, Multiplicative products of $\eta-$ functions, Contemp. Math., 45 (1985), p. 89–98.

[4] A.J.F. Biagioli, The construction of modular forms as products of transforms of the Dedekind eta-function, Acta Arithm., 54 (1990), № 4, p. 273–300.

[5] H. Cohen, J. Oesterle, Dimensions des espaces de formes modulaires, In: J.P. Serre, D.B. Zagier (eds.), Modular Functions of One Variable VI. Lecture Notes in Mathematics, vol 627. Springer, Berlin, Heidelberg, 1977. P. 69–78.