Vestnik of Samara University. Natural Science SeriesVestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University1068910.18287/2541-7525-2021-27-4-7-13Research ArticleON THE NATURE OF ADDITIONAL SPACE AT CUTTING OF SPACES OF CUSP FORMSVoskresenskayaG. V.<p>Doctor of Physical and Mathematical Sciences, professor of the Department of Algebra<br />and Geometry</p>galvosk@mail.ruhttps://orcid.org/0000-0002-6288-5372Samara National Research University211220212747131310202213102022Copyright © 2021, Voskresenskaya G.V.2021<p style="text-align: justify;">In the article we study a space of cusp forms by the method of cutting. This space is a direct sum of the subspace of forms divided by the fixed cusp form named the cutting function and the additional space. If the additional space is zero we have the situation of exact cutting. In common case the cutting is not exact and it is important to research the nature of the additional space. We prove that the basis of the additional space can be described by the space of cusp forms of small weight. This weight is not more than 14 and often is equal to 4. We give examples of all cutting functions for all levels. We prove the theorem about the basis of the additional space to the space of cusp forms in the space of modular forms of the same level, weight and character. We use properties of eta-products, Biagioli formula for orders in cusps and Cohen — Oesterle formula for dimensions.</p>модулярные формыпараболические формыэта-функция Дедекиндапараболические вершиныряды Эйзенштейнаструктурные теоремыформула Коэна — Остерлеформула Биаджиолиmodular formscusp formsDedekind eta–functioncuspsEisenstein seriesstructure theoremCohen — Oesterle formulaBiagioli formula[Ono K. The web of modularity: arithmetic of the coefficients of modular forms and q-series. A.M.S., Providence, 2004, 216 p. DOI: http://doi.org/10.1090/CBMS%2F102.][Koblitz N. Introduction To Elliptic Curves and Modular Forms. Moscow: Mir, 1988, 320 p. Available at: http://ega-math.narod.ru/Books/Koblitz.htm (in Russ.)][Knapp A. Elliptic Curves. Moscow: Faktorial Press, 2004, 488 p. Available at: http://ega-math.narod.ru/Books/Knapp.djv (in Russ.)][Voskresenskaya G.V. Dedekind _—Function in Modern Research. Journal of Mathematical Sciences, 2018, vol. 235, pp. 788–833. DOI: http://doi.org/10.1007/s10958-018-4093-5. (English; Russian original)][Voskresenskaya G.V. Exact Cutting in Spaces of Cusp Forms with Characters. Mathematical Notes, 2018, vol. 103, no. 6, pp. 881–891. DOI: http://doi.org/10.1134/S0001434618050243. (English; Russian original).][Voskresenskaya G.V. MacKay functions in spaces of higher levels. Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia = Vestnik of Samara University. Natural Science Series, 2018, vol. 24, issue 4, pp. 13–18. DOI: http://doi.org/10.18287/2541-7525-2018-24-4-13-18 (in Russ.)][Gordon B., Sinor D. Multiplicative properties of _−products. In: Alladi K. (eds) Number Theory, Madras 1987. Lecture Notes in Mathematics, vol 1395. Springer, Berlin, Heidelberg, 1987, vol. 1395, pp. 173–200. DOI: http://doi.org/10.1007/BFb0086404.][Dummit D.,Кisilevsky H., МасKay J. Multiplicative products of _− functions. Contemporary Mathematics, 1985, vol. 45, pp. 89–98.][Cohen H., Oesterle J. Dimensions des espaces de formes modulaires. In: Lecture Notes in Mathematics, 1976, Vol. 627, pp. 69–78. DOI: http://doi.org/10.1007/BFB0065297.][Biagioli A.J.F. The construction of modular forms as products of transforms of the Dedekind eta-function. Acta Arithmetica, 1990, vol. LIV, no 4, pp. 273–300. DOI: http://doi.org/10.4064/AA-54-4-273-300.]