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from Part I - Modular forms and their variants
Published online by Cambridge University Press: 28 November 2024
In this chapter, we shall discuss modular forms for the congruence subgroups introduced in Chapter 6. We shall obtain the dimension formulas for the corresponding rings of modular forms and cusp forms, describe the fields of modular functions on the modular curves introduced in Chapter 6, and construct the associated Eisenstein series. Throughout the chapter, we shall make use of the correspondence between modular forms and differential forms, viewed as sections of holomorphic line bundles on the compact Riemann surface of the modular curve. We shall provide concrete examples of modular forms for the standard congruence subgroups and apply the results to the theorems of Lagrange and Jacobi on counting the number of representations of an integer as a sum of squares.
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