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The Primary Role of Modular Equations

In: Number Theory: New York Seminar 1991–1995

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  • Harvey Cohn

    (City College (CUNY), Department o Mathematics)

Abstract

Recent developments indicate that the modular equation, historically the end product of modular function theory, should be considered as a starting point. For instance, while a Fourier series for a modular function has coefficients which determine a modular equation, conversely, modular functions are also determined by a modular equation. It is therefore of interest to ask what are the a priori determining features of a modular equation. Some features are revealed for the classic (one variable) case relating to normality of the parametrization. Other features are revealed for the Hilbert (two variable) case relating to the diagonals. A complete answer is still elusive, but a reasonable goal would be to find singular moduli directly (without modular invariants).

Suggested Citation

  • Harvey Cohn, 1996. "The Primary Role of Modular Equations," Springer Books, in: David V. Chudnovsky & Gregory V. Chudnovsky & Melvyn B. Nathanson (ed.), Number Theory: New York Seminar 1991–1995, chapter 3, pages 19-41, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-2418-1_3
    DOI: 10.1007/978-1-4612-2418-1_3
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