Abstract
ABSTRACTFor any square-free integer N such that the “moonshine group” Γ0(N)+ has genus zero, the Monstrous Moonshine Conjectures relate the Hauptmodul of Γ0(N)+ to certain McKay–Thompson series associated to the representation theory of the Fischer–Griess monster group. In particular, the Hauptmoduli admits a q-expansion which has integer coefficients. In this article, we study the holomorphic function theory associated to higher genus groups Γ0(N)+. For all such arithmetic groups of genus up to and including three, we prove that the corresponding function field admits two generators whose q-expansions have integer coefficients, has lead coefficient equal to one, and has minimal order of pole at infinity. As corollary, we derive a polynomial relation which defines the underlying projective curve, and we deduce whether i∞ is a Weierstrass point. Our method of proof is based on modular forms and includes extensive computer assistance, which, at times, applied Gauss elimination to matrices with thousands of entries, each one of which was a rational number whose numerator and denominator were thousands of digits in length.
Highlights
For any square-free N, the function field associated to any positive genus g group 0(N)+ admits two generators whose q-expansions have integer coefficients after the lead coefficient has been normalized to equal one
It is our hope that someone will recognize some patterns in the q-expansions we present in this article, as well as in [Jorgenson et al preprint-a] and [Jorgenson et al in preparation-b], and perhaps, higher genus moonshine will manifest itself
The set S is passed in symbolic notation to PARI/GP [The PARI Group 11] which computes in rational arithmetic the q-expansions of all elements of S up to any given positive order κ and forms the matrix AS
Summary
The action of the discrete group PSL(2, Z) on the hyperbolic upper half plane H yields a quotient space PSL(2, Z)\H which has genus zero and one cusp. The unique function obtained by setting of c−1 = 1 and c0 = 744 is known as the j-invariant, which we denote by j(z). For quite some time it has been known that the j-invariant has importance far beyond the setting of automorphic forms and the uniformization theorem as applied to PSL(2, Z)\H. The seminal work of Gross–Zagier [Gross and Zagier 85] studies the factorization of the j-function evaluated at imaginary quadratic integers, yielding numbers known as singular moduli, from which we have an abundance of current research which reaches in various directions of algebraic and arithmetic number theory. There is so much richness in the arithmetic properties of the j-invariant that we are unable to provide an exhaustive list, but rather ask the reader to accept the above-mentioned examples as indicating the important role played by the j-invariant in number theory and algebraic geometry
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