Rational structures on the space of cusp forms on Γ0(N) through the rationality of periods

초록

Kohnen and Zagier studied two types of cusp forms F-k(,D,1,A)+/- on SL2(Z) defined by binary quadratic forms. They showed the cusp forms F-k,(+)(D,1,A) have rational even periods, while F-k,(-)(D,1,A) have rational odd periods. In this paper, we examine analogous cusp forms F-k(,D,N,A)+/- on Gamma(0)(N) for higher levels N. Using locally harmonic Maass forms on Gamma(0)(N), we explicitly compute the even (respectively, odd) period polynomials of F-k,(+)(D,N,A) (respectively, F-k,(-)(D,N,A)). Consequently, we establish that the space of cusp forms of weight 2k on Gamma(0)(N) has rational structures, attributed to the periods of these cusp forms.

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