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Abstract: We present here simple trace formulas for Hecke operators $T kp$ for all$p>3$ on $S k\Gamma 03$ and $S k\Gamma 09$, the spaces of cusp forms ofweight $k$ and levels 3 and 9. These formulas can be expressed in terms ofspecial values of Gaussian hypergeometric series and lend themselves to simplerecursive expressions in terms of traces of Hecke operators on spaces of lowerweight. Along the way, we show how to express the traces of Frobenius of afamily of elliptic curves with 3-torsion as special values of a Gaussianhypergeometric series over $\mathbb{F} q$, when $q\equiv 1 \pmod{3}$. We alsouse these formulas to provide a simple expression for the Fourier coefficientsof $\eta3z^8$, the unique normalized cusp form of weight 4 and level 9, andthen show that the number of points on a certain threefold is expressible interms of these coefficients.



Author: Catherine Lennon

Source: https://arxiv.org/



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