A Trace Formula for Certain Hecke Operators and Gaussian Hypergeometric Functions

We present here simple trace formulas for Hecke operators $T_k(p)$ for all $p>3$ on $S_k(\Gamma_0(3))$ and $S_k(\Gamma_0(9))$, the spaces of cusp forms of weight $k$ and levels 3 and 9. These formulas can be expressed in terms of special values of Gaussian hypergeometric series and lend themselves to simple recursive expressions in terms of traces of Hecke operators on spaces of lower weight. Along the way, we show how to express the traces of Frobenius of a family of elliptic curves with 3-torsion as special values of a Gaussian hypergeometric series over $\mathbb{F}_q$, when $q\equiv 1 \pmod{3}$. We also use these formulas to provide a simple expression for the Fourier coefficients of $\eta(3z)^8$, the unique normalized cusp form of weight 4 and level 9, and then show that the number of points on a certain threefold is expressible in terms of these coefficients.Comment: 28 page

COMPUTING THE ℓ-POWER TORSION OF AN ELLIPTIC CURVE OVER A FINITE FIELD

Abstract. The algorithm we develop outputs the order and the structure, including generators, of the ℓ-Sylow subgroup of the group of rational points of an elliptic curve defined over a finite field. To do this, we do not assume any knowledge of the group order. We are able to choose points in such a way that a linear number of successive ℓ-divisions leads to generators of the subgroup under consideration. After the computation of a couple of polynomials, each division step relies on finding rational roots of polynomials of degree ℓ. We specify in complete detail the case ℓ = 3, when the complexity of each trisection is given by the computation of cubic roots in finite fields. 1

COMPUTING THE ℓ-POWER TORSION OF AN ELLIPTIC CURVE OVER A FINITE FIELD

Abstract. The algorithm we develop outputs the order and the structure, including generators, of the ℓ-Sylow subgroup of the group of rational points of an elliptic curve defined over a finite field. To do this, we do not assume any knowledge of the group order. We are able to choose points in such a way that a linear number of successive ℓ-divisions leads to generators of the subgroup under consideration. After the computation of a couple of polynomials, each division step relies on finding rational roots of polynomials of degree ℓ. We specify in complete detail the case ℓ = 3, when the complexity of each trisection is given by the computation of cubic roots in finite fields. 1