Hankel-transform Somos-4 conjecture for elliptic-curve division polynomials

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Let E=E(a,b,c,d)E=E(a,b,c,d) be the elliptic curve

y2+axy+by=x3+cx2+dx,y^2+axy+by=x^3+cx^2+dx,

with nonzero discriminant, and let (0,0)(0,0) be the reference point. Let hnh_n be the Hankel transform of the sequence generated by

g(x)=2b4xAx4+Bx3+Cx2+Dx+1+Fx2+Gx−1,g(x)=\frac{2b^4x}{\sqrt{Ax^4+Bx^3+Cx^2+Dx+1}+Fx^2+Gx-1},

where A,B,C,D,F,GA,B,C,D,F,G are the coefficients specified in the paper. Hankel-transform Somos-4 conjecture. The sequence hnbn2−2n\frac{h_n}{b^{n^2-2n}} is a (b2,abd−b2c+d2)(b^2,abd-b^2c+d^2) Somos-44 sequence which coincides with the division polynomial sequence of EE. This conjecture connects Hankel transforms and Somos sequences with division polynomials of elliptic curves; the source gives no resolution, so its status remains open.

References

Primary source

Paul Barry, “Integer sequences from elliptic curves”, arXiv:2306.05025 (2023).

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