Denominator conjecture for modular polynomials in the b'-invariants

Let bi(Ω):=θi(Ω/2)/θ0(Ω/2)b'_i(\Omega):=\theta_i(\Omega/2)/\theta_0(\Omega/2), and let LpQ[b1,b2,b3]L'_p\in\mathbb{Q}[b'_1,b'_2,b'_3] be a polynomial describing the locus of principally polarized abelian surfaces modulo Γ(2,4)\Gamma(2,4) that are (p,p)(p,p)-isogenous to a principally polarized abelian surface Ω\Omega which is isogenous to a product of two elliptic curves by the (2,2)(2,2)-isogeny ΩΩ/2\Omega\to\Omega/2 and satisfies θ0(Ω/2)=0\theta_0(\Omega/2)=0.

Denominator conjecture. The polynomial LpL'_p is the denominator of every coefficient of the three modular polynomials in the functions b1,b2,b3b'_1,b'_2,b'_3.

The preceding proposition shows that the denominators are divisible by a polynomial describing this locus, and computations for p=3,5,7p=3,5,7 support the assertion that the same polynomial LpL'_p is the denominator for all coefficients. The source does not state a resolution status.

Sources & referencesView supporting material

Primary source

Enea Milio, “A quasi-linear time algorithm for computing modular polynomials in dimension 2”, arXiv:1411.0409 (2015).

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