Mod-p vanishing conjecture for the relative quantum connection

Let pp be a prime satisfying the condition denoted by, let nn and ff be the parameters appearing there, and let ∇u∂q\nabla_{u\partial_q} be the relative quantum connection on SHu,q∗(M,D;Z)\mathit{SH}^*_{u,q}(M,D;\mathbb{Z}). The expression f(∇u∂qp)f(\nabla_{u\partial_q}^p) is the polynomial in the pp-th power of this connection appearing in the source.

Mod-pp vanishing conjecture. Assume pp is chosen so that holds. Then

qnpf(∇u∂qp):SHu,q∗(M,D;Z)⟶SHu,q∗(M,D;Z)q^{np}f(\nabla_{u\partial_q}^p):\mathit{SH}^*_{u,q}(M,D;\mathbb{Z})\longrightarrow\mathit{SH}^*_{u,q}(M,D;\mathbb{Z})

is zero modulo pp. This is proposed as a common strengthening of two preceding torsion statements. The supplied text does not establish the strengthening, so it remains open.

References

Primary source

Dan Pomerleano and Paul Seidel, “The quantum connection and its mod p reduction”, arXiv:2606.28256 (2026).

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