Functional equation conjecture for zeta functions of arithmetic schemes

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Let X\mathcal X be a regular scheme of dimension dd, proper and flat over Spec⁡(Z)\operatorname{Spec}(\mathbb Z). Let A(X)A(\mathcal X) be its Bloch conductor and let

ζ(X‾,s):=ζ(X,s)⋅ζ(X∞,s)\zeta(\overline{\mathcal X},s):=\zeta(\mathcal X,s)\cdot\zeta(\mathcal X_\infty,s)

be the completed zeta function, where ζ(X∞,s)\zeta(\mathcal X_\infty,s) is the archimedean Euler factor.

Functional equation conjecture. The function ζ(X,s)\zeta(\mathcal X,s) should have a meromorphic continuation to all s∈Cs\in\mathbb C and satisfy

A(X)(d−s)/2⋅ζ(X‾,d−s)=±A(X)s/2⋅ζ(X‾,s).A(\mathcal{X})^{(d-s)/2}\cdot \zeta(\overline{\mathcal{X}},d-s)=\pm A(\mathcal{X})^{s/2} \cdot \zeta(\overline{\mathcal{X}},s).

This is the expected functional equation for the completed zeta function and is used in the paper to compare the special value conjecture at nn and d−nd-n. The supplied text gives no evidence that this conjecture has been resolved.

References

Primary source

Matthias Flach and Baptiste Morin, “Compatibility of Special value conjectures with the functional equation of Zeta functions”, arXiv:2005.04829 (2020).

Additional references

2 papers in this index state this conjecture (2010–2020). The statement above is taken from the most recent of them; the others are arXiv:1002.0554.

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