Functional equation conjecture for zeta functions of arithmetic schemes

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 sCs\in\mathbb C and satisfy

A(X)(ds)/2ζ(X,ds)=±A(X)s/2ζ(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 dnd-n. The supplied text gives no evidence that this conjecture has been resolved.

Sources & referencesView supporting material

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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