Tamagawa number conjecture for SLnSL_n over a function field

Let F=C(X)F=\mathbb C(X) be the function field associated with the Riemann surface MM, let A\mathbb A be its adelic ring, and let K\mathbb K be the specified maximal compact subgroup. Let τ\tau denote the Tamagawa measure and m(K)\mathrm{m}(\mathbb K) the mass of K\mathbb K. Tamagawa number conjecture. There exist natural measures such that

τ(SLn(F)\SLn(A))=1\tau\big(SL_n(F)\backslash SL_n(\mathbb A)\big)=1

and

m(K)=ζ^M(1)ζ^M(2)ζ^M(n).\mathrm{m}(\mathbb K)=\widehat\zeta_M(1)\widehat\zeta_M(2)\cdots\widehat\zeta_M(n).

The conjecture seeks a Tamagawa-number normalization and a zeta-value formula for the maximal compact subgroup. The source gives no resolution.

Sources & referencesView supporting material

Primary source

Lin Weng, “General Uniformity of Zeta Functions”, arXiv:1209.4515 (2012).

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