Mass formula of division algebras over global function fields (Q413419)

From the text: The authors give two proofs of the mass formula for definite central division algebras over global function fields, due to \textit{M. Denert} and \textit{J. Van Geel} [Math. Ann. 282, No. 3, 379--393 (1988; Zbl 0627.16003)]. NEWLINENEWLINELet \(K\) be a global function field with constant field \(\mathbb F_q\). Fix a place \(\infty\) of \(K\), referred to as the place at infinity. Let \(A\) be the subring of functions in \(K\) regular everywhere outside \(\infty\). Let \(B\) be a definite central division algebra of dimension \(r^2\) over \(K\). Let \(R\) be a maximal \(A\)-order in \(B\) and let \(G'\) be the multiplicative group of \(R\), regarded as a group scheme over \(A\). Denote by \(\hat A\) the pro-finite completion of \(A\), which is the maximal open compact topological subring of the ring \(\mathbb A^\infty_K\) of finite adeles of \(K\). The mass associated to the double coset space \(G'(K)\backslash G'(\mathbb A^\infty_K)/G'(\hat A)\)is defined as NEWLINE\[NEWLINE\text{Mass}(G',G'(\hat A)):= \sum_{i=1}^h | \Gamma_i|^{-1}, \quad \Gamma_i:= G'(K)\cap c_i G'(\hat A)c_i^{-1},\tag{1}NEWLINE\]NEWLINE where \(c_1,\ldots, c_h\) are complete representatives for the double coset space. Then the authors prove the following result.NEWLINENEWLINENEWLINETheorem 1.1. We have NEWLINE\[NEWLINE\text{Mass}(G',G'(\hat A))=\frac{\#\text{Pic}(A)}{q-1}\cdot\prod_{i=1}^{r-1} \zeta_K(-i)\cdot\prod_{v\in S}\lambda_v,\tag{2}NEWLINE\]NEWLINENEWLINEwhere \(\text{Pic}(A)\) is the Picard group of \(A\), \(\zeta_K(s)=\prod_v(1-N(v)^{-s})^{-1}\) is the zeta function of \(K\), \(S\) is the finite subset of ramified places for \(B\) and NEWLINE\[NEWLINE\lambda_v=\prod_{_{\substack{ 1\leq i\leq r-1\\ d_v\nmid i}}} (N(v)^i-1),\tag{3}NEWLINE\]NEWLINE where \(d_v\) is the index of the central simple algebra \(B_v = B\otimes_K K_v\).NEWLINENEWLINE NEWLINEThe first proof is based on a calculation of Tamagawa measures. The second proof is based on analytic methods, in which the authors establish the relationship directly between the mass and the value of the associated zeta function at zero.