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On the Deuring-Shafarevich formula. (English) Zbl 1237.14034
The author gives a new proof of the Deuring-Shafarevich formula. Let \(K\) be a global function field over a finite field of characteristic \(p\). Let \(L/K\) be a geometric cyclic extension of degree \(p\). Let \(\lambda_L\) (resp. \(\lambda_K\)) denote the Hasse-Witt invariant of \(L\) (resp. \(K\)). The formula mentioned above has the following form: \(\lambda_L-1=p(\lambda_K-1)+\sum_{P\in S_K}(e_P-1)\deg_KP\), where \(S_K\) is the set of primes of \(K\), \(e_P\) is the ramification index of \(P\) in \(L/K\), and \(\deg_KP\) is the degree of \(P\). He proves this formula by using the fact that \(\lambda_K=\deg\bar{Z}_K(X)\), where \(\bar{Z}_K(X)\in\mathbb{F}_p[X]\) denotes the reduction of the zeta polynomial \(Z_K(X)\), and by focusing on the behavior of the polynomial under the field extension.

MSC:
14H40 Jacobians, Prym varieties
11M38 Zeta and \(L\)-functions in characteristic \(p\)
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