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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 $$p$$
##### Keywords:
Jacobians; Zeta functions; Hasse-Witt invariant
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##### References:
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