zbMATH — the first resource for mathematics

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.

14H40 Jacobians, Prym varieties
11M38 Zeta and \(L\)-functions in characteristic \(p\)
Full Text: DOI
[1] Crew, Richard M., Etale \(p\)-covers in characteristic \(p\), Compositio Math., 52 (1984), no. 1, 31-45. · Zbl 0558.14009
[2] M. Deuring, Automorphismen und Divisorenklassen der Ordnung \(\iota\) in algebraischen Funktionenkörpern, Math. Ann., 113 (1937), 208-215. · Zbl 0014.29301
[3] Hayes, D. R., Explicit class field theory for rational function fields, Trans. Amer. Math. Soc., 189 (1974), 77-91. · Zbl 0292.12018
[4] Guo, L. and Shu, L., Class numbers of cyclotomic function fields., Trans. Amer. Math. Soc. 351 (1999), no., 11 , 4445-4467. · Zbl 0929.11049
[5] Katz, M. and Messing, W., Some consequences of the Riemann hypothesis for varieties over finite fields. Invent. Math., 23 (1974), 73-77. · Zbl 0275.14011
[6] Madan, Manohar L., On a theorem of M. Deuring and I. R. Shafarevich., Manuscripta Math., 23 (1977), no. 1, 91-102. · Zbl 0369.12011
[7] Rosen, Michael, Number Theory in Function Fields , Springer-Verlag, Berlin, 2002. · Zbl 1043.11079
[8] I. R. Shafarevich, On \(p\)-Extensions, Amer. Math. Soc. Trans. Series II, 4 (1954), 59-71.
[9] Subrao, Doré, The \(p\)-rank of Artin-Schreier curves, Manuscripta Math., 16 (1975), no. 2, 169-193. · Zbl 0321.14017
[10] Sur l’image de l’application d’Abel-Jacobi de Bloch, Bull. Soc. Math. France 116 (1988), no., 1 , 69-101. · Zbl 0686.14006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.