Riv. Mat. Univ. Parma, Vol. 13, No. 1, 2022

Cornelius Greither [a]

A note on normal bases for the square root of the codifferent in local fields

Pages: 87–109
Received: 12 April 2021
Accepted in revised form: 14 September 2021
Mathematics Subject Classification: 11R33, 11S23.
Keywords: Normal integral bases, inverse different, elliptic functions.
Authors address:
[a]: Fakultät Informatik, Universität der Bundeswehr München, 85577 Neubiberg, Germany.

Full Text (PDF)

Abstract: We consider \(p\)-elementary weakly ramified extensions \(M/K\) of \(p\)-adic fields. (Weak ramification is usually wild, but in some sense "not too wild".) It has been known that for odd primes \(p\), the inverse square root \(A_{M/K}\) of the different of \(M/K\) has a normal integral basis, which opens up a nice parallel to the classical theory of tame extensions. In this note, we extend results of Pickett, concerning the existence of normal integral bases for \(A\) with extra qualities. We assume \(K\) to be unramified over \(\mathbb Q_p\) and construct a self-dual normal basis in the case \(M/K\) is cyclic (Pickett treated the case \(K=\mathbb Q_p\)). We give an explicit construction of a normal basis, presumably no longer self-dual, when \(M/K\) is general \(p\)-elementary. The note ends with a few comments concerning the global situation, which is far less understood.

L. Childs, Taming wild extensions: Hopf algebras and local Galois module theory, Math. Surveys Monogr., 80, AMS, Providence, 2000. MR1767499
B. Erez, The Galois structure of the trace form in extensions of odd prime degree, J. Algebra 118 (1988), 438-446. MR0969683
I. B. Fesenko and S. V. Vostokov, Local fields and their extensions, Transl. Math. Monogr., 121, AMS, Providence, 2002. MR1915966
A. Fröhlich, Galois module structure of algebraic integers, Ergeb. Math. Grenzgeb. (3), 1, Springer-Verlag, Berlin, 1983. MR0717033
C. Greither, Cyclic Galois extensions of commutative rings, Lecture Notes in Math., 1534, Springer-Verlag, Berlin, 1992. MR1222646
C. Greither, Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring, Math. Z. 210 (1992), 37-67. MR1161169
C. Greither, Elliptic functions and normal bases for the root of the inverse different, unpublished manuscript, 2016.
H. Johnston, Explicit integral Galois module structure of weakly ramified extensions of local fields, Proc. Amer. Math. Soc. 143 (2015), 5059-5071. MR3411126
E. J. Pickett, Explicit construction of self-dual integral normal bases for the square-root of the inverse different, J. Number Theory 129 (2009), 1773-1785. MR2524194
E. J. Pickett and S. Vinatier, Self-dual integral normal bases and Galois module structure, Compos. Math. 149 (2013), 1175-1202. MR3078643
R. Schertz, Complex multiplication, New Math. Monogr., 15, Cambridge University Press, Cambridge, 2010. MR2641876
J.-P. Serre, Corps locaux, Hermann, Paris, 1968. MR0354618
J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Grad. Texts in Math., 151, Springer-Verlag, New York, 1994. MR1312368
J. Tate, Les conjectures de Stark sur les fonctions \(L\) d'Artin en \(s=0\), Progr. Math., 47, Birkhäuser, Boston, 1984. MR0782485
S. Ullom, Integral normal bases in Galois extensions of local fields, Nagoya Math. J. 39 (1970), 141-148. MR0263790
V. S. Vostokov, A normal base of an ideal of a local field, (Russian), Rings and modules, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 64 (1976), 64-68. MR0450245
B. F. Wyman, Wildly ramified gamma extensions, Amer. J. Math. 91 (1969), 135-152. MR0241386

Home Riv.Mat.Univ.Parma