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On the square root of the inverse different
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 11 January 2023, pp. 283-318
- Print publication:
- February 2024
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- Article
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Let
$F_{\pi }$ be a finite Galois-algebra extension of a number field F, with group G. Suppose that 
$F_{\pi }/F$ is weakly ramified and that the square root 
$A_\pi $ of the inverse different 
$\mathfrak {D}_{\pi }^{-1}$ is defined. (This latter condition holds if, for example, 
$|G|$ is odd.) Erez has conjectured that the class 
$(A_\pi )$ of 
$A_\pi $ in the locally free class group 
$\operatorname {\mathrm {Cl}}(\mathbf {Z} G)$ of 
$\mathbf {Z} G$ is equal to the Cassou–Noguès–Fröhlich root number class 
$W(F_{\pi }/F)$ associated with 
$F_\pi /F$. This conjecture has been verified in many cases. We establish a precise formula for 
$(A_\pi )$ in terms of 
$W(F_{\pi }/F)$ in all cases where 
$A_\pi $ is defined and 
$F_\pi /F$ is tame, and are thereby able to deduce that, in general, 
$(A_\pi )$ is not equal to 
$W(F_\pi /F)$.
