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In Joe Silverman's "The Arithmetic of Ecliptic Curves" he talks a lot about the integral domain $\overline{K}(C)$. On page $27$ he suddenly decides to chose an element $f$ from some set $\overline{K}(C)^*$. What does this notation mean? After page $27$ he begins using $\overline{K}(C)^*$ a lot, and I haven't been able to pick up what it is from context clues.

Milo Moses
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2 Answers2

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$\bar{K}(C)$ is the function field of $C$ over the algebraically closure of $K$ (as recalled in the beginning of chapter).

$\bar{K}(C)^*$ is the multiplicative group of units of $\bar{K}(C)$.

The notation $R^*$ for the multiplicative group of units of a ring $R$ is standard. If $R$ is a field then $R^*=R-\{0\}$.

user10354138
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  • Thank you so much! I also appreciate you using the term "function field of the algebraic closure"; I knew what to call the function field, but I always referred to $\overline{K}(C)$ as "the analogue of the function field to the algebraic closure" – Milo Moses Feb 22 '21 at 23:17
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There is already a correct answer, but I will just add here that the notation is defined in Chapter I, in a Definition right after Example 1.3.3. where Silverman defines the affine coordinate ring and the function field of a variety.

Álvaro Lozano-Robledo
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