Abstract
We study the Gross conjecture for the cyclotomic function field extension k(Λf)/k where k = Fq(t) is the rational function field and f is a monic polynomial in Fq[t]. We prove the conjecture in the Fermat curve case(i.e., when f = t(t - 1)) by a direct calculation. We also prove the case when f is irreducible, which is analogous to the Weil reciprocity law. In the general case, we manage to show the weak version of the Gross conjecture here.
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References
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Yi, O. The Gross conjecture over rational function fields. Sci. China Ser. A-Math. 48, 1609–1617 (2005). https://doi.org/10.1360/04ys0230
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DOI: https://doi.org/10.1360/04ys0230