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Class groups of global function fields with certain splitting behaviors of the infinite prime
- Class groups of global function fields with certain splitting behaviors of the infinite prime
- Lee Y.
- Ewha Authors
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- Proceedings of the American Mathematical Society
- Proceedings of the American Mathematical Society vol. 137, no. 2, pp. 415 - 424
- SCIE; SCOPUS
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- For certain two cases of splitting behaviors of the prime at infinity with unit rank r, given positive integers m,n, we construct infinitely many global function fields K such that the ideal class group of K of degree m over F(T) has n-rank at least m - r - 1 and the prime at infinity splits in K as given, where F denotes a finite field and T a transcendental element over F. In detail, for positive integers m, n and r with 0 ≤ r ≤ m - 1 and a given signature (ei,fi), 1 ≤ i ≤ r + 1, such that Σr+1i=1 eifi = m, in the following two cases where ei is arbitrary and fi = 1 for each i, or ei = 1 and fi's are the same for each i, we construct infinitely many global function fields K of degree m over F(T) such that the ideal class group of K contains a subgroup isomorphic to (ℤ/nℤ) m-r-1and the prime at infinity p;∞ splits into r + 1 primes β 1, β2, • • •, P r+1 in K with e(βi/p;∞) = e i and f(βi/p;infin;) = fi for 1 ≤ i ≤ r + 1 (so, K is of unit rank r). © 2008 American Mathematical Society.
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