Class groups and L-series of function fields
نویسندگان
چکیده
منابع مشابه
The distribution of class groups of function fields
For any sufficiently general family of curves over a finite field Fq and any elementary abelian `-group H with ` relatively prime to q, we give an explicit formula for the proportion of curves C for which Jac(C)[`](Fq) ∼= H. In doing so, we prove a conjecture of Friedman and Washington. In 1983, Cohen and Lenstra introduced heuristics [5] to explain statistical observations about class groups o...
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The author has computed the class groups of all complex quadratic number fields Q(\f^~D) °f discriminant D for 0 < D < 4000000. In so doing, it was found that the first occurrences of rank three in the 3-Sylow subgroup are D = 3321607 = prime, class group C(3) x C(3) x C(9.7) (C(n) a cyclic group of order n), and D = 3640387 = 421.8647, class group C(3) X C(3) X C(9.2). The author has also foun...
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In the previous lecture we proved the Kronecker-Weber theorem: every abelian extension L of Q lies in a cyclotomic extension Q(ζm)/Q. The isomorphism Gal(Q(ζm)/Q) ' (Z/mZ)× allows us to view Gal(L/Q) as a quotient of (Z/mZ)×. Conversely, for each quotient H of (Z/mZ)×, there is a subfield L of Q(ζm) for which H ' Gal(L/Q). We now want make the correspondence between H and L explicit, and then g...
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We find a lower bound on the number of imaginary quadratic extensions of the function field Fq(T ) whose class groups have an element of a fixed order. More precisely, let q ≥ 5 be a power of an odd prime and let g be a fixed positive integer ≥ 3. There are q 1 2+ 1 g ) polynomials D ∈ Fq[T ] with deg(D) ≤ ` such that the class groups of the quadratic extensions Fq(T, √ D) have an element of or...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1986
ISSN: 0022-314X
DOI: 10.1016/0022-314x(86)90068-5