94 research outputs found
L-series and their 2-adic and 3-adic valuations at s=1 attached to CM elliptic curves
series attached to two classical families of elliptic curves with complex
multiplications are studied over number fields, formulae for their special
values at bound of the values, and criterion of reaching the bound are
given. Let be elliptic curves over the Gaussian
field K=\Q(\sqrt{-1}), with or , where are
distinct primes in . A formula for special values of Hecke series
attached to such curves expressed by Weierstrass function are given; a
lower bound of 2-adic valuations of these values of Hecke series as well as
a criterion for reaching these bounds are obtained. Furthermore, let be elliptic curves over the quadratic field
\Q(\sqrt{-3}) with where are distinct primes of \Q(\sqrt{-3}), similar results as above but
for valuation are also obtained. These results are consistent with the
predictions of the conjecture of Birch and Swinnerton-Dyer, and develop some
results in recent literature for more special case and for valuation
Cyclic quartic fields and genus theory of their subfields
AbstractLet k = Q(βu) (u β 1 squarefree), K any possible cyclic quartic field containing k. A close relation is established between K and the genus group of k. In particular: (1) Each K can be written uniquely as K = Q(βvwΞ·), where Ξ· is fixed in k and satisfies Ξ· βͺ’ 1, (Ξ·) = U2βu, |U2| = |(βu)|, (v, u) = 1, v β Z is squarefree, w|u, 0 < w < βu. Thus if u β a2 + b2, there is no K β k. If u = a2 + b2 then for each fixed v there are 2g β 1K β k, where g is the number of prime divisors of u. (2) Kk has a relative integral basis (RIB) (i.e., OK is free over Ok) iff N(Ξ΅0) = β1 and w = 1, where Ξ΅0 is the fundamental unit of k, (or, equivalently, iff K = Q(βvΞ΅0βu), (v, u) = 1). (3) A RIB is constructed explicitly whenever it exists. (4) disc(K) is given. In particular, the following results are special cases of (2): (i) Narkiewicz showed in 1974 that Kk has a RIB if u is a prime; (ii) Edgar and Peterson (J. Number Theory 12 (1980), 77β83) showed that for u composite there is at least one K β k having no RIB. Besides, it follows from (4) that the classification and integral basis of K given by Albert (Ann. of Math. 31 (1930), 381β418) are wrong
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