L-series and their 2-adic and 3-adic valuations at s=1 attached to CM elliptic curves

$L-$series attached to two classical families of elliptic curves with complex multiplications are studied over number fields, formulae for their special values at $s=1,$ bound of the values, and criterion of reaching the bound are given. Let $E_1: y^{2}=x^{3}-D_1 x$ be elliptic curves over the Gaussian field K=\Q(\sqrt{-1}), with $D_1 =\pi_{1} ... \pi_{n}$ or $D_1 =\pi_{1} ^{2}... \pi_{r} ^{2} \pi_{r+1} ... \pi_{n}$, where $\pi_{1}, ..., \pi_{n}$ are distinct primes in $K$. A formula for special values of Hecke $L-$series attached to such curves expressed by Weierstrass $\wp-$function are given; a lower bound of 2-adic valuations of these values of Hecke $L-$series as well as a criterion for reaching these bounds are obtained. Furthermore, let $E_{2}: y^{2}=x^{3}-2^{4}3^{3}D_2^{2}$ be elliptic curves over the quadratic field \Q(\sqrt{-3}) with $D_2 =\pi_{1} ... \pi_{n},$ where $\pi_{1}, ..., \pi_{n}$ are distinct primes of \Q(\sqrt{-3}), similar results as above but for $3-adic$ 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 $2-adic$ valuation

Cyclic quartic fields and genus theory of their subfields

AbstractLet k = Q(√u) (u ≠ 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(√vwη), where η is fixed in k and satisfies η ⪢ 1, (η) = U2√u, |U2| = |(√u)|, (v, u) = 1, v ∈ Z is squarefree, w|u, 0 < w < √u. Thus if u ≠ a2 + b2, there is no K ⊃ k. If u = a2 + b2 then for each fixed v there are 2g − 1K ⊃ 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(ε0) = −1 and w = 1, where ε0 is the fundamental unit of k, (or, equivalently, iff K = Q(√vε0√u), (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), 77–83) showed that for u composite there is at least one K ⊃ 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), 381–418) are wrong