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

Abstract

$L-$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.