L-series and their 2-adic and 3-adic valuations at s=1 attached to CM elliptic curves, by Qiu Derong and Zhang Xianke

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} \cdots \pi _{n}\ $ or $\ D_1 =\pi _{1} ^{2 }\cdots \pi _{r} ^{2} \pi _{r+1} \cdots \pi _{n}$, where $\pi _{1},\ \cdots ,\ \pi _{n}$ 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} \cdots \pi _{n}, \, $ where $\pi _{1} , \ \cdots ,\ \pi _{n}$ 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.

Qiu Derong and Zhang Xianke <xianke@tsinghua.edu.cn>