### 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>