Squareness in the special $L$-value

Amod Agashe

Let $N$ be a prime and let~$A$ be a quotient of~$J_0(N)$ over~$\Q$ associated to a newform~$f$ such that the special $L$-value of~$A$ (at $s=1$) is non-zero. Suppose that the algebraic part of special $L$-value is divisible by an odd prime~$q$ such that $q$ does not divide the numerator of~$\frac{N-1}{12}$. Then the Birch and Swinnerton-Dyer conjecture predicts that $q^2$ divides the algebraic part of special $L$ value of~$A$, as well as the order of the Shafarevich-Tate group. Under a $\bmod\ q$ non-vanishing hypothesis on special $L$-values of twists of~$A$, we show that~$q^2$ does divide the algebraic part of the special $L$-value of~$A$ and the Birch and Swinnerton-Dyer conjectural order of the Shafarevich-Tate group of~$A$. This gives theoretical evidence towards the second part of the Birch and Swinnerton-Dyer conjecture. We also give a formula for the algebraic part of the special $L$-value of~$A$ over suitable quadratic imaginary fields which shows that this algebraic part is a perfect square away from two.