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- W2775682907 abstract "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 the special $L$-value of $A$ 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 indeed 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 $A$. We also give a formula for the algebraic part of the special $L$-value of $A$ over suitable quadratic imaginary fields in terms of the free abelian group on isomorphism classes of supersingular elliptic curves in characteristic $N$ (equivalently over conjugacy classes of maximal orders in the definite quaternion algebra over $Q$ ramified at $N$ and $infty$) which shows that this algebraic part is a perfect square away from two." @default.
- W2775682907 created "2017-12-22" @default.
- W2775682907 creator A5091041300 @default.
- W2775682907 date "2008-10-29" @default.
- W2775682907 modified "2023-09-27" @default.
- W2775682907 title "Squareness in the special $L$-value" @default.
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