Congruences of modular forms and Selmer groups

We show that the congruence modulo $11$ between the normalized cusp form $\Delta$ of weight $12$ and the normalized cusp form of weight $2$ and level $11$ `descends' to a congruence between forms of weights $13/2$ and $3/2$. Combining Waldspurger’s theorem with the Bloch-Kato conjecture we predict the existence of elements of order $11$ in Selmer groups for certain quadratic twists of $\Delta$. These are then constructed using rational points on twists of the elliptic curve $X_0(11)$, assuming the Birch and Swinnerton-Dyer conjecture on the rank. Everything generalizes to forms of weights $2+10s$ in an $11$-adic family, to congruences modulo higher powers of $11$, and to other elliptic curves over $\QQ$ of prime conductor $p\equiv 3\pmod{4}$ such that $L(E_{-p},1)\neq 0$ and $p\nmid\ord_p(j(E))$.

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Published 1 January 2001