Quadratic Twists of Elliptic Curves

In this paper, we show that Tian's induction method can be generalised to study the Birch-Swinnerton-Dyer conjecture for the quadratic twists, both with global root number $+1$ and with global root number $-1$, of certain elliptic curves $E$ defined over $\mathbb Q$. In particular, for the curve $E = X_0(49)$ we prove the following results. Let $q_1, \ldots, q_r$ be distinct primes which are congruent to $1$ modulo $4$ and inert in the field $F = \mathbb Q(\sqrt{-7})$, and let $E^{(R)}$ be the twist of $E$ by the quadratic extension $\mathbb Q(\sqrt{R})/\mathbb Q$, where $R=q_1\ldots q_r$. Then we show that the complex L-series of $E^{(R)}$ does not vanish at $s=1$, and the full Birch-Swinnerton-Dyer conjecture is true for $E^{(R)}$. Let $l_0$ be a prime number which is congruent to $3$ modulo $4$, and is such that $7$ splits in the field $K = \mathbb Q(\sqrt{-l_0})$. If we assume in addition that all of the primes $q_1, \ldots, q_r$ are inert in $K$ as well as in $F$, then we prove that the complex $L$-series of the twist of $E$ by $\mathbb Q(\sqrt{-l_0R})/\mathbb Q$ always has a simple zero at $s=1$. Similar results are obtained for certain other elliptic curves defined over $\mathbb Q$.

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