Number theory seminar in memory of F. Momose
On the Heegner index of an elliptic curve over $\mathbb{Q}$.
Presenters
Carlos Castaño-Bernard
Abstract
In this talk we discuss an approach to study the re ned version of the conjecture of Birch and Swinnerton-Dyer for $E$ of rank one over $\mathbb{Q}$. Our approach is based on a result of Gross, Kohnen and Zagier that says that this conjecture is essentially equivalent to $[E(\mathbb{Q}) : \mathbb{Z}P]^2 = c_E \cdot n_E \cdot m_E \cdot |(E)|$, where $(E)$ is the Tate-Safarevic group of $E$, $c_E$ is Manin's constant, $P$ is the trace of a Heegner point with minimal height, $n_E$ is the index of the canonical subgroup of $H_1(E(\mathbb{C});\mathbb{Z})^-$, and $m_E$ is the product of the Tamagawa numbers $c_p = |\Phi_{E,p}(p)|$, where $\Phi_{E,p}$ denotes the component group of $E$ at $p$, and $p$ runs through the set of finite primes $p$. So it seems natural to ask whether the "missing" component group $\Phi_{E,\infty} = E(\mathbb{R})/E(\mathbb{R})^0$ of $E$ at $p = \infty$ encodes information related in some way to the index $[E(\mathbb{Q}) : \mathbb{Z}P]$. So let us consider the canonical map $\phi_{E,\infty} : E(\mathbb{Q})\rightarrow \Phi_{E,\infty}$, and let $n_{E,p} = p^\alpha$ , where $p^\alpha \| n_E$ with $p$ prime. We conjecture that if $E$ is an elliptic curve of rank one and prime conductor $N$, then $n_{E,2} = 2^\alpha$ , where $\alpha \in \{0, 2\}$ and, moreover, $$n_{E,2} = \left\{ \begin{array}{l} |coker (\phi_{E,\infty})|^2, \text{ if $(E)[2]$ is trivial,} \\ |(E)[2]|, \text{ otherwise.} \end{array} \right.$$ We discuss an approach to study this conjecture which involves the intersection numbers of X^+_0 (N)(\mathbb{R}) (which may be regarded as a generalised Heegner geodesic cycle, certainly in $H_1(X^+_ 0 (N)(\mathbb{C}),\mathbb{Z})^+)$ with the Heegner geodesic cycles in $H_1(X^+_0 (N)(\mathbb{C}),\mathbb{Z})^-$. (The latter cycles yield the canonical subgroup of $H_1(E(\mathbb{C}),\mathbb{Z})^-$ via the modular parametrisation.) Here $X^+_0 (N)$ is the Atkin-Lehner quotient $X^+_0 (N) = X_0(N)/\omega_N$ of the modular curve $X_0(N)$, and the superindex in homology denotes the $\pm 1$-eigenspace with respect to the action induced by complex conjugation.
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