On the local divisibility of Heegner points. (English) Zbl 1276.11091

Goldfeld, Dorian (ed.) et al., Number theory, analysis and geometry. In memory of Serge Lang. Berlin: Springer (ISBN 978-1-4614-1259-5/hbk; 978-1-4614-1260-1/ebook). 215-241 (2012).
From the text: We relate the local \(\ell \)-divisibility of a Heegner point on an elliptic curve of conductor \(N\), at a prime \(p\) which is inert in the imaginary quadratic field, to the first \(\ell\)-descent on a related abelian variety of level \(Np\).
Heegner points on the modular curve \(X_0(N)\), and their images on elliptic curve factors \(E\) of the Jacobian, enjoy many remarkable properties. These points are the moduli of level structures with endomorphisms by the ring of integers of an imaginary quadratic field \(K\). Their traces to \(E(K)\) have height given by the first derivative at \(s=1\) of the \(L\)-function of \(E\) over \(K\) (cf. the first author and D. B. Zagier [Invent. Math. 84, 225–320 (1986; Zbl 0608.14019)]), and their \(\ell\)-divisibility in the Mordell-Weil group controls the first \(\ell\)-descent on \(E\) over \(K\) (cf. the first author [L-functions and arithmetic (Durham, 1989), London Math. Soc. Lect. Notes 153, 235–256 (1991; Zbl 0743.14021)]).
In this paper, we show how their \(\ell\)-divisibility in the local group \(E(K_p)\), where \(p\) is a prime that is inert in \(K\), often determines a first descent over \(K\) on a related abelian variety \(A\) over \(\mathbb Q\). The abelian variety \(A\) is associated to a modular form of weight 2 and level \(Np\) that is obtained by Ribet’s level-raising theorem from the modular form of level \(N\) associated to \(E\). This descent result is Theorem 2 below. To prove the descent theorem, we compare the local conditions defining a certain Selmer group for \(A\) with those defining the \(\ell\)-Selmer group for \(E\). The conditions agree at places of \(K\) prime to \(p\), and at \(p\) the condition changes from the unramified local condition to a transverse condition. The parity lemma proved (5.3) then compares the ranks of the corresponding Selmer groups in terms of the \(\ell\)-divisibility of \(P\) in \(E(K_p)\) and allows us to understand a first descent on \(A/K\) based on Kolyvagin’s determination of the first \(\ell\)-descent on \(E/K\).
For the entire collection see [Zbl 1230.00036].


11G05 Elliptic curves over global fields
11G07 Elliptic curves over local fields
11G10 Abelian varieties of dimension \(> 1\)
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