Statistics for anticyclotomic Iwasawa invariants of elliptic curves (Q6562901)

Let \((E,K,p)\) be a triple with an elliptic curve \(E\) defined over a quadratic imaginary field \(K\) and with good reduction at the odd prime \(p\). Let \(K_\infty\) be the anticyclotomic \(\mathbb{Z}_p\)-extension of \(K\) and define \(\mathcal{S}(E/K_\infty)\) to be the Pontrjagin dual of the \(p\)-Selmer group of \(E\) over \(K_\infty\). The group \(\mathcal{S}(E/K_\infty)\) is a \(\Lambda:=\mathbb{Z}_p[[\text{Gal}(K_\infty/K)]]\)-module and, whenever it is finitely generated and torsion, its Iwasawa invariants \(\mu\) and \(\lambda\) provide information on the rank of \(E(K)\). Let \(N_E\) be the conductor of \(E\) and write it as \(N_E^+N_E^-\), where \(N_E^+\) (resp. \(N_E^-\)) is divisible by primes which split (resp. are inert) in \(K\). Several papers have studied the \(\Lambda\) structure of \(\mathcal{S}(E/K_\infty)\) in this setting, in particular for rank 0 curves in the definite case (i.e., when \(N_E^-\) is squarefree and divisible by an odd number of primes) and for rank 1 curves in the indefinite case (with the presence of Heegner points).\N\NIn the definite setting, fixing two elements in the triple \((E,K,p)\) and letting the third one vary the authors provide conditions (on the \(p\)-torsion of \(E\), on the representation associated to the \(p\)-torsion, on the finiteness of the Tate-Shafarevich group, and so on) which yield a torsion module \(\mathcal{S}(E/K_\infty)\) with null \(\mu\) and \(\lambda\) invariants. Many conditions on \(E\) and/or \(K\) and/or \(p\) are derived from known results on the structure of \(\mathcal{S}(E/K_\infty)\) and the ``additional'' ones are tailored in order to allow the authors to compute the Euler characteristic with which they obtain the Iwasawa invariants. Then, they give estimates on the density of elliptic curves or quadratic imaginary fields or primes verifying the conditions. For example: fixing \(E\) and \(p\), they find that the proportion of primes \(\ell\) such that \(\mathcal{S}(E/\mathbb{Q}(\sqrt{-\ell})_\infty)\) is torsion with trivial \(\mu\) invariant is \(\frac{1}{2^{k+1}}\), where \(k\) is the number of primes \(q|N_E\) such that \(q\equiv \pm 1 \pmod{p}\) and the residual representation in \(p\) is unramified at \(q\).\N\NIn the indefinite case, the presence of Heegner points forces the authors to use certain auxiliary Selmer groups whose Pontrjagin dual can actually be torsion and whose \(\lambda\) invariant is an upper bound for the \(\lambda\) invariant of \(\mathcal{S}(E/K_\infty)\). They use Euler characteristic of these auxiliary Selmer groups to find conditions forcing the \(\lambda\) invariant to be 0 (in a somehow restrictive setting, starting with \(N_E^-=1\)). Then, they proceed to estimate again the density of elliptic curves or quadratic imaginary fields or primes verifying those conditions.