If $R(E/\Bbb Q_{\infty})$ is the fine Selmer group and $Y(E/\Bbb Q_{\infty})$ is its dual, then we know that $Y(E/\Bbb Q_{\infty})$ is a finitely generated $\Lambda$-module and by a theorem of Kato, it is also torsion. My understanding is that it should thus make sense to want to attach a $p$-adic L-function.

Can we say what this $p$-adic L-function is? Or does it just follow (trivially) from the Iwasawa main conjecture?


You leave out a bit of information, which I will fill in here; I hope correctly.

Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}_{\infty}$ be the cyclotomic $\mathbb{Z}_p$-extension for an odd prime $p$. The fine (also called strict) Selmer group is the kernel of the usual $p$-primary Selmer group to the local cohomology of $E[p^{\infty}]$ at all places above $p$. Kato proves indeed in his theorem 13.4.1 in Astérisque 295 that its dual $Y$ over $\mathbb{Q}_\infty$ is a finitely generated, torsion $\Lambda$-module where $\Lambda$ is the usual Iwasawa algebra for $\mathbb{Q}_\infty/\mathbb{Q}$.

His formulation of the main conjecture 12.10 states that the characteristic series of $Y$ (denoted by $\mathbf{H}^2(T)$ in Kato) should be equal to that of $\mathbf{H}^1(T)/Z(T)$ where, first, $\mathbf{H}^1(T)$ is the projective limits of global Galois cohomology $H^1(G_S(\mathbb{Q}_n), T)$ unramified outside the usual set $S$ of bad places and places above $p$, and secondly $Z(T)$ is its submodule generated by certain zeta elements; and $T$ could be the Tate module of $E$.

Further $\mathbf{H}^1(T)$ is a torsion-free $\Lambda$-module of rank $1$. In fact it is usually free, but in the presence of a $p$-isogeny defined over $\mathbb{Q}$ on $E$ it may be isomorphic to the maximal ideal in $\Lambda$ instead. Let us suppose that it is free. Then there is a particular element $z_{\gamma}$ of Kato's Euler system that generates $Z(T)$. So we can call $z_{\gamma}$ the "$p$-adic $L$-function of $Y$" if we wish. If the reduction is good ordinary, we know that its image under the composition of mapping it into the singular local cohomology $\mathbf{H}^1_s(T)$ and the Coleman map to $\Lambda$ is the usual $p$-adic $L$-function constructed directly from modular symbols. In the supersingular and multiplicative case, we have similar links. Then the main conjecture is equivalent to the usual formulations and known in quite a few cases now.

A long time ago I did some computations of the characteristic series of $Y$ and in particular its leading term. It is conjectured and often known that $\mu=0$ for $Y$. In ther large majority of cases one can show that the leading term is a unit, which implies that it is just $T^{r-1}$ times a unit where $T$ is now the variable in $\Lambda=\mathbb{Z}_p[\![T]\!]$ and $r$ is the rank of $E$ over $\mathbb{Q}$. I found a case where the rank jumps from $2$ to $6$ in the first layer of the $\mathbb{Z}_3$-extension, which gives $T$ times a simple cyclotomic factor. For all I know it could be that the Tate-Shafarevich part of the fine Selmer group is always bounded and so there is never anything more interesting. For rank $0$ curves, this would say that the Euler system $z_{\gamma}$ is a generator of $\mathbf{H}^1(T)$ in a very large number of cases even when the $p$-adic $L$-function is genuinely interesting. (This is all very long ago and my recollection of things is not so precise anymore. Probably there are people out there knowing more about the Euler system that could correct me.)

  • $\begingroup$ Thank you for adding this detail. What had inspired me to ask this question was something a little different though. I was thinking that if one expects that \mu=0 for $Y$, then there might be an analogue of the proof of Sinnott (and Schneps) to at least construct the associated $p$-adic $L$-function. I remember that the proof of Schneps starts with constructing rational functions on certain elliptic curves and at this point I don't know what would be the right analogue, but that is probably because I haven't thought enough. $\endgroup$ – debanjana Jun 19 '18 at 14:31
  • $\begingroup$ @debanjana I am not sure I understand what you mean. For the usual Selmer group you want an analytic function interpolating values of the complex L-function. For the fine Selmer group, whose chracteristic series is a factor of the characteristic series of the usual Selmer group, I fear there is no better analogue than what I described above. But maybe someone will correct me and spell out some modified interpolation property $\endgroup$ – Chris Wuthrich Jun 19 '18 at 22:30

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