What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?

What is known about the structure of $J_{0}(N)$ over $\mathbb{Q}[\mu_{p^{\infty}}]$?

More generally, what do we know about $J_{0}(N)$ over $\mathbb{Q}[\mu_{p^{\infty}},k^{1/p^{n}}]$, where $k\in\mathbb{Z}$?

Does it have infinite rank? Does it have finite torsion?

I am especially interested in the case where $N$ is the conductor of an elliptic curve with additive reduction over $\mathbb{Q}$ at $p$. (This curve acquires, of course, a Néron model with semistable reduction st $p$ over $\mathbb{Q}[\mu_{p^{\infty}}]$.)

EDIT: In view of the previous comments, I would like to ask whether the $rank({J}_{0}(N))$ is still finite over $$K(k)\colon=\lim_{\stackrel{\rightarrow}{n}}(\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{{\frac{{1}}{{{p}^{n}}}}}])$$

• Much is known over $\mathbf{Q}(\mu_{p^\infty})$. That the torsion is finite follows from work of Serre in the 1970s. It is also true that the rank is finite, but this is a very much deeper result (due to Kato in 2004). – David Loeffler Dec 14 '15 at 16:47
• Adding to David Loeffler's comment: I do not know about the case of additive reduction at p for the the latter (which I believe is open), but for an elliptic curve E having good ordinary reduction at p, you might also find some relevant theorems in Darmon-Tian, "Heegner points over towers of Kummer extensions", Canad. J. Math. 62, No. 5 (2010) 1060-1082 (at least when k is p-power free). See Proposition 1.4 (assuming B+S-D), Theorem 1.8, or Theorem 1.9 for instance. Roughly, the arithmetic of E here exhibits similar behaviour to that of E in the Z_p^2 extension of an imaginary quadratic field. – jvo Dec 14 '15 at 17:01
• @jvo Can't we easily arrange that k is p-power-free WLOG? Did you maybe mean "when k is coprime to p"? – David Loeffler Dec 14 '15 at 18:53
• @David Loeffler Yes, WLOG! Thanks for pointing this out, I was writing in haste without thinking. – jvo Dec 14 '15 at 19:19
• $X_0(49)$ has additive, but potentially good, reduction at $7$, and doesn't acquire a good reduction Neron model over $\mathbb Q(\mu_{7^{\infty}}).$ – tracing Dec 15 '15 at 4:00

Theorem 14.4 of Kato's paper in Asterisque 295 (2004), on page 236, says:

Let $A$ be an abelian variety over $\mathbf{Q}$ such that there is a surjective homomorphism $J_1(N) \to A$ for some integer $N$. Then for any $m \ge 1$, $\bigcup_n A(\mathbf{Q}(\mu_{m^n}))$ is a finitely-generated abelian group.

(I'll leave the case of Kummer extensions to others, but this answers the question in the title.)

• Thanks. I have edited the title now. Will accept your answer as soon as someone answers the Kummer case. – The Thin Whistler Dec 15 '15 at 10:11

The case of Kummer extensions is not yet completely understood, though partial results are obtained by Darmon-Tian in "Heegner points over towers of Kummer Extensions", Canad. J. Math. 62 (5) 2010, 1060 - 1080, following conjectures made in Dokchitser-Dokchitser "Computations in non-commutative Iwasawa theory," Proc. London Math. Soc. 94 (1) 2007, 211-272 (for instance). The results here are a bit tricky to summarize succinctly: There is a "root number dichotomy" which amounts to characterizing the forced vanishing of central values by the functional equation, and this must be considered a priori. The settings of forced vanishing typically correspond to Heegner-like growth of Mordell-Weil rank, as in the setting of anticyclotomic extensions, and this is described nicely in Darmon-Tian (who study a variation coming from varying parametrizations by Shimura curves). Anyhow, one predicts systematic growth of rank in these settings, and boundedness of rank in the others (i.e. when there is not forced vanishing coming from the functional equation). Also, results in this direction (including Kato + Rohrlich) are typically shown by combining some kind of analytic nonvanishing results for central values (such as Rohrlich's "On L-functions of elliptic curves in cyclotomic towers", Inventiones (75) 1984, 409 - 423) with an "Euler system" construction/argument (such as Kato's, as alluded to in David Loeffler's answer).

• Correct, at least as far as I know. The nonvanishing theorems are blind to the reduction type, but the Euler systems and p-adic L-functions constructions (which are typically only available for good ordinary, multiplicative, or special cases of good supersingular reduction) depend greatly upon it. – jvo Dec 15 '15 at 14:32
• So here I stand - poor fool! - once more, \\ no wiser than I was before. \\ (Goethe) – The Thin Whistler Dec 15 '15 at 16:30
• Well … be bold, and mighty forces will come to your aid :) – jvo Dec 15 '15 at 16:39
• Sorry, my bad: I had forgotten how Kato works, and should have looked more carefully before posting ... The results of Darmon-Tian are all stated for good ordinary reduction, though it is not clear to me whether this is strictly necessary for their arguments. These results for Kummer towers are fragmentary in any case, though I imagine more could be done in this direction now. – jvo Dec 15 '15 at 22:57
• @jdh: Thanks for checking this out! There is a copy here at the moment: webusers.imj-prg.fr/~riccardo.brasca/pages/files/K2.pdf. I seem to recall that the deduction requires arguments from Kato's "Euler Systems, Iwasawa Theory, and Selmer Groups" too. Colmez and Scholl have also written expository accounts of this construction/argument: See Colmez, "La conjecture de Birch et Swinnerton-Dyer p-adique" and Scholl, "An introduction to Kato's Euler systems". – jvo Dec 16 '15 at 10:37