jvo
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Member for 14 years, 7 months
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Reference Request: Beilinson-Bloch conjecture in terms of Beilinson regulator isomorphism
You might find Peter Schneider's "Introduction to the Beilinson conjectures" helpful, specifically §§ 5-6 (Conjectures I, II, and III). See: "Introduction to the Beilinson conjectures" (with M. Rapoport, N. Schappacher) in Beilinson's Conjecture on Special Values of L-Functions, Perspectives in Math. 4, Academic Press 1988.
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Points $\alpha_n$ of $A$ over the $m_n$-th layer in a $\mathbb{Z}_p$-ext. of $K$, where $A$ is an Ab. var. and $m_n$ is strictly increasing
@ David Loeffler: Agreed! People working in the analytic theory of automorphic forms have completely analogous intuition here concerning vanishing/nonvanishing behaviour. What I am proposing is not a solution to the problem of course, but rather a source of intuition. And while it is rare, I suppose it is not obvious that Heegner-like growth should necessarily be confined to the CM setting in this framework.
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Points $\alpha_n$ of $A$ over the $m_n$-th layer in a $\mathbb{Z}_p$-ext. of $K$, where $A$ is an Ab. var. and $m_n$ is strictly increasing
I think it's still an interesting question to ask, even if the conditions need to be modified (as I agree that the sequence is unlikely to exist in general). Heuristically at least, and subject to various standard/folklore conjectures, one could try to reformulate it (very speculatively) in terms of L-values -- namely central derivative values of self-dual GL(2)*GL(2) Rankin-Selberg L-functions in families with symmetric functional equation(s), where much of the anticyclotomic phenomena is expected (i.e. conjectured) to carry over granted special/generic conditions on the root number(s).
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Good references for K-theory of modular curves?
Beilinson's "Higher regulators and values of L-functions", J. Sov. Math. 30 (1985) is supposed to be a good (general) reference here.
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What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?
@jdh: Too bad! I would offer you a full copy myself if I had one. (I am currently waiting to get full library access at my new university). Also,100 km is quite a trek (!) -- though perhaps only the beginning of it if you do go through Kato's entire argument …
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What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?
@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".
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What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?
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.
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What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?
Well … be bold, and mighty forces will come to your aid :)
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What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?
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.