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L-INVARIANTS OF TATE CURVES

In Sections 2 to 4 of this paper we describe a computation expressing the Linvariant of Greenberg in an explicit form for the symmetric n-th power for n ≤ 2 of the Tate module of an elliptic curve over a totally real field F . In the last section, we generalize our computation to the case where n > 2. Our result is conjectural if n > 2 in the sense that if we have the expected structure theorem of the nearly ordinary Galois deformation ring of the symmetric power, it is likely that the conjecture holds. Indeed, after having written this paper, the author found a proof (under mild assumptions) of this implication: Conjecture 5.4 (on the structure of the deformation ring) implies Conjecture 1.3, which will be described in our forthcoming papers [H07a] and [H07b]. Though the discussion in [H07a] and [H07b] of the material treated in Corollary 3.5 and Section 5 of this paper is simpler (and Galois cohomological), we keep our original approach of this paper because (i) it is intriguing to have two essentially different proofs of Corollary 3.5 (one automorphic presented in this paper and the other Galois cohomological given in [H07b] Corollary 1.11) and (ii) our approach in Section 5 to the factor L(m) might indicate its independence of m, though the discussion in Section 5 is less elaborated than the computation in the proof of [H07a] Theorem 1.14. Our conjecture generalizes the conjecture of Mazur-Tate-Teitelbaum [MTT] in the case of n = 1 and F = Q, and our computation generalizes the one by Greenberg-Stevens [GS1] again in the case of n = 1 and F = Q. A principal point of the conjecture is that the L-invariant of a power of a Tate curve is basically independent of n.