arXiv 1007.22132010-07-13math.NT

On Greenberg's $L$-invariant of the symmetric sixth power of an ordinary cusp form

Robert Harron

备注：: 20 pages, submitted
发表信息：: Compositio Math. 148 (2012) 1033-1050
DOI：: 10.1112/S0010437X12000176

英文摘要：

We derive a formula for Greenberg's $L$-invariant of Tate twists of the symmetric sixth power of an ordinary non-CM cuspidal newform of weight $\geq4$, under some technical assumptions. This requires a "sufficiently rich" Galois deformation of the symmetric cube which we obtain from the symmetric cube lift to $\GSp(4)_{/\QQ}$ of Ramakrishnan--Shahidi and the Hida theory of this group developed by Tilouine--Urban. The $L$-invariant is expressed in terms of derivatives of Frobenius eigenvalues varying in the Hida family. Our result suggests that one could compute Greenberg's $L$-invariant of all symmetric powers by using appropriate functorial transfers and Hida theory on higher rank groups.

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