Siegel modular forms of genus $2$ attached to elliptic curves

Ramakrishnan, Dinakar; Shahidi, Freydoon

doi:10.4310/mrl.2007.v14.n2.a13

Cited by 32 publications

(41 citation statements)

References 46 publications

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“…The precise form of the question is now: Does there exist an automorphic form F := g h on GL 4 /Q, whose L-function equals L * (s, g ⊗ h) after removing the ramified factors? This was finally positive solved by Ramakrishnan [13] (see also the recent results of Ramkrishnan and Shahidi [14]). Since Hecke L-functions are spinor L-functions of elliptic modular forms, it seems to be natural to go one step further and ask about modularity if one exchanges one of the elliptic modular forms in the Ramakrishnan setting by a Siegel Hecke eigenform G of degree 2.…”

Section: Introductionmentioning

confidence: 89%

On the modularity of the GL2-twisted spinor L-function

Heim

2009

Abh. Math. Semin. Univ. Hambg.

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In modern number theory there are famous theorems on the modularity of Dirichlet series attached to geometric or arithmetic objects. There is Hecke's converse theorem, Wiles proof of the Taniyama-Shimura conjecture, and Fermat's Last Theorem to name a few. In this article in the spirit of the Langlands philosophy we raise the question on the modularity of the GL 2 -twisted spinor L-function Z G⊗h (s) related to automorphic forms G, h on the symplectic group GSp 2 and GL 2 . This leads to several promising results and finally culminates into a precise very general conjecture. This gives new insights into the Miyawaki conjecture on spinor L-functions of modular forms. We indicate how this topic is related to Ramakrishnan's work on the modularity of the Rankin-Selberg L-series.

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Section: Introductionmentioning

confidence: 89%

On the modularity of the GL2-twisted spinor L-function

Heim

2009

Abh. Math. Semin. Univ. Hambg.

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“…Recently, F. Shahidi encouraged us to write up a complete proof, which has an important application to the proof of the existence of certain stable Siegel modular forms of weight three on GSp 4 . See [Ramakrishnan and Shahidi 2007] for details.…”

Section: Dihua Jiang and David Soudrymentioning

confidence: 99%

The multiplicity-one theorem for generic automorphic forms of GSp(4)

Jiang¹,

Soudry²

2007

Pacific J. Math.

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“…The new ingredient we bring to the table is the idea to use a functorial transfer of Sym n/2 f to a higher rank group, use Hida theory there, and hope that the additional variables in the Hida family provide non-trivial Galois cohomology classes. In Theorem A, we show that this works for n = 6 using the symmetric cube lift of Ramakrishnan-Shahidi [RS07] (under certain technical assumptions). This provides hope that such a strategy would yield formulas for Greenberg's L-invariant for all symmetric powers in the crystalline case.…”

Section: Introductionmentioning

confidence: 97%

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

Harron¹

2012

Compositio Math.

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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 4, 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) /Q of RamakrishnanShahidi 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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Siegel modular forms of genus $2$ attached to elliptic curves

Cited by 32 publications

References 46 publications

On the modularity of the GL2-twisted spinor L-function

On the modularity of the GL2-twisted spinor L-function

The multiplicity-one theorem for generic automorphic forms of GSp(4)

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

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