dc.contributor.author Delbourgo, Daniel en_NZ dc.contributor.author Gilmore, Hamish Julian en_NZ dc.date.accessioned 2019-05-27T00:28:19Z dc.date.available 2019-03-23 en_NZ dc.date.available 2019-05-27T00:28:19Z dc.date.issued 2019 en_NZ dc.identifier.citation Delbourgo, D., & Gilmore, H. (2019). Computing L-Invariants for the Symmetric Square of an Elliptic Curve. Experimental Mathematics. https://doi.org/10.1080/10586458.2018.1490936 en dc.identifier.issn 1058-6458 en_NZ dc.identifier.uri https://hdl.handle.net/10289/12559 dc.description.abstract Let E be an elliptic curve over Q, and p≠2 a prime of good ordinary reduction. The p-adic L-function for Sym²E always vanishes at s = 1, even though the complex L-function does not have a zero there. The L-invariant itself appears on the right-hand side of the formula ddsLp(Sym²E,s)∣∣∣s=1=Lp(Sym²E)×(1−α⁻²ᵖ)(1−pα⁻²ᵖ)×L∞(Sym²E,1)(2πi)⁻¹Ω+EΩ-E where X²−aᵖ(E)X+p=(X−αᵖ)(X−βᵖ) with αp∈Z×ᵖ. We first devise a method to calculate Lp(Sym²E) effectively, then show it is non-trivial for all elliptic curves E of conductor NE≤300 with 4|NE, and almost all ordinary primes p < 17. Hence, in these cases at least, the order of the zero in Lp(Sym²E,s) at s = 1 is exactly one. dc.format.mimetype application/pdf dc.language.iso en en_NZ dc.publisher Taylor & Francis en_NZ dc.rights This is an author’s accepted version of an article published in the journal: Experimental Mathematics. © 2019 Taylor & Francis Group, LLC dc.subject Science & Technology en_NZ dc.subject Physical Sciences en_NZ dc.subject Mathematics en_NZ dc.subject Elliptic curves en_NZ dc.subject Iwasawa theory en_NZ dc.subject L-functions en_NZ dc.subject deformation theory en_NZ dc.subject modular forms en_NZ dc.subject ADIC L-FUNCTIONS en_NZ dc.subject AUTOMORPHIC REPRESENTATIONS en_NZ dc.title Computing L-Invariants for the Symmetric Square of an Elliptic Curve en_NZ dc.type Journal Article dc.identifier.doi 10.1080/10586458.2018.1490936 en_NZ dc.relation.isPartOf Experimental Mathematics en_NZ pubs.elements-id 224903 pubs.publication-status Published en_NZ dc.identifier.eissn 1944-950X en_NZ
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