STARK POINTS AND -ADIC ITERATED INTEGRALS ATTACHED TO MODULAR FORMS OF WEIGHT ONE
| Autor: | Henri Darmon, Alan G. B. Lauder, Victor Rotger |
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| Přispěvatelé: | Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. TN - Grup de Recerca en Teoria de Nombres |
| Rok vydání: | 2015 |
| Předmět: |
Statistics and Probability
Ring (mathematics) Pure mathematics Grups discontinus Algebra and Number Theory Mathematics::Number Theory Modular form Galois group Formal group Algebraic number field Dihedral group Geometria algèbrica--Aritmètica Elliptic curve 11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry) [Classificació AMS] 11 Number theory::11F Discontinuous groups and automorphic forms [Classificació AMS] Discrete Mathematics and Combinatorics Quadratic field Arithmetical algebraic geometry Geometry and Topology Matemàtiques i estadística::Àlgebra::Teoria de nombres [Àrees temàtiques de la UPC] Discontinuous groups Mathematical Physics Analysis Mathematics |
| Zdroj: | Recercat. Dipósit de la Recerca de Catalunya instname UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) |
| ISSN: | 2050-5086 |
| Popis: | Let$E$be an elliptic curve over$\mathbb{Q}$, and let${\it\varrho}_{\flat }$and${\it\varrho}_{\sharp }$be odd two-dimensional Artin representations for which${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$is self-dual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms$f$,$g$, and$h$of respective weights two, one, and one, giving rise to$E$,${\it\varrho}_{\flat }$, and${\it\varrho}_{\sharp }$via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain$p$-adic iterated integralsattached to the triple$(f,g,h)$, which are$p$-adic avatars of the leading term of the Hasse–Weil–Artin$L$-series$L(E,{\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp },s)$when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on$E$—referred to asStark points—which are defined over the number field cut out by${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$. This formula can be viewed as an elliptic curve analogue of Stark’s conjecture on units attached to weight-one forms. It is proved when$g$and$h$are binary theta series attached to a common imaginary quadratic field in which$p$splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing$p$-adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell–Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark–Heegner points attached to Shintani-type cycles on${\mathcal{H}}_{p}\times {\mathcal{H}}$), and extensions of$\mathbb{Q}$with Galois group a central extension of the dihedral group$D_{2n}$or of one of the exceptional subgroups$A_{4}$,$S_{4}$, and$A_{5}$of$\mathbf{PGL}_{2}(\mathbb{C})$. |
| Databáze: | OpenAIRE |
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