p-adic L-functions, p-adic Gross-Zagier formulas and plectic points
In this work we generalize the construction of p-adic anticyclotomic L-functions associated to an elliptic curve E/F and a quadratic extension K/F, by defining a measure µ_f^p attached to K/F and an automorphic form. In the case of parallel 2, the automorphic form is associated with an elliptic curv...
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| Tipo de recurso: | tesis doctoral |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/367334 |
| Acceso en línea: | https://hdl.handle.net/2117/367334 https://dx.doi.org/10.5821/dissertation-2117-367334 |
| Access Level: | acceso abierto |
| Palabra clave: | Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | In this work we generalize the construction of p-adic anticyclotomic L-functions associated to an elliptic curve E/F and a quadratic extension K/F, by defining a measure µ_f^p attached to K/F and an automorphic form. In the case of parallel 2, the automorphic form is associated with an elliptic curve E/F. The first main result is a p-adic Gross-Zagier formula: if E has split multiplicative reduction at p and p does not split at K/F, we compute the first derivative of the p-adic L-function by relating it with the conjugate difference of a Darmon point twisted by a character ¿. The proof uses the reciprocity map provided by class field theory as a natural way to interpret conjugate differences of points in E(Kp) as elements in the augmentation ideal for the aluation at the character ¿. This generalizes a result of Bertolini and Darmon. With a similar argument, after discovering the work of Fornea and ehrmann on plectic points, we prove an exceptional zero formula which relates a higher order derivative of In this work we generalize the construction of p-adic anticyclotomic L-functions associated to an elliptic curve E/F and a quadratic extension K/F, by defining a measure µ_f^p attached to K/F and an automorphic form. In the case of parallel 2, the automorphic form is associated with an elliptic curve E/F. The first main result is a p-adic Gross-Zagier formula: if E has split multiplicative reduction at p and p does not split at K/F, we compute the first derivative of the p-adic L-function by relating it with the conjugate difference of a Darmon point twisted by a character ¿. The proof uses the reciprocity map provided by class field theory as a natural way to interpret conjugate differences of points in E(Kp) as elements in the augmentation ideal for the evaluation at the character ¿. This generalizes a result of Bertolini and Darmon. With a similar argument, after discovering the work of Fornea and Gehrmann on plectic points, we prove an exceptional zero formula which relates a higher order derivative of µ_f^S with plectic points. We find an interpolating measure µ_F^p for µ_f^p attached to an interpolating Hida family F for f. Here µ_F^p can be regarded as a two variable p-adic L-function, which now includes the weight as a variable. Then we define the Hida-Rankin p-adic L-function Lp(f^p, ¿, k) as the restriction of µ_F^p to the weight space. Finally, we prove a formula which relates the weight-leading term of Lp(f^p, ¿, k) with plectic points. In short, the leading term is an explicit constant times Euler factors times the logarithm of the trace of a plectic point. This formula is a generalization of a result of Longo, Kimball and Hu, which has been used to prove the rationality of a Darmon point under some hypotheses. |
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