2024-03-29T13:54:39Zhttps://www.tdx.cat/oai/requestoai:www.tdx.cat:10803/6741932022-05-09T08:20:58Zcom_10803_183col_10803_185
nam a 5i 4500
p-adic L-functions, p-adic Gross-Zagier formulas and plectic points
[Barcelona] :
Universitat Politècnica de Catalunya,
2022
Accés lliure
http://hdl.handle.net/10803/674193
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Hernández Barrios, Víctor,
autor
Matemàtica aplicada,
degree
1 recurs en línia (105 pàgines)
Tesi
Doctorat
Universitat Politècnica de Catalunya. Facultat de Matemàtiques i Estadística
2022
Universitat Politècnica de Catalunya. Facultat de Matemàtiques i Estadística
Tesis i dissertacions electròniques
Rotger Cerdà, Víctor,
supervisor acadèmic
Molina Blanco, Santiago,
supervisor acadèmic
TDX
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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