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Construction of Anticyclotomic Euler Systems Using Diagonal Cycles.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	Construction of Anticyclotomic Euler Systems Using Diagonal Cycles./
Author:	Alonso Rodriguez, Raul.
Description:	1 online resource (185 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 84-12, Section: B.
Contained By:	Dissertations Abstracts International84-12B.
Subject:	Mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9798379718206

Construction of Anticyclotomic Euler Systems Using Diagonal Cycles.
Alonso Rodriguez, Raul.

Construction of Anticyclotomic Euler Systems Using Diagonal Cycles. - 1 online resource (185 pages)

Source: Dissertations Abstracts International, Volume: 84-12, Section: B.

Thesis (Ph.D.)--Princeton University, 2023.

Includes bibliographical references

In this thesis, we construct a new anticyclotomic Euler system for the four-dimensional Galois representation attached to two modular forms and a Hecke character of an imaginary quadratic field. To state the results more precisely, let g and h be newforms of weights l ≥ m of the same parity and let ψ be a Hecke character of an imaginary quadratic field K of infinity-type (1 − k, 0) for some even integer k ≥ 2. Assume that the product of the characters of g, h and the CM-form attached to ψ is trivial. Let p be a prime which splits in K. We then study the p-adic GK-representation V := Vg ⊗ Vh(ψ−1 )(1 − c), where c = (k + l + m − 2)/2. Combining a geometric construction using modified diagonal cycles in the product of three modular curves with a result of Lei-Loeffler-Zerbes, we obtain cohomology classes over ring class field extensions of K, and we prove that they form a split anticyclotomic Euler system in the sense of Jetchev-Nekovar-Skinner.The bottom Λ-adic class of our Euler system is related to a one-variable specialization of the triple product p-adic L-function constructed by Darmon-Rotger via a reciprocity law proved by Bertolini-Seveso-Venerucci and Darmon-Rotger. This reciprocity law, together with the Euler-system machinery developed by Jetchev- Nekovar-Skinner, allows us to deduce, under some additional hypotheses, different cases of the Bloch-Kato conjecture for the representation V in analytic rank zero and one. As a different application, we also give two equivalent formulations of an Iwasawa-Greenberg Main Conjecture in this setting and prove one divisibility.When h = g∗, i.e., the modular form obtained by conjugating the Fourier coefficients of g, we obtain an Euler system for the three-dimensional GK-representation V' := ad0 (Vg)(ψ −1 )(1 − k/2) ⊂ V and use it to derive similar applications towards the Bloch-Kato conjecture in analytic rank zero and one and towards a divisibility of an Iwasawa-Greenberg Main Conjecture.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9798379718206Subjects--Topical Terms:

527692
Mathematics.
Subjects--Index Terms:

Euler systemsIndex Terms--Genre/Form:

554714
Electronic books.

Construction of Anticyclotomic Euler Systems Using Diagonal Cycles.
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520 $a In this thesis, we construct a new anticyclotomic Euler system for the four-dimensional Galois representation attached to two modular forms and a Hecke character of an imaginary quadratic field. To state the results more precisely, let g and h be newforms of weights l ≥ m of the same parity and let ψ be a Hecke character of an imaginary quadratic field K of infinity-type (1 − k, 0) for some even integer k ≥ 2. Assume that the product of the characters of g, h and the CM-form attached to ψ is trivial. Let p be a prime which splits in K. We then study the p-adic GK-representation V := Vg ⊗ Vh(ψ−1 )(1 − c), where c = (k + l + m − 2)/2. Combining a geometric construction using modified diagonal cycles in the product of three modular curves with a result of Lei-Loeffler-Zerbes, we obtain cohomology classes over ring class field extensions of K, and we prove that they form a split anticyclotomic Euler system in the sense of Jetchev-Nekovar-Skinner.The bottom Λ-adic class of our Euler system is related to a one-variable specialization of the triple product p-adic L-function constructed by Darmon-Rotger via a reciprocity law proved by Bertolini-Seveso-Venerucci and Darmon-Rotger. This reciprocity law, together with the Euler-system machinery developed by Jetchev- Nekovar-Skinner, allows us to deduce, under some additional hypotheses, different cases of the Bloch-Kato conjecture for the representation V in analytic rank zero and one. As a different application, we also give two equivalent formulations of an Iwasawa-Greenberg Main Conjecture in this setting and prove one divisibility.When h = g∗, i.e., the modular form obtained by conjugating the Fourier coefficients of g, we obtain an Euler system for the three-dimensional GK-representation V' := ad0 (Vg)(ψ −1 )(1 − k/2) ⊂ V and use it to derive similar applications towards the Bloch-Kato conjecture in analytic rank zero and one and towards a divisibility of an Iwasawa-Greenberg Main Conjecture.
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653 $a Iwasawa theory
653 $a Euler system
653 $a Quadratic field
653 $a Cohomology
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