國立高雄大學圖資館 |

Language: English

Back

Construction of Anticyclotomic Euler Systems Using Diagonal Cycles.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Construction of Anticyclotomic Euler Systems Using Diagonal Cycles.
Author:	Alonso Rodriguez, Raul.
Published:	Ann Arbor : ProQuest Dissertations & Theses, 2023
Description:	185 p.
Notes:	Source: Dissertations Abstracts International, Volume: 84-12, Section: B.
Notes:	Advisor: Skinner, Christopher.
Contained By:	Dissertations Abstracts International84-12B.
Subject:	Mathematics.
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=30490630
ISBN:	9798379718206

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

Construction of Anticyclotomic Euler Systems Using Diagonal Cycles. - Ann Arbor : ProQuest Dissertations & Theses, 2023 - 185 p.

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

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

This item must not be sold to any third party vendors.

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.

ISBN: 9798379718206Subjects--Topical Terms:

184409
Mathematics.
Subjects--Index Terms:

Euler systems

Construction of Anticyclotomic Euler Systems Using Diagonal Cycles.
LDR:03206nmm a2200385 4500 001 655772
005 20240414211926.5
006 m o d
007 cr#unu||||||||
008 240620s2023 ||||||||||||||||| ||eng d
020 $a 9798379718206
035 $a (MiAaPQ)AAI30490630
035 $a AAI30490630
040 $a MiAaPQ $c MiAaPQ
100 1 $a Alonso Rodriguez, Raul. $3 966849
245 1 0 $a Construction of Anticyclotomic Euler Systems Using Diagonal Cycles.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2023
300 $a 185 p.
500 $a Source: Dissertations Abstracts International, Volume: 84-12, Section: B.
500 $a Advisor: Skinner, Christopher.
502 $a Thesis (Ph.D.)--Princeton University, 2023.
506 $a This item must not be sold to any third party vendors.
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.
590 $a School code: 0181.
650 4 $a Mathematics. $3 184409
650 4 $a Applied mathematics. $3 377601
653 $a Euler systems
653 $a Iwasawa theory
653 $a Euler system
653 $a Quadratic field
653 $a Cohomology
690 $a 0405
690 $a 0364
710 2 $a Princeton University. $b Mathematics. $3 966839
773 0 $t Dissertations Abstracts International $g 84-12B.
790 $a 0181
791 $a Ph.D.
792 $a 2023
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=30490630