André–oort Conjecture and Nonvanishing of Central -Values Over Hilbert Class fields

Advances in Mathematics

Tian

2020

Self Cite

Let F be a totally real number field and A a modular GL 2 -type abelian variety over F . Let K/F be a CM quadratic extension. Let χ be a class group character over K such that the Rankin-Selberg convolution L(s, A, χ) is self-dual with root number −1. We show that the number of class group characters χ with bounded ramification such that L ′ (1, A, χ) = 0 increases with the absolute value of the discriminant of K.We also consider a rather general rank zero situation. Let π be a cuspidal cohomological automorphic representation over GL 2 (A F ). Let χ be a Hecke character over K such that the Rankin-Selberg convolution L(s, π, χ) is self-dual with root number 1. We show that the number of Hecke characters χ with fixed ∞-type and bounded ramification such that L(1/2, π, χ) = 0 increases with the absolute value of the discriminant of K.The Gross-Zagier formula and the Waldspurger formula relate the question to horizontal nonvanishing of Heegner points and toric periods, respectively. For both situations, the strategy is geometric relying on the Zariski density of CM points on self-products of a quaternionic Shimura variety. The recent result [29, 34, 1] on the André-Oort conjecture is accordingly fundamental to the approach.

Section: It Induces a Galois Representationmentioning

confidence: 65%

“…We prove the horizontal non-vanishing of toric periods (Theorem 4.8). The approach is based on the Hilbert modular case in [2].…”

Section: Main Resultmentioning

confidence: 99%

“…As indicated earlier, a special case of Theorem 1.7 was proven in [2]. In fact, our motivation partly came from [2].…”

Section: Introductionmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 95%

“…The initial attempt was to seek analogue of the non-vanishing in the special case to the case of root number −1. Our approach is based on the root number +1 case in [2].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Horizontal non-vanishing of Heegner points and toric periods

Advances in Mathematics

Tian

2020

Self Cite

On the non-triviality of the $p$-adic Abel–Jacobi image of generalised Heegner cycles modulo $p$, I: Modular curves

2020

J. Algebraic Geom.

Generalised Heegner cycles are associated with a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension K/Q. Let p be an odd prime split in K/Q and l = p an odd unramified prime. We prove the non-triviality of the p-adic Abel-Jacobi image of generalised Heegner cycles modulo p over the Z l -anticylotomic extension of K. The result is an evidence for the refined Bloch-Beilinson and the Bloch-Kato conjecture. In the case of weight two, it provides a refinement of the results of Cornut and Vatsal on Mazur's conjectures regarding the non-triviality of Heegner points over the Z l -anticylotomic extension of K.

p-rigidity and Iwasawa μ-invariants

Hida

2017

Alg. Number Th.

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Let F be a totally real field with ring of integers O and p be an odd prime unramified in F. Let p be a prime above p. We prove that a mod p Hilbert modular form associated to F is determined by its restriction to the partial Serre-Tate deformation space Gm ⊗ Op (p-rigidity). Let K/F be an imaginary quadratic CM extension such that each prime of F above p splits in K and λ a Hecke character of K. Partly based on p-rigidity, we prove that the µ-invariant of anticyclotomic Katz p-adic L-function of λ equals the µ-invariant of the full anticyclotomic Katz p-adic L-function of λ. An analogue holds for a class of Rankin-Selberg p-adic L-functions. When λ is self-dual with the root number −1, we prove that the µ-invariant of the cyclotomic derivatives of Katz p-adic L-function of λ equals the µ-invariant of the cyclotomic derivatives of Katz p-adic L-function of λ. Based on previous works of authors and Hsieh, we consequently obtain a formula for the µ-invariant of these p-adic L-functions and derivatives, in most of the cases. We also prove a p-version of a conjecture of Gillard, namely the vanishing of the µ-invariant of Katz p-adic L-function of λ.