Article records
https://feeds.library.caltech.edu/people/Burungale-Ashay-A/article.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 30 Nov 2023 17:52:33 +0000On the non-triviality of the p-adic Abel–Jacobi image of generalised Heegner cycles modulo p, I: Modular curves
https://resolver.caltech.edu/CaltechAUTHORS:20200312-112116706
Authors: Burungale, Ashay A.
Year: 2020
DOI: 10.1090/jag/748
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 let l≠p be 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 evidence for the refined Bloch-Beilinson and the Bloch-Kato conjecture. In the case of weight two and l an ordinary prime, it provides a non-trivial refinement of the results of Cornut and Vatsal on Mazur's conjecture regarding the non-triviality of Heegner points over the Z_l -anticylotomic extension of K. In the case of weight two and l a supersingular prime, it settles Mazur's conjecture earlier known just for l ordinary.https://authors.library.caltech.edu/records/5n08d-y2286Horizontal non-vanishing of Heegner points and toric periods
https://resolver.caltech.edu/CaltechAUTHORS:20200102-101210515
Authors: Burungale, Ashay A.; Tian, Ye
Year: 2020
DOI: 10.1016/j.aim.2019.106938
Let F be a totally real number field and A a modular GL₂-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 GL₂ (AF). 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 non-vanishing 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 [26], [31], [1] on the André–Oort conjecture is accordingly fundamental to the approach.https://authors.library.caltech.edu/records/pby48-75q47p-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin
https://resolver.caltech.edu/CaltechAUTHORS:20191104-080339248
Authors: Burungale, Ashay A.; Tian, Ye
Year: 2020
DOI: 10.1007/s00222-019-00929-7
Let E be a CM elliptic curve over the rationals and p > 3 a good ordinary prime for E. We show that
Corank_(Z_p)Sel_(p^∞)(E/_Q) = 1 ⟹ ord_(s=1)L(s,E/_Q) = 1
for the p^∞-Selmer group Sel_(p^∞)(E/_Q) and the complex L-function L(s,E/_Q). In particular, the Tate–Shafarevich group X(E/_Q) is finite whenever corank_(Z_p)Selp^∞(E/_Q) = 1. We also prove an analogous p-converse for CM abelian varieties arising from weight two elliptic CM modular forms with trivial central character. For non-CM elliptic curves over the rationals, first general results towards such a p-converse theorem are independently due to Skinner (A converse to a theorem of Gross, Zagier and Kolyvagin, arXiv:1405.7294, 2014) and Zhang (Camb J Math 2(2):191–253, 2014).https://authors.library.caltech.edu/records/va3nw-phx39Quantitative non-vanishing of Dirichlet L-values modulo p
https://resolver.caltech.edu/CaltechAUTHORS:20200710-102009367
Authors: Burungale, Ashay; Sun, Hae-Sang
Year: 2020
DOI: 10.1007/s00208-020-02017-1
Let p be an odd prime and k a non-negative integer. Let N be a positive integer such that p∤N and λ a Dirichlet character modulo N. We obtain quantitative lower bounds for the number of Dirichlet character χ modulo F with the critical Dirichlet L-value L(−k,λ_χ) being p-indivisible. Here F→∞ with (N,F)=1 and p∤Fϕ(F). We explore the indivisibility via an algebraic and a homological approach. The latter leads to a bound of the form F^(1/2). The p-indivisibility yields results on the distribution of the associated p-Selmer ranks. We also consider an Iwasawa variant. It leads to an explicit upper bound on the lowest conductor of the characters factoring through the Iwasawa Z_ℓ-extension of Q for an odd prime ℓ≠p with the corresponding critical L-value twists being p-indivisible.https://authors.library.caltech.edu/records/ax54e-sse24On the non-vanishing of p-adic heights on CM abelian varieties, and the arithmetic of Katz p-adic L-functions
https://resolver.caltech.edu/CaltechAUTHORS:20210528-104049998
Authors: Burungale, Ashay A.; Disegni, Daniel
Year: 2021
DOI: 10.5802/aif.3381
Let B be a simple CM abelian variety over a CM field E, p a rational prime. Suppose that B has potentially ordinary reduction above p and is self-dual with root number −1. Under some further conditions, we prove the generic non-vanishing of (cyclotomic) p-adic heights on B along anticyclotomic Zp-extensions of E. This provides evidence towards Schneider's conjecture on the non-vanishing of p-adic heights. For CM elliptic curves over Q, the result was previously known as a consequence of works of Bertrand, Gross–Zagier and Rohrlich in the 1980s. Our proof is based on non-vanishing results for Katz p-adic L-functions and a Gross–Zagier formula relating the latter to families of rational points on B.https://authors.library.caltech.edu/records/68ph6-am867Rubin's conjecture on local units in the anticyclotomic tower at inert primes
https://resolver.caltech.edu/CaltechAUTHORS:20211208-951535000
Authors: Burungale, Ashay; Kobayashi, Shinichi; Ota, Kazuto
Year: 2021
DOI: 10.4007/annals.2021.194.3.8
We prove a fundamental conjecture of Rubin on the structure of local units in the anticyclotomic ℤ_p-extension of the unramified quadratic extension of ℚ_p for p ≥ 5 a prime.https://authors.library.caltech.edu/records/8svwt-jn031A proof of Perrin-Riou's Heegner point main conjecture
https://resolver.caltech.edu/CaltechAUTHORS:20211130-203141758
Authors: Burungale, Ashay; Castella, Francesc; Kim, Chan-Ho
Year: 2021
DOI: 10.2140/ant.2021.15.1627
Let E∕Q be an elliptic curve of conductor N, let p > 3 be a prime where E has good ordinary reduction, and let K be an imaginary quadratic field satisfying the Heegner hypothesis. In 1987, Perrin-Riou formulated an Iwasawa main conjecture for the Tate–Shafarevich group of E over the anticyclotomic Zp-extension of K in terms of Heegner points.
In this paper, we give a proof of Perrin-Riou's conjecture under mild hypotheses. Our proof builds on Howard's theory of bipartite Euler systems and Wei Zhang's work on Kolyvagin's conjecture. In the case when p splits in K, we also obtain a proof of the Iwasawa–Greenberg main conjecture for the p-adic L-functions of Bertolini, Darmon and Prasanna.https://authors.library.caltech.edu/records/zq3sc-8jx23The even parity Goldfeld conjecture: Congruent number elliptic curves
https://resolver.caltech.edu/CaltechAUTHORS:20210714-164422990
Authors: Burungale, Ashay; Tian, Ye
Year: 2022
DOI: 10.1016/j.jnt.2021.05.001
In 1979 Goldfeld conjectured: 50% of the quadratic twists of an elliptic curve defined over the rationals have analytic rank zero. In this expository article we present a few recent developments towards the conjecture, especially its first instance - the congruent number elliptic curves.https://authors.library.caltech.edu/records/nawb8-1r938p∞-Selmer groups and rational points on CM elliptic curves
https://resolver.caltech.edu/CaltechAUTHORS:20220802-744239000
Authors: Burungale, Ashay; Castella, Francesc; Skinner, Christopher; Tian, Ye
Year: 2022
DOI: 10.1007/s40316-022-00203-y
Let E/Q be a CM elliptic curve and p a prime of good ordinary reduction for E. We show that if Sel_(p∞) (E/Q) has Zₚ-corank one, then E(Q) has a point of infinite order. The non-torsion point arises from a Heegner point, and thus ordₛ₌₁ L(E,s) = 1, yielding a p-converse to a theorem of Gross–Zagier, Kolyvagin, and Rubin in the spirit of [49, 54]. For p > 3, this gives a new proof of the main result of [12], which our approach extends to all primes. The approach generalizes to CM elliptic curves over totally real fields [4].https://authors.library.caltech.edu/records/1791g-a9y92A p-adic Waldspurger Formula and the Conjecture of Birch and Swinnerton-Dyer
https://resolver.caltech.edu/CaltechAUTHORS:20221028-653838600.6
Authors: Burungale, Ashay A.
Year: 2022
DOI: 10.1007/s41745-022-00313-0
About a decade ago Bertolini–Darmon–Prasanna proved a p-adic Waldspurger formula, which expresses values of an anticyclotomic p-adic L-function associated to an elliptic curve E/ℚ outside its defining range of interpolation in terms of the p-adic logarithm of Heegner points on E. In the ensuing decade an insight of Skinner based on the p-adic Waldspurger formula has initiated a progress towards the Birch and Swinnerton-Dyer conjecture for elliptic curves over ℚ, especially rank one aspects. In this note we outline some of this recent progress.https://authors.library.caltech.edu/records/cdmzn-kdq67