[
{
"id": "authors:cdmzn-kdq67",
"collection": "authors",
"collection_id": "cdmzn-kdq67",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221028-653838600.6",
"type": "article",
"title": "A p-adic Waldspurger Formula and the Conjecture of Birch and Swinnerton-Dyer",
"author": [
{
"family_name": "Burungale",
"given_name": "Ashay A.",
"orcid": "0000-0002-2469-2115",
"clpid": "Burungale-Ashay-A"
}
],
"abstract": "About a decade ago Bertolini\u2013Darmon\u2013Prasanna proved a p-adic Waldspurger formula, which expresses values of an anticyclotomic p-adic L-function associated to an elliptic curve E/\u211a 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 \u211a, especially rank one aspects. In this note we outline some of this recent progress.",
"doi": "10.1007/s41745-022-00313-0",
"issn": "0970-4140",
"publisher": "Springer Science and Business Media LLC",
"publication": "Journal of the Indian Institute of Science",
"publication_date": "2022-09-15",
"series_number": "3",
"volume": "102",
"issue": "3",
"pages": "885-894"
},
{
"id": "authors:1791g-a9y92",
"collection": "authors",
"collection_id": "1791g-a9y92",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220802-744239000",
"type": "article",
"title": "p\u221e-Selmer groups and rational points on CM elliptic curves",
"author": [
{
"family_name": "Burungale",
"given_name": "Ashay",
"orcid": "0000-0002-2469-2115",
"clpid": "Burungale-Ashay-A"
},
{
"family_name": "Castella",
"given_name": "Francesc",
"orcid": "0000-0002-5532-2387",
"clpid": "Castella-Francesc"
},
{
"family_name": "Skinner",
"given_name": "Christopher",
"clpid": "Skinner-Christopher"
},
{
"family_name": "Tian",
"given_name": "Ye",
"clpid": "Tian-Ye"
}
],
"abstract": "Let E/Q be a CM elliptic curve and p a prime of good ordinary reduction for E. We show that if Sel_(p\u221e) (E/Q) has Z\u209a-corank one, then E(Q) has a point of infinite order. The non-torsion point arises from a Heegner point, and thus ord\u209b\u208c\u2081 L(E,s) = 1, yielding a p-converse to a theorem of Gross\u2013Zagier, 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].",
"doi": "10.1007/s40316-022-00203-y",
"issn": "2195-4755",
"publisher": "Springer",
"publication": "Annales math\u00e9matiques du Qu\u00e9bec",
"publication_date": "2022-08-02"
},
{
"id": "authors:nawb8-1r938",
"collection": "authors",
"collection_id": "nawb8-1r938",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210714-164422990",
"type": "article",
"title": "The even parity Goldfeld conjecture: Congruent number elliptic curves",
"author": [
{
"family_name": "Burungale",
"given_name": "Ashay",
"orcid": "0000-0002-2469-2115",
"clpid": "Burungale-Ashay-A"
},
{
"family_name": "Tian",
"given_name": "Ye",
"clpid": "Tian-Ye"
}
],
"abstract": "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.",
"doi": "10.1016/j.jnt.2021.05.001",
"issn": "0022-314X",
"publisher": "Elsevier",
"publication": "Journal of Number Theory",
"publication_date": "2022-01",
"volume": "230",
"pages": "161-195"
},
{
"id": "authors:zq3sc-8jx23",
"collection": "authors",
"collection_id": "zq3sc-8jx23",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211130-203141758",
"type": "article",
"title": "A proof of Perrin-Riou's Heegner point main conjecture",
"author": [
{
"family_name": "Burungale",
"given_name": "Ashay",
"orcid": "0000-0002-2469-2115",
"clpid": "Burungale-Ashay-A"
},
{
"family_name": "Castella",
"given_name": "Francesc",
"orcid": "0000-0002-5532-2387",
"clpid": "Castella-Francesc"
},
{
"family_name": "Kim",
"given_name": "Chan-Ho",
"orcid": "0000-0002-5932-9391",
"clpid": "Kim-Chan-Ho"
}
],
"abstract": "Let E\u2215Q 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\u2013Shafarevich group of E over the anticyclotomic Zp-extension of K in terms of Heegner points.\nIn 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\u2013Greenberg main conjecture for the p-adic L-functions of Bertolini, Darmon and Prasanna.",
"doi": "10.2140/ant.2021.15.1627",
"issn": "1944-7833",
"publisher": "Mathematical Sciences Publishers",
"publication": "Algebra & Number Theory",
"publication_date": "2021-11-02",
"series_number": "7",
"volume": "15",
"issue": "7",
"pages": "1627-1653"
},
{
"id": "authors:8svwt-jn031",
"collection": "authors",
"collection_id": "8svwt-jn031",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211208-951535000",
"type": "article",
"title": "Rubin's conjecture on local units in the anticyclotomic tower at inert primes",
"author": [
{
"family_name": "Burungale",
"given_name": "Ashay",
"orcid": "0000-0002-2469-2115",
"clpid": "Burungale-Ashay-A"
},
{
"family_name": "Kobayashi",
"given_name": "Shinichi",
"clpid": "Kobayashi-Shinichi"
},
{
"family_name": "Ota",
"given_name": "Kazuto",
"clpid": "Ota-Kazuto"
}
],
"abstract": "We prove a fundamental conjecture of Rubin on the structure of local units in the anticyclotomic \u2124_p-extension of the unramified quadratic extension of \u211a_p for p \u2265 5 a prime.",
"doi": "10.4007/annals.2021.194.3.8",
"issn": "0003-486X",
"publisher": "Annals of Mathematics",
"publication": "Annals of Mathematics",
"publication_date": "2021-11",
"series_number": "3",
"volume": "194",
"issue": "3",
"pages": "943-966"
},
{
"id": "authors:68ph6-am867",
"collection": "authors",
"collection_id": "68ph6-am867",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210528-104049998",
"type": "article",
"title": "On the non-vanishing of p-adic heights on CM abelian varieties, and the arithmetic of Katz p-adic L-functions",
"author": [
{
"family_name": "Burungale",
"given_name": "Ashay A.",
"clpid": "Burungale-Ashay-A"
},
{
"family_name": "Disegni",
"given_name": "Daniel",
"clpid": "Disegni-Daniel"
}
],
"abstract": "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 \u22121. 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\u2013Zagier and Rohrlich in the 1980s. Our proof is based on non-vanishing results for Katz p-adic L-functions and a Gross\u2013Zagier formula relating the latter to families of rational points on B.",
"doi": "10.5802/aif.3381",
"issn": "1777-5310",
"publisher": "Association des Annales de l'Institut Fourier",
"publication": "Annales de l'Institut Fourier",
"publication_date": "2021-04-15",
"series_number": "5",
"volume": "70",
"issue": "5",
"pages": "2077-2101"
},
{
"id": "authors:ax54e-sse24",
"collection": "authors",
"collection_id": "ax54e-sse24",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200710-102009367",
"type": "article",
"title": "Quantitative non-vanishing of Dirichlet L-values modulo p",
"author": [
{
"family_name": "Burungale",
"given_name": "Ashay",
"orcid": "0000-0002-2469-2115",
"clpid": "Burungale-Ashay-A"
},
{
"family_name": "Sun",
"given_name": "Hae-Sang",
"clpid": "Sun-Hae-Sang"
}
],
"abstract": "Let p be an odd prime and k a non-negative integer. Let N be a positive integer such that p\u2224N and \u03bb a Dirichlet character modulo N. We obtain quantitative lower bounds for the number of Dirichlet character \u03c7 modulo F with the critical Dirichlet L-value L(\u2212k,\u03bb_\u03c7) being p-indivisible. Here F\u2192\u221e with (N,F)=1 and p\u2224F\u03d5(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_\u2113-extension of Q for an odd prime \u2113\u2260p with the corresponding critical L-value twists being p-indivisible.",
"doi": "10.1007/s00208-020-02017-1",
"issn": "0025-5831",
"publisher": "Springer",
"publication": "Mathematische Annalen",
"publication_date": "2020-10",
"volume": "378",
"pages": "317-358"
},
{
"id": "authors:va3nw-phx39",
"collection": "authors",
"collection_id": "va3nw-phx39",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191104-080339248",
"type": "article",
"title": "p-converse to a theorem of Gross\u2013Zagier, Kolyvagin and Rubin",
"author": [
{
"family_name": "Burungale",
"given_name": "Ashay A.",
"orcid": "0000-0002-2469-2115",
"clpid": "Burungale-Ashay-A"
},
{
"family_name": "Tian",
"given_name": "Ye",
"clpid": "Tian-Ye"
}
],
"abstract": "Let E be a CM elliptic curve over the rationals and p > 3 a good ordinary prime for E. We show that\nCorank_(Z_p)Sel_(p^\u221e)(E/_Q) = 1 \u27f9 ord_(s=1)L(s,E/_Q) = 1\nfor the p^\u221e-Selmer group Sel_(p^\u221e)(E/_Q) and the complex L-function L(s,E/_Q). In particular, the Tate\u2013Shafarevich group X(E/_Q) is finite whenever corank_(Z_p)Selp^\u221e(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\u2013253, 2014).",
"doi": "10.1007/s00222-019-00929-7",
"issn": "0020-9910",
"publisher": "Springer",
"publication": "Inventiones Mathematicae",
"publication_date": "2020-04",
"volume": "220",
"pages": "211-253"
},
{
"id": "authors:pby48-75q47",
"collection": "authors",
"collection_id": "pby48-75q47",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200102-101210515",
"type": "article",
"title": "Horizontal non-vanishing of Heegner points and toric periods",
"author": [
{
"family_name": "Burungale",
"given_name": "Ashay A.",
"orcid": "0000-0002-2469-2115",
"clpid": "Burungale-Ashay-A"
},
{
"family_name": "Tian",
"given_name": "Ye",
"clpid": "Tian-Ye"
}
],
"abstract": "Let F be a totally real number field and A a modular GL\u2082-type abelian variety over F. Let K/F be a CM quadratic extension. Let \u03c7 be a class group character over K such that the Rankin-Selberg convolution L(s,A,\u03c7) is self-dual with root number \u22121. We show that the number of class group characters \u03c7 with bounded ramification such that L\u2032(1,A,\u03c7)\u22600 increases with the absolute value of the discriminant of K.\nWe also consider a rather general rank zero situation. Let \u03c0 be a cuspidal cohomological automorphic representation over GL\u2082 (AF). Let \u03c7 be a Hecke character over K such that the Rankin\u2013Selberg convolution L(s,\u03c0,\u03c7) is self-dual with root number 1. We show that the number of Hecke characters \u03c7 with fixed \u221e-type and bounded ramification such that L(1/2,\u03c0,\u03c7)\u22600 increases with the absolute value of the discriminant of K.\nThe Gross\u2013Zagier 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 Andr\u00e9\u2013Oort conjecture is accordingly fundamental to the approach.",
"doi": "10.1016/j.aim.2019.106938",
"issn": "0001-8708",
"publisher": "Elsevier",
"publication": "Advances in Mathematics",
"publication_date": "2020-03-04",
"volume": "362",
"pages": "Art. No. 106938"
},
{
"id": "authors:5n08d-y2286",
"collection": "authors",
"collection_id": "5n08d-y2286",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200312-112116706",
"type": "article",
"title": "On the non-triviality of the p-adic Abel\u2013Jacobi image of generalised Heegner cycles modulo p, I: Modular curves",
"author": [
{
"family_name": "Burungale",
"given_name": "Ashay A.",
"orcid": "0000-0002-2469-2115",
"clpid": "Burungale-Ashay-A"
}
],
"abstract": "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\u2260p 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.",
"doi": "10.1090/jag/748",
"issn": "1056-3911",
"publisher": "American Mathematical Society",
"publication": "Journal of Algebraic Geometry",
"publication_date": "2020-01-08",
"series_number": "2",
"volume": "29",
"issue": "2",
"pages": "329-371"
}
]