Arithmetic properties of the Shimura–Shintani–Waldspurger correspondence

Prasanna, Kartik

doi:10.1007/s00222-008-0169-z

Cited by 17 publications

(20 citation statements)

References 34 publications

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“…(40) In our application, we will take ℓF = d ds |s=0ν s for a character ν : GF → 1 + pZp. (41) This is different from the operator bearing the same name in [40].…”

Section: )mentioning

confidence: 99%

p-adic heights of Heegner points on Shimura curves

Disegni¹

2015

Algebra Number Theory

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Let f be a primitive Hilbert modular form of parallel weight 2 and level N for the totally real field F , and let p be a rational prime coprime to 2N . If f is ordinary at p and E is a CM extension of F of relative discriminant ∆ prime to N p, we give an explicit construction of the p-adic Rankin-Selberg L-function Lp(fE, ·). When the sign of its functional equation is −1, we show, under the assumption that all primes ℘|p are principal ideals of OF which split in OE, that its central derivative is given by the p-adic height of a Heegner point on the abelian variety A associated with f .This p-adic Gross-Zagier formula generalises the result obtained by Perrin-Riou when F = Q and (N, E) satisfies the so-called Heegner condition. We deduce applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjectures for A.

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“…(40) In our application, we will take ℓF = d ds |s=0ν s for a character ν : GF → 1 + pZp. (41) This is different from the operator bearing the same name in [40].…”

Section: )mentioning

confidence: 99%

p-adic heights of Heegner points on Shimura curves

Disegni¹

2015

Algebra Number Theory

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“…Classical Atkin-Lehner via geometry. Let's start by recalling the geometric construction of the Atkin-Lehner operator w , following Conrad [20].…”

Section: Regularity Conditions On the Hecke Algebramentioning

confidence: 99%

Newforms mod 𝑝 in squarefree level with applications to Monsky’s Hecke-stable filtration

Deo

Medvedovsky

2019

Trans. Amer. Math. Soc. Ser. B

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We propose an algebraic definition of the space of-new mod-p modular forms for Γ 0 (N) in the case that is prime to N , which naturally generalizes to a notion of newforms modulo p in squarefree level. We use this notion of newforms to interpret the Hecke algebras on the graded pieces of the space of mod-2 level-3 modular forms described by Paul Monsky. Along the way, we describe a renormalized version of the Atkin-Lehner involution: no longer an involution, it is an automorphism of the algebra of modular forms, even in characteristic p.

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“…} occurs as the set of squares of Fourier coefficients of a modular form h S inπ S . Having made this observation, one can apply the results of [9,10] to compare the p-adic valuations of the forms h S as S varies. In particular we get the following result (Theorem 2.2).…”

Section: On P-adic Properties Of Central L-values Of Quadratic Twistsmentioning

confidence: 99%

“…Also, the author certainly expects that the terms δ ′ S0 in (1.1) can be replaced by δ S0 . The only obstacle to doing so is that one needs to rework some of the computations in [9,10] with less restrictive assumptions at the prime 2.…”

Section: On P-adic Properties Of Central L-values Of Quadratic Twistsmentioning

confidence: 99%

“…First, instead of directly using Waldspurger's result (which is really about ratios of Fourier coefficients), we build on Waldspurger's work by studying in detail the p-integrality of the theta correspondence for the dual pair ( SL 2 , PB × ), where B is an indefinite quaternion algebra over Q. (The reader is referred to the forthcoming articles [9,10] Curve 401 issues.) Secondly, we exploit crucially the structure of the Waldspurger packet on SL 2 corresponding to the automorphic representation π of PGL 2 associated with E, as is explained below.…”

Section: Introductionmentioning

confidence: 99%

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On p-Adic Properties of Central L-Values of Quadratic Twists of an Elliptic Curve

Prasanna¹

2010

Can. j. math.

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We study p-indivisibility of the central values L(1, Ed) of quadratic twists Ed of a semi-stable elliptic curve E of conductor N. A consideration of the conjecture of Birch and Swinnerton-Dyer shows that the set of quadratic discriminants d splits naturally into several families ℱS, indexed by subsets S of the primes dividing N. Let δS = gcdd∈ℱSL(1, Ed)alg, where L(1, Ed)alg denotes the algebraic part of the central L-value, L(1, Ed). Our main theorem relates the p-adic valuations of δS as S varies. As a consequence we present an application to a refined version of a question of Kolyvagin. Finally we explain an intriguing (albeit speculative) relation betweenWaldspurger packets on and congruences of modular forms of integral and half-integral weight. In this context, we formulate a conjecture on congruences of half-integral weight forms and explain its relevance to the problem of p-indivisibility of L-values of quadratic twists.

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Arithmetic properties of the Shimura–Shintani–Waldspurger correspondence

Cited by 17 publications

References 34 publications

p-adic heights of Heegner points on Shimura curves

p-adic heights of Heegner points on Shimura curves

Newforms mod 𝑝 in squarefree level with applications to Monsky’s Hecke-stable filtration

On p-Adic Properties of Central L-Values of Quadratic Twists of an Elliptic Curve

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