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

Prasanna, Kartik

doi:10.4153/cjm-2010-023-2

Cited by 4 publications

(3 citation statements)

References 17 publications

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“…We expect that one can always find such E * when the two necessary conditions (c p (E)'s are odd for p = 2 and a 2 (E) is even) are satisfied, and so we expect that Theorem 1.12 applies to a large positive proportion of elliptic curves E. Showing the existence of such E * amounts to showing that the value of the anticyclotomic p-adic Lfunction at the trivial character is nonvanishing mod p among quadratic twist families for p = 2. This nonvanishing mod p result seems to be more difficult and we do not address it here (but when p 5, see Prasanna [63] and the recent work of Burungale-Hida-Tian [10]).…”

Section: Examplesmentioning

confidence: 99%

Goldfeld’s Conjecture and Congruences Between Heegner Points

Kriz

2019

Forum of Mathematics, Sigma

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Given an elliptic curve $E$ over $\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (respectively 1). We show that this conjecture holds whenever $E$ has a rational 3-isogeny. We also prove the analogous result for the sextic twists of $j$-invariant 0 curves. For a more general elliptic curve $E$, we show that the number of quadratic twists of $E$ up to twisting discriminant $X$ of analytic rank 0 (respectively 1) is $\gg X/\log ^{5/6}X$, improving the current best general bound toward Goldfeld’s conjecture due to Ono–Skinner (respectively Perelli–Pomykala). To prove these results, we establish a congruence formula between $p$-adic logarithms of Heegner points and apply it in the special cases $p=3$ and $p=2$ to construct the desired twists explicitly. As a by-product, we also prove the corresponding $p$-part of the Birch and Swinnerton–Dyer conjecture for these explicit twists.

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Section: Examplesmentioning

confidence: 99%

Goldfeld’s Conjecture and Congruences Between Heegner Points

Kriz

2019

Forum of Mathematics, Sigma

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“…We expect that one can always find such E * when the two necessary conditions (c p (E)'s are odd for p = 2 and a 2 (E) is even) are satisfied, and so we expect that Theorem 1.4 applies to a large positive proportion of elliptic curves E. Showing the existence of such E * amounts to showing that the value of the anticyclotomic p-adic L-function at the trivial character is nonvanishing mod p among quadratic twists families for p = 2. This nonvanishing mod p result seems to be more difficult and we do not address it here (but when p ≥ 5 see Prasanna [Pra10] and the forthcoming work of Burungale-Hida-Tian).…”

Section: (Mod 2)mentioning

confidence: 99%

Congruences between Heegner points and quadratic twists of elliptic curves

Kriz¹,

Li²

2016

Preprint

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We establish a congruence formula between p-adic logarithms of Heegner points for two elliptic curves with the same mod p Galois representation. As a first application, we use the congruence formula when p = 2 to explicitly construct many quadratic twists of analytic rank zero (resp. one) for a wide class of elliptic curves E. We show that the number of twists of E up to twisting discriminant X of analytic rank zero (resp. one) is ≫ X/ log 5/6 X, improving the current best general bound towards Goldfeld's conjecture due to Ono-Skinner (resp. Perelli-Pomykala). We also prove the 2-part of the Birch and Swinnerton-Dyer conjecture for many rank zero and rank one twists of E, which was only recently established for specific CM elliptic curves E. |d|

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“…Then, by Proposition 2.2, is if q divides the order of X(E −D ). Now there is no clear reason for q to divide the order of X(E −D ) for every D. Kolyvagin has asked whether for a given elliptic curve A and a prime q, there is a twist of A such that q does not divide the order of the Shafarevich-Tate group of the twist (see Question A in [Pra08]). We are interested in the same question, but with the added restrictions that the level N is prime, the special L-value of the twist is nonzero, and that (E, −D) satisfies the hypothesis (**).…”

Section: Special L-values Over Qmentioning

confidence: 99%

Squareness in the Special L-Value and Special L-Values of Twists

Agashe

2010

Int. J. Number Theory

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Let N be a prime and let A be a quotient of J 0 (N ) over Q associated to a newform such that the special L-value of A (at s = 1) is non-zero. Suppose that the algebraic part of the special L-value of A is divisible by an odd prime q such that q does not divide the numerator of N −1 12 . Then the Birch and Swinnerton-Dyer conjecture predicts that the q-adic valuations of the algebraic part of the special L-value of A and of the order of the Shafarevich-Tate group are both positive even numbers. Under a certain mod q non-vanishing hypothesis on special L-values of twists of A, we show that the q-adic valuations of the algebraic part of the special L-value of A and of the Birch and Swinnerton-Dyer conjectural order of the Shafarevich-Tate group of A are both positive even numbers. We also give a formula for the algebraic part of the special L-value of A over quadratic imaginary fields K in terms of the free abelian group on isomorphism classes of supersingular elliptic curves in characteristic N (equivalently, over conjugacy classes of maximal orders in the definite quaternion algebra over Q ramified at N and ∞) which shows that this algebraic part is a perfect square up to powers of the prime two and of primes dividing the discriminant of K. Finally, for an optimal elliptic curve E, we give a formula for the special L-value of the twist E −D of E by a negative fundamental discriminant −D, which shows that this special L-value * I would like to modify the title and include the words "and special L-values of twists". I have included the earlier title in case it was used as an identifier for the paper.We also call the ratio L(A/F,1) Ω(A/F ) , the algebraic part of the special L-value of A f over F ; in the contexts where we shall use this, it is known that the ratio is a rational number (and hence an algebraic number).If N is a positive integer, then let X 0 (N ) denote the modular curve over Q associated to Γ 0 (N ), and let J 0 (N ) be its Jacobian. Let T denote the subring of endomorphisms of J 0 (N ) generated by the Hecke operators (usually denoted T ℓ for ℓ ∤ N and U p for p | N ). If f is a newform of weight 2 on Γ 0 (N ), then let I f = Ann T f and let A f denote the quotient abelian variety J 0 (N )/I f J 0 (N ) over Q. We also denote by L(f, s) the L-function associated to f and by L(A f , s) the L-function associated to A f . It is known thatΩ(A f ) is a rational number. Now fix a newform f of weight 2 on Γ 0 (N ) such that L(A f , 1) = 0. Then by [KL89], A f (Q) has rank zero and X(A f ) is finite. Thus the second part of the Birch and Swinnerton-Dyer conjecture becomes:Conjecture 1.2 (Birch and Swinnerton-Dyer).

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

Cited by 4 publications

References 17 publications

Goldfeld’s Conjecture and Congruences Between Heegner Points

Goldfeld’s Conjecture and Congruences Between Heegner Points

Congruences between Heegner points and quadratic twists of elliptic curves

Squareness in the Special L-Value and Special L-Values of Twists

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