The modular degree, congruence primes, and multiplicity one

Agashe, Amod; Ribet, Kenneth A.; Stein, William

doi:10.1007/978-1-4614-1260-1_2

Cited by 26 publications

(59 citation statements)

References 32 publications

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“…The result in this form follows from [34] and [16]: see [33,Theorem 6.4] for details. Explicit formulas relating L K ( f , χ , 1) and L( f , χ ) can be found in [13] in a special case and in [35] in the greatest generality. 2…”

Section: Consequences Of the Gross-zhang Formulamentioning

confidence: 99%

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On the vanishing of Selmer groups for elliptic curves over ring class fields

Longo

Vigni

2010

Journal of Number Theory

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Let E be a rational elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let H be the ring class field of K of conductor c prime to ND with Galois group G over K. Fix a complex character \chi of G. Our main result is that if the special value of the \chi-twisted L-function of E/K is non-zero then the tensor product (with respect to \chi) of the p-Selmer group of E/H with W over Z[G] is 0 for all but finitely many primes p, where W is a suitable finite extension of Z_p containing the values of \chi. Our work extends results of Bertolini and Darmon to almost all non-ordinary primes p and also offers alternative proofs of a \chi-twisted version of the Birch and Swinnerton-Dyer conjecture for E over H (Bertolini and Darmon) and of the vanishing of the p-Selmer group of E/K for almost all p (Kolyvagin) in the case of analytic rank zero

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Section: Consequences Of the Gross-zhang Formulamentioning

confidence: 99%

“…More precisely, we offer an alternative proof of Theorem 1.4 (Kolyvagin). If L K (E, 1) = 0 then the Mordell-Weil group E(K ) is finite and Sel p (E/K ) = X p (E/K ) = 0 for all but finitely many primes p.…”

Section: Introductionmentioning

confidence: 99%

On the vanishing of Selmer groups for elliptic curves over ring class fields

Longo

Vigni

2010

Journal of Number Theory

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“…If J 0 is an abelian subvariety of J that is preserved by End J, then by Ann TnQ J 0 we mean the kernel of the image of T n Q in End J 0 n Q. Note that End J preserves J g (e.g., see [6], §3) and Ann TnQ J g ¼ Ann TnQ S ½g . If T is a subset of the set of Galois conjugacy classes of newforms of some level dividing N, then let S T ¼ L ½g A T S ½g , I T ¼ Ann T S T , and let J T denote the abelian subvariety of J generated by the J g for all ½g A T. Then J T is isogenous to Q ½g A T J g , hence is preserved by End J and moreover, Ann TnQ J T ¼ Ann TnQ S T .…”

Section: Relating the Factor To An Intersectionmentioning

confidence: 98%

A visible factor of the special L-value

Agashe¹

2010

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Let A be a quotient of J 0 ðNÞ associated to a newform f such that the special L-value of A (at s ¼ 1) is non-zero. We give a formula for the ratio of the special L-value to the real period of A that expresses this ratio as a rational number. We extract an integer factor from the numerator of this formula; this factor is non-trivial in general and is related to certain congruences of f with eigenforms of positive analytic rank. We use the techniques of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime q divides this factor, then q divides either the order of the Shafarevich-Tate group or the order of a component group of A. Suppose p is an odd prime such that p 2 does not divide N, p does not divide the order of the rational torsion subgroup of A, and f is congruent modulo a prime ideal over p to an eigenform whose associated abelian variety has positive Mordell-Weil rank. Then we show that p divides the factor mentioned above; in particular, p divides the numerator of the ratio of the special L-value to the real period of A. Both of these results are as implied by the second part of the Birch and Swinnerton-Dyer conjecture, and thus provide theoretical evidence towards the conjecture.

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“…The congruence number is closely related to deg φ f E , the modular degree of f E , which is the degree of the minimal parametrization φ f E : X 0 (p) → E of the strong Weil elliptic curve E /Q associated with f E (E is isogenous to E but they may not be equal). In general, deg φ f E |D E , and if the conductor of E is prime, we have that deg φ f E = D E (see [1]). …”

Section: Theorem 1 Let E/q Be An Elliptic Curve Of Prime Conductor P mentioning

confidence: 99%

“…The eigenspace with eigenvalue −1 is the orthogonal complement of φ i in the trace zero subspace B 0 of B (since for f ∈ B 0 we have For b ∈ M ij let w 1 ∈ W − and w 2 ∈ W + be such that b = w 1 + w 2 . Then (b) = −w 1 + w 2 ∈ M ij , and 2w 1 …”

Section: Proposition 21 Let L = P Be An Odd Prime Such Thatmentioning

confidence: 99%

Supersingular zeros of divisor polynomials of elliptic curves of prime conductor

Kazalicki

Kohen

2017

Res Math Sci

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For a prime number p, we study the zeros modulo p of divisor polynomials of rational elliptic curves E of conductor p. Ono (CBMS regional conference series in mathematics, 2003, vol 102, p. 118) made the observation that these zeros are often j-invariants of supersingular elliptic curves over F p . We show that these supersingular zeros are in bijection with zeros modulo p of an associated quaternionic modular form v E . This allows us to prove that if the root number of E is −1 then all supersingular j-invariants of elliptic curves defined over F p are zeros of the corresponding divisor polynomial. If the root number is 1, we study the discrepancy between rank 0 and higher rank elliptic curves, as in the latter case the amount of supersingular zeros in F p seems to be larger. In order to partially explain this phenomenon, we conjecture that when E has positive rank the values of the coefficients of v E corresponding to supersingular elliptic curves defined over F p are even. We prove this conjecture in the case when the discriminant of E is positive, and obtain several other results that are of independent interest.

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The modular degree, congruence primes, and multiplicity one

Cited by 26 publications

References 32 publications

On the vanishing of Selmer groups for elliptic curves over ring class fields

On the vanishing of Selmer groups for elliptic curves over ring class fields

A visible factor of the special L-value

Supersingular zeros of divisor polynomials of elliptic curves of prime conductor

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