On the Vanishing of Twisted<i>L</i>-Functions of Elliptic Curves

David, Chantal; Fearnley, Jack; Kisilevsky, Hershy

doi:10.1080/10586458.2004.10504532

Cited by 37 publications

(49 citation statements)

References 27 publications

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“…As in the Kramarz-Zagier case, the percentage of curves with analytic rank ≥ 2 was in the 20% range but did seem to be going down. Similar computations [DFK04] have also been undertaken for twists by other (complex) Dirichlet characters, which are related to ranks over number fields. Finally, Fermigier [Fer96] investigated specializations of various (about 100) elliptic curves defined over Q(t) and found that typically 10%-20% of the specializations had excess rank that could not be explained simply from parity.…”

Section: Conjecturesmentioning

confidence: 97%

Average ranks of elliptic curves: Tension between data and conjecture

Bektemirov¹,

Mazur

Stein

et al. 2007

Bull. Amer. Math. Soc.

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Abstract. Rational points on elliptic curves are the gems of arithmetic: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are to algebraic geometry. A rational point in just the right context, at one place in the theory, can inhibit and control-thanks to ideas of Kolyvagin-the existence of rational points and other mathematical structures elsewhere. Despite all that we know about these objects, the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today.We have a network of heuristics and conjectures regarding rational points, and we have massive data accumulated to exhibit instances of the phenomena. Generally, we would expect that our data support our conjectures, and if not, we lose faith in our conjectures. But here there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either. We discuss various aspects of this story, including recent heuristics and data that attempt to resolve this mystery. We shall try to convince the reader that, despite seeming discrepancy, data and conjecture are, in fact, in harmony.

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Section: Conjecturesmentioning

confidence: 97%

Average ranks of elliptic curves: Tension between data and conjecture

Bektemirov¹,

Mazur

Stein

et al. 2007

Bull. Amer. Math. Soc.

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“…This is done by using random matrix theory to predict the value distribution of the L-values at the critical point and then the discretization of these values [38,35,25] to calculate a probability of the L-value being zero. See also David, Fearnley and Kisilevsky [12] for a similar use of random matrix theory to predict frequency of vanishing at the critical point amongst families of elliptic curve L-functions twisted by cubic characters.…”

Section: Random Matrix Theory and Number Theorymentioning

confidence: 99%

Derivatives of random matrix characteristic polynomials with applications to elliptic curves

Snaith¹

2005

J. Phys. A: Math. Gen.

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The value distribution of derivatives of characteristic polynomials of matrices from SO(N ) is calculated at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. We consider subsets of matrices from SO(N ) that are constrained to have n eigenvalues equal to 1, and investigate the first non-zero derivative of the characteristic polynomial at that point. The connection between the values of random matrix characteristic polynomials and values of L-functions in families has been well-established. The motivation for this work is the expectation that through this connection with L-functions derived from families of elliptic curves, and using the Birch and Swinnerton-Dyer conjecture to relate values of the L-functions to the rank of elliptic curves, random matrix theory will be useful in probing important questions concerning these ranks.

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“…In a large number of cases, and with high accuracy, the distribution of zeros of automorphic L-functions coincide with the distribution of eigenvalues of random matrices. See [37,85] for numerical investigations and conjectures and see [40,49,50,53,68,82,84] and the references therein for theoretical results.…”

Section: Introductionmentioning

confidence: 99%

Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$ L -functions

2015

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We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let G be a reductive group over a number field F which admits discrete series representations at infinity. Let L G = G Gal(F/F) be the associated L-group and r : L G → GL(d, C) a continuous homomorphism which is irreducible and does not factor through Gal(F/F). The families under consideration consist of discrete automorphic representations of G(A F ) of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato-Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak (Prog Math 70:321-331, 1987) and Serre (J Am Math Soc 10(1):75-102, 1997). As an application we study the distribution of the lowlying zeros of the associated family of L-functions L(s, π, r ), assuming from the principle of functoriality that these L-functions are automorphic. We find that the distribution of the 1-level densities coincides with the distribution of the 1-level densities of eigenvalues of one of the unitary, symplectic and orthogonal ensembles, in accordance with the Katz-Sarnak heuristics. We provide a criterion based on the Frobenius-Schur indicator to determine this symmetry type. If r is not isomorphic to its dual r ∨ then the symmetry type is unitary. Otherwise there is a bilinear form on C d which realizes the isomorphism between r and r ∨ . If the bilinear form is symmetric (resp. alternating) then r is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).

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On the Vanishing of TwistedL-Functions of Elliptic Curves

Cited by 37 publications

References 27 publications

Average ranks of elliptic curves: Tension between data and conjecture

Average ranks of elliptic curves: Tension between data and conjecture

Derivatives of random matrix characteristic polynomials with applications to elliptic curves

Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$ L -functions

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