Minoration de rangs de courbes elliptiques

Templier, Nicolas

doi:10.1016/j.crma.2008.10.016

Cited by 4 publications

(2 citation statements)

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“…Such a polynomial is irreducible precisely when it has no rational roots. We set Templier-Tsimerman [44, Thm 1], generalizing and extending earlier work of Blomer [1] and Templier [43], showed that sums like (1.4) but with ϕ k essentially fixed (i.e., independent of k) exhibit cancellation provided that ϕ k is non-dihedral. One might extrapolate such results to the following conjecture.…”

Section: Resultsmentioning

confidence: 99%

“…Remark 1.10. Holowinsky's approach differs significantly from that of most literature on the shifted convolution problem (split or non-split), see for instance [24, §4.4] and [2,1,43,44]. The cited works seek to achieve power savings estimates for sums like (1.2) and (1.4) by exploiting cancellation coming from the variation of the sign of the Hecke eigenvalues, but with the automorphic form ϕ k held essentially fixed (i.e., independent of k) as the length of the sum increases.…”

Section: Resultsmentioning

confidence: 99%

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Quadratic Hecke sums and mass equidistribution

Nelson

2020

Preprint

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We consider the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke eigenvalues summed over quadratic progressions. Our reduction provides an analogue for the compact case of a criterion established by Luo-Sarnak for the case of the non-compact modular surface. The novelty is that known proofs of such criteria have depended crucially upon Fourier expansions, which are not available in the compact case. Unconditionally, we establish a twisted variant of the Holowinsky-Soundararajan theorem involving restrictions of normalized Hilbert modular forms arising via base change.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Quadratic Hecke sums and mass equidistribution

Nelson

2020

Preprint

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Quadratic Hecke Sums and Mass Equidistribution

Nelson

2021

International Mathematics Research Notices

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We consider the analogue of the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke eigenvalues summed over quadratic progressions. Our reduction provides an analogue for the compact case of a criterion established by Luo–Sarnak for the case of the non-compact modular surface. The novelty is that known proofs of such criteria have depended crucially upon Fourier expansions, which are not available in the compact case. Unconditionally, we establish a twisted variant of the Holowinsky–Soundararajan theorem involving restrictions of normalized Hilbert modular forms arising via base change.

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Sur le rang des courbes elliptiques sur les corps de classes de Hilbert

Templier

2011

Compositio Math.

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Let E/Q be an elliptic curve and let D < 0 be a sufficiently large fundamental discriminant. If E(Q) contains Heegner points of discriminant D, those points generate a subgroup of rank at least |D| δ , where δ > 0 is an absolute constant. This result is compatible with the Birch and Swinnerton-Dyer conjecture. RésuméSoit E/Q une courbe elliptique. Soit D < 0 un discriminant fondamental suffisamment grand. Si E(Q) contient des points de Heegner de discriminant D, ces points engendrent un sous-groupe dont le rang est supérieurà |D| δ , où δ > 0 est une constante absolue. Ce résultat est en accord avec la conjecture de Birch et Swinnerton-Dyer.

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Minoration de rangs de courbes elliptiques

Cited by 4 publications

References 20 publications

Quadratic Hecke sums and mass equidistribution

Quadratic Hecke sums and mass equidistribution

Quadratic Hecke Sums and Mass Equidistribution

Sur le rang des courbes elliptiques sur les corps de classes de Hilbert

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