Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-08-07T10:44:10.939617Z
Paper Citation Record · LEDGER
As of 12 August 2026, this Paper Citation Record lists 47 of 47 outbound references and 0 inbound Pith citation observations for arXiv:2608.06286.
A citation records a reference. It does not transfer a finding from one paper to another.
Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-08-07T10:44:10.939617Z
One-hop event checks from named stored sources.
Source: scholarly_work_events, retraction_status_cache, observed 2026-08-12T06:34:41.77262+00:00
Pith citing papers itemized under the disclosed page cap.
Source: paper_references, paper_reference_links
A source-named dated measurement, never combined with another source.
Source: cited_works
47 of 47 outbound references displayed
External citation measurements
No source-named external measurement is stored.
Observation 4277a7e8-fedf-40ba-9831-3c31d2d6c487 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Integers expressible as the sum of two rational cubes
Reference 1
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 2e2fe861-4a07-4b62-b6d3-fffc6b36d2db · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work
Reference 2
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 7527f668-badf-4f19-a6c7-3754f25ebb2a · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ BrumerThe average rank of elliptic curves I, Invent Math109(1992), 445–472
Reference 3
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation f2df79fb-133b-4e6a-bc0f-d1c1f7decade · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Burungale and Ye Tian,A rank zerop-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin, preprint, arXiv:2506.03465 (2025)
Reference 4
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 6c9c957f-fee4-497d-8082-66d1597dc3b0 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves
Reference 5
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation beee2ca2-f714-4403-997c-3dabf4de7e4d · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work
Reference 6
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation cc8b8317-22d1-48cb-8b91-39b1e5c962ec · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Comeau-Lapointe,One-level density of the family of twists of an elliptic curve over function fields, J
Reference 7
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 3c27f42f-a9ed-48c5-9bec-53d8da7edde9 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Davenport,Multiplicative Number Theory, Second Edition, Springer-Verlag, New York, 1980
Reference 8
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 4cba3018-ac31-4874-b018-16c76c91fb87 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ David, A
Reference 9
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 22051e1d-12e6-4362-b979-cbe3070945b1 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ David, L
Reference 10
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 21e93f2c-91a8-4862-8780-3ad27b2602c3 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ David, A
Reference 11
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 723a40ca-486d-46ca-ad8d-d706c557b854 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Nonvanishing of $L$--functions associated to fixed order characters over function fields
Reference 12
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation eb3bae1d-4e1e-4584-aefc-fb5054670eae · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ David and A
Reference 13
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation ce581e39-7794-4dc0-aa66-34a68dd78c89 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Non-orthogonality of the cubic and quartic large sieves via Rankin-Selberg
Reference 14
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 26d2f9f9-7eaf-4110-89fc-1f97c3fb9d3d · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Diaconu,Mean square values of HeckeL–series formed withr–th order characters, Invent
Reference 15
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation d18fb8b9-9377-412a-9c4d-045dd747ef1c · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters
Reference 16
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 62c59fe5-0ade-48f5-a91d-10b0af5743b6 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Dunn and M
Reference 17
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation e1621626-1c8c-456d-8608-c951c86d4748 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Fiorilli,A conditional determination of the average rank of elliptic curves, J
Reference 18
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 4a9ede83-f529-4d00-81ad-ae1c73553f88 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Fiorilli, J
Reference 19
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation b4c80384-8222-4a28-b8a2-2954ae502e4c · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Gao and L
Reference 20
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation c45664e6-3ea8-40b2-af72-8a7061a18d09 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Gao and L
Reference 21
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation e071c088-e123-4502-a62d-16cc7ea9cec3 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Goldfeld,Conjectures on elliptic curves over quadratic fields, Number theory, Proc
Reference 22
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation a87f2d62-6012-43e0-830c-9bcb44daab9f · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work
Reference 23
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 979508b1-c926-4153-a3ac-5e1f0f577840 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work
Reference 24
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 4126650e-87cf-4242-89ab-e178275eebcc · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work
Reference 25
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 2e889ab3-1e16-41d1-9202-d73d21d090cd · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work
Reference 26
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation de7aa2ec-03d9-45cc-b6dc-3f05ee004330 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Ireland and M
Reference 27
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 5758b3f3-b140-4683-b9ff-a8dc86446912 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Iwaniec and E
Reference 28
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 8ad3b5e7-f993-40aa-8ba3-1562d1eee352 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Iwaniec, W
Reference 29
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 3fa11a0e-ece8-4c71-96b6-70e64b77b23a · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Katz and P
Reference 30
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 775d8e0c-4f9d-4b85-b7a6-479b3f5a52df · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Katz and P
Reference 31
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation e790ee25-9109-4d0f-8676-6662189af79b · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Sums of rational cubes and the $3$-Selmer group
Reference 32
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 650158fb-fc4b-4e32-bf2d-8912a0926096 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Lemmermeyer,Reciprocity laws: From Euler to Eisenstein, Springer Monogr
Reference 33
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 26c33594-b2c6-4554-a143-40bd9317718f · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Meisner and A
Reference 34
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 913a3255-a610-4d1f-a2ad-d6b4a95bff60 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work
Reference 35
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 50706faf-2b61-4dbb-966c-512a9e768bc6 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Onodera,Bound for the sum involving the Jacobi symbol inZ[i], Funct
Reference 36
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation fec25044-b155-414a-8a49-c02368fd36a0 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work
Reference 37
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 40f78295-6095-4f23-8d2f-65b05784b8c8 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work
Reference 38
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation ea2ca14d-d177-4688-b011-fc0f573e8b01 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Phillips,Average analytic ranks of elliptic curves over number fields, Forum Math
Reference 39
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 85e6c0fe-048f-45d7-b389-6d106fd11463 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Sarnak, S
Reference 40
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation a8a172a6-2164-443e-8bb5-1a87910d388f · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Smith,The distribution ofℓ ∞-Selmer groups in degreeℓtwist families
Reference 41
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 7b4c0809-3b9a-47b6-820c-93034de350bc · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Smith,The distribution ofℓ ∞-Selmer groups in degreeℓtwist families
Reference 42
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation f3711460-04ea-4f0b-8f12-170ce133a429 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ The Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture
Reference 43
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 8e31da23-ade4-4968-8f85-133d02e2e10f · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Soundararajan,Nonvanishing of quadratic DirichletL-functions ats= 1 2 , Ann
Reference 44
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 0ef28c0e-b8b4-44ef-8681-188bea13fd59 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Suzuki,Some results on the coefficients of the biquadratic theta series, J
Reference 45
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 16b06a8a-9ace-4043-94ad-ac7c4542a9d0 · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work
Reference 46
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 6b874718-81af-444d-bb5b-d5881f39e98a · outbound
Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work
Reference 47
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
No inbound Pith citation observations are available.