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Paper Citation Record · LEDGER

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$

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.

pith.paper-citation-record.v1
2608.06286 v1

Coverage vector

measured 47 of 47 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-07T10:44:10.939617Z

measured 47 of 47 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-12T06:34:41.77262+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: cited_works

Reference resolution

47 of 47 outbound references displayed

  • verified exact5
  • verified fuzzy27
  • unresolved15
  • parse uncertain0
  • malformed identifier0
  • metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 4277a7e8-fedf-40ba-9831-3c31d2d6c487 · outbound

This paper cites Integers expressible as the sum of two rational cubes.

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

Resolution
unresolved
no resolver link, observed 2026-08-07T10:44:07.660270Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

source=pdf_text observed=2026-08-07T10:44:07.660270Z digest=sha256:7a6561b31d6e4af1e9eac08d2d012f872d25a877ec7d1ee8adcc721caf29ffba

Observation 2e2fe861-4a07-4b62-b6d3-fffc6b36d2db · outbound

This paper cites an unresolved cited work.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work

Reference 2

Resolution
unresolved
raw_fallback, observed 2026-08-07T10:44:17.811709Z

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.

source=pdf_text observed=2026-08-07T10:44:07.698721Z digest=sha256:6e6cb84c6e1a752c65564137393fe7df1795f94af2922e37de61c09ed6f4a1e8

Observation 7527f668-badf-4f19-a6c7-3754f25ebb2a · outbound

This paper cites BrumerThe average rank of elliptic curves I, Invent Math109(1992), 445–472.

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

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:17.677877Z

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.

source=pdf_text observed=2026-08-07T10:44:07.778165Z digest=sha256:52a1058c1e0dce94fe50c41159bd5b7aa49dc9ff3bb4631622255ba8e27f1eca

Observation f2df79fb-133b-4e6a-bc0f-d1c1f7decade · outbound

This paper cites Burungale and Ye Tian,A rank zerop-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin, preprint, arXiv:2506.03465 (2025).

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

Resolution
verified exact
raw_fallback, observed 2026-08-07T10:44:11.753957Z

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.

source=pdf_text observed=2026-08-07T10:44:07.858347Z digest=sha256:f705ba195c5b212367f4495f70c225c2183988e8475f403a30c9cf5c7d5e1b4a

Observation 6c9c957f-fee4-497d-8082-66d1597dc3b0 · outbound

This paper cites Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves.

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

Resolution
verified exact
local_arxiv, observed 2026-08-07T10:44:11.572026Z

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.

source=pdf_text observed=2026-08-07T10:44:07.921961Z digest=sha256:3e04e33cc6226a9e35f3c9526f4ce0e881afe7085e468b17453af981e60f6e0e

Observation beee2ca2-f714-4403-997c-3dabf4de7e4d · outbound

This paper cites an unresolved cited work.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work

Reference 6

Resolution
unresolved
raw_fallback, observed 2026-08-07T10:44:17.485221Z

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.

source=pdf_text observed=2026-08-07T10:44:08.001232Z digest=sha256:d65d6ae2462a283056ca6072f3eb93275d323b74f90357ffe5570dcf27c1447c

Observation cc8b8317-22d1-48cb-8b91-39b1e5c962ec · outbound

This paper cites Comeau-Lapointe,One-level density of the family of twists of an elliptic curve over function fields, J.

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

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:17.297187Z

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.

source=pdf_text observed=2026-08-07T10:44:08.054790Z digest=sha256:5e621fcc3ea3135ebe1ca52950a4335a50286132d26204258bc3ae6e038f5a40

Observation 3c27f42f-a9ed-48c5-9bec-53d8da7edde9 · outbound

This paper cites Davenport,Multiplicative Number Theory, Second Edition, Springer-Verlag, New York, 1980.

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

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:17.110202Z

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.

source=pdf_text observed=2026-08-07T10:44:08.126976Z digest=sha256:7332469845cdb4426747d4ce99a713a954b8d076adaebe699fa725eccfe36146

Observation 4cba3018-ac31-4874-b018-16c76c91fb87 · outbound

This paper cites David, A.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ David, A

Reference 9

Resolution
unresolved
no resolver link, observed 2026-08-07T10:44:08.190385Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

source=pdf_text observed=2026-08-07T10:44:08.190385Z digest=sha256:f383c5eb9135713ad3cd2071192846f04da6237ea74c26133053b7a5988faa5f

Observation 22051e1d-12e6-4362-b979-cbe3070945b1 · outbound

This paper cites David, L.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ David, L

Reference 10

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:16.924960Z

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.

source=pdf_text observed=2026-08-07T10:44:08.280774Z digest=sha256:267c1e4c5bff4c7679934073d11e50c2e1c10022d20f8d73a93fdad39c3c2932

Observation 21e93f2c-91a8-4862-8780-3ad27b2602c3 · outbound

This paper cites David, A.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ David, A

Reference 11

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:16.738870Z

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.

source=pdf_text observed=2026-08-07T10:44:08.348249Z digest=sha256:95aba2e63f3a4349cdf6b35a7c0f72e11eaecadd6467c2c87a9dc8c85815af03

Observation 723a40ca-486d-46ca-ad8d-d706c557b854 · outbound

This paper cites Nonvanishing of $L$--functions associated to fixed order characters over function fields.

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

Resolution
verified exact
local_arxiv, observed 2026-08-07T10:44:11.368853Z

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.

source=pdf_text observed=2026-08-07T10:44:08.425719Z digest=sha256:ac088068fb6a0844b5bf0e2bbb9b6f99b345ea3a3402ce17d68efe41c33a06b2

Observation eb3bae1d-4e1e-4584-aefc-fb5054670eae · outbound

This paper cites David and A.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ David and A

Reference 13

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:16.616136Z

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.

source=pdf_text observed=2026-08-07T10:44:08.492379Z digest=sha256:cf764281fd5276574080b11a1e88d3331174ad52ceb150572f3ba5d8a301551f

Observation ce581e39-7794-4dc0-aa66-34a68dd78c89 · outbound

This paper cites Non-orthogonality of the cubic and quartic large sieves via Rankin-Selberg.

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

Resolution
verified exact
local_arxiv, observed 2026-08-07T10:44:11.232558Z

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.

source=pdf_text observed=2026-08-07T10:44:08.569362Z digest=sha256:cde933631311146ec635f856c06a4058a4a1c2c8bac81eb868ead0cac09f67be

Observation 26d2f9f9-7eaf-4110-89fc-1f97c3fb9d3d · outbound

This paper cites Diaconu,Mean square values of HeckeL–series formed withr–th order characters, Invent.

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

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:16.484629Z

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.

source=pdf_text observed=2026-08-07T10:44:08.628194Z digest=sha256:711a60b847691ec19b91210b2637dc25b673e8fae171c0c5f0b10c729a49356f

Observation d18fb8b9-9377-412a-9c4d-045dd747ef1c · outbound

This paper cites On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters.

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

Resolution
verified exact
local_arxiv, observed 2026-08-07T10:44:11.077060Z

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.

source=pdf_text observed=2026-08-07T10:44:08.708999Z digest=sha256:f15906a1a06c7f99ba5dc5e2f17d96668e88516db4584b960ce58e1356ee7ae0

Observation 62c59fe5-0ade-48f5-a91d-10b0af5743b6 · outbound

This paper cites Dunn and M.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Dunn and M

Reference 17

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:16.337967Z

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.

source=pdf_text observed=2026-08-07T10:44:08.785156Z digest=sha256:1cd3664724e139d6c51d9fc4d7ce0add20abc144157be654018f68ea0ed2abff

Observation e1621626-1c8c-456d-8608-c951c86d4748 · outbound

This paper cites Fiorilli,A conditional determination of the average rank of elliptic curves, J.

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

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:16.230193Z

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.

source=pdf_text observed=2026-08-07T10:44:08.876881Z digest=sha256:8244727407a0269a52b46a627807bd0c6f1da7fcedc32b588c9467b9100de84e

Observation 4a9ede83-f529-4d00-81ad-ae1c73553f88 · outbound

This paper cites Fiorilli, J.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Fiorilli, J

Reference 19

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:16.110312Z

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.

source=pdf_text observed=2026-08-07T10:44:08.926868Z digest=sha256:cbc8b85e055bf95deda4570e9f563ec2a3d4508e714d00ff22787a2304d3f638

Observation b4c80384-8222-4a28-b8a2-2954ae502e4c · outbound

This paper cites Gao and L.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Gao and L

Reference 20

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:16.005378Z

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.

source=pdf_text observed=2026-08-07T10:44:08.995483Z digest=sha256:2996f1e0378ea13e067fd2ed97af4f49efc5fb99267b844345232b9e2fd128d1

Observation c45664e6-3ea8-40b2-af72-8a7061a18d09 · outbound

This paper cites Gao and L.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Gao and L

Reference 21

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:15.880782Z

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.

source=pdf_text observed=2026-08-07T10:44:09.094593Z digest=sha256:59f6a576f51f630f8dd15bbad1380c4f1541042b535edf543ccdbe1b56f6bd34

Observation e071c088-e123-4502-a62d-16cc7ea9cec3 · outbound

This paper cites Goldfeld,Conjectures on elliptic curves over quadratic fields, Number theory, Proc.

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

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:15.740002Z

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.

source=pdf_text observed=2026-08-07T10:44:09.168016Z digest=sha256:a90b31d044761f19560a4cda8c79b408db0f331a96756cc7e27672526c45e389

Observation a87f2d62-6012-43e0-830c-9bcb44daab9f · outbound

This paper cites an unresolved cited work.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work

Reference 23

Resolution
unresolved
raw_fallback, observed 2026-08-07T10:44:15.616711Z

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.

source=pdf_text observed=2026-08-07T10:44:09.237922Z digest=sha256:0a0621005714a7e16e03be7ed6e6c3857676485282102091a3267c093e437ad6

Observation 979508b1-c926-4153-a3ac-5e1f0f577840 · outbound

This paper cites an unresolved cited work.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work

Reference 24

Resolution
unresolved
raw_fallback, observed 2026-08-07T10:44:15.508155Z

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.

source=pdf_text observed=2026-08-07T10:44:09.286100Z digest=sha256:b2d93217ccb7fb7d217f208ae03e4d10441e7f4124d47b0e75e1e052ef90e14b

Observation 4126650e-87cf-4242-89ab-e178275eebcc · outbound

This paper cites an unresolved cited work.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work

Reference 25

Resolution
unresolved
raw_fallback, observed 2026-08-07T10:44:15.316442Z

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.

source=pdf_text observed=2026-08-07T10:44:09.372550Z digest=sha256:acfe74345bc8c816f6683b8329629cd0dcfd795a0a8f3845a29678c3856dbc87

Observation 2e889ab3-1e16-41d1-9202-d73d21d090cd · outbound

This paper cites an unresolved cited work.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work

Reference 26

Resolution
unresolved
raw_fallback, observed 2026-08-07T10:44:15.134641Z

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.

source=pdf_text observed=2026-08-07T10:44:09.451377Z digest=sha256:9cea0e512eaa840e00867f47b3f3e4d246a077e5e9db650db208fc3287310051

Observation de7aa2ec-03d9-45cc-b6dc-3f05ee004330 · outbound

This paper cites Ireland and M.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Ireland and M

Reference 27

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:14.989754Z

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.

source=pdf_text observed=2026-08-07T10:44:09.538912Z digest=sha256:baf77875e715ec61c3ccf6b63df4f37f917d1f838c3af94f78130aa304ed9b13

Observation 5758b3f3-b140-4683-b9ff-a8dc86446912 · outbound

This paper cites Iwaniec and E.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Iwaniec and E

Reference 28

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:14.825435Z

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.

source=pdf_text observed=2026-08-07T10:44:09.618081Z digest=sha256:033e5b71fd62e7012bf9c8540c0a6a5134bd7429ec02eb5a2eafeb2662912cdc

Observation 8ad3b5e7-f993-40aa-8ba3-1562d1eee352 · outbound

This paper cites Iwaniec, W.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Iwaniec, W

Reference 29

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:14.680983Z

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.

source=pdf_text observed=2026-08-07T10:44:09.679750Z digest=sha256:895dd17d8f8cd199e258a365f430d9d8d017483883b432375d47789a8b102414

Observation 3fa11a0e-ece8-4c71-96b6-70e64b77b23a · outbound

This paper cites Katz and P.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Katz and P

Reference 30

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:14.524328Z

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.

source=pdf_text observed=2026-08-07T10:44:09.737879Z digest=sha256:ed4a040cf8085a72ec7169274971c323a29c42731b10b44e8184638cfc08c60c

Observation 775d8e0c-4f9d-4b85-b7a6-479b3f5a52df · outbound

This paper cites Katz and P.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Katz and P

Reference 31

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:14.379556Z

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.

source=pdf_text observed=2026-08-07T10:44:09.808027Z digest=sha256:e9a1588c7975d8037e22db24f7f8ebbd6b5f1c289b705e4b94d85160259bc7f2

Observation e790ee25-9109-4d0f-8676-6662189af79b · outbound

This paper cites Sums of rational cubes and the $3$-Selmer group.

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

Resolution
unresolved
no resolver link, observed 2026-08-07T10:44:09.887153Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

source=pdf_text observed=2026-08-07T10:44:09.887153Z digest=sha256:8fa7741cbb6a91a27fa382aaa87aa291ea919c2a5f50ca3e90812a23a5791230

Observation 650158fb-fc4b-4e32-bf2d-8912a0926096 · outbound

This paper cites Lemmermeyer,Reciprocity laws: From Euler to Eisenstein, Springer Monogr.

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

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:14.187488Z

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.

source=pdf_text observed=2026-08-07T10:44:09.977057Z digest=sha256:2ccce25d809855579ef4d6c5af35fb72e13e0185181d8ddd017dbb4b79e7ac3f

Observation 26c33594-b2c6-4554-a143-40bd9317718f · outbound

This paper cites Meisner and A.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Meisner and A

Reference 34

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:13.986329Z

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.

source=pdf_text observed=2026-08-07T10:44:10.050614Z digest=sha256:7f7e006ff5b7249408eac40f3be4784441ed0ac8befb203c0ee99a35346b9d4c

Observation 913a3255-a610-4d1f-a2ad-d6b4a95bff60 · outbound

This paper cites an unresolved cited work.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work

Reference 35

Resolution
unresolved
raw_fallback, observed 2026-08-07T10:44:13.817606Z

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.

source=pdf_text observed=2026-08-07T10:44:10.116875Z digest=sha256:a93954cf7464c74b465eaf1d9d831d0bc9c8d37a4dccedab61cf78189c1acfe3

Observation 50706faf-2b61-4dbb-966c-512a9e768bc6 · outbound

This paper cites Onodera,Bound for the sum involving the Jacobi symbol inZ[i], Funct.

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

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:13.635825Z

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.

source=pdf_text observed=2026-08-07T10:44:10.194994Z digest=sha256:bbb8c34b9d9ae9b667e139e9c12954ea72f02a912c519218d048d7c72e356abb

Observation fec25044-b155-414a-8a49-c02368fd36a0 · outbound

This paper cites an unresolved cited work.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work

Reference 37

Resolution
unresolved
raw_fallback, observed 2026-08-07T10:44:13.467939Z

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.

source=pdf_text observed=2026-08-07T10:44:10.261798Z digest=sha256:dc12f29cd55dd7e3bca31ace16fceb8fe010f1b540891bcf1402794ea576fd6f

Observation 40f78295-6095-4f23-8d2f-65b05784b8c8 · outbound

This paper cites an unresolved cited work.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work

Reference 38

Resolution
unresolved
raw_fallback, observed 2026-08-07T10:44:13.276408Z

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.

source=pdf_text observed=2026-08-07T10:44:10.333971Z digest=sha256:ec9339b3bfb8d26f1f58e02c8bf5f53c7d0afd2ae3273353894447d1c227f7ea

Observation ea2ca14d-d177-4688-b011-fc0f573e8b01 · outbound

This paper cites Phillips,Average analytic ranks of elliptic curves over number fields, Forum Math.

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

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:13.083295Z

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.

source=pdf_text observed=2026-08-07T10:44:10.408188Z digest=sha256:afb227df9b6c83858d593b9ed546aac384f22dc4fbcbf16250abab7676fa76a7

Observation 85e6c0fe-048f-45d7-b389-6d106fd11463 · outbound

This paper cites Sarnak, S.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Sarnak, S

Reference 40

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:12.895579Z

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.

source=pdf_text observed=2026-08-07T10:44:10.480718Z digest=sha256:51372aa3df6bcff12400a146ec562820d21b4d40adabc69cee93c80bc36b9454

Observation a8a172a6-2164-443e-8bb5-1a87910d388f · outbound

This paper cites Smith,The distribution ofℓ ∞-Selmer groups in degreeℓtwist families.

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

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:12.746885Z

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.

source=pdf_text observed=2026-08-07T10:44:10.537906Z digest=sha256:4316dbe9ec99f7687e374a44cef9cf742daddd70460cd6c3811d9c3ef140572d

Observation 7b4c0809-3b9a-47b6-820c-93034de350bc · outbound

This paper cites Smith,The distribution ofℓ ∞-Selmer groups in degreeℓtwist families.

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

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:12.565052Z

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.

source=pdf_text observed=2026-08-07T10:44:10.603758Z digest=sha256:c255b37ee16a8266ce32be4f5e5770a7c7111e4d4f23c64c0b289e79f7509bc6

Observation f3711460-04ea-4f0b-8f12-170ce133a429 · outbound

This paper cites The Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture.

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

Resolution
unresolved
no resolver link, observed 2026-08-07T10:44:10.661681Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

source=pdf_text observed=2026-08-07T10:44:10.661681Z digest=sha256:74234b5dff503681e1b227ea9ea04a11fd10dc8f91812355409411d1ff5cb913

Observation 8e31da23-ade4-4968-8f85-133d02e2e10f · outbound

This paper cites Soundararajan,Nonvanishing of quadratic DirichletL-functions ats= 1 2 , Ann.

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

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:12.400372Z

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.

source=pdf_text observed=2026-08-07T10:44:10.723573Z digest=sha256:33c5835ea4727581d1342b88359a465fc9065f93e9d3daa2e11920ccef8deb8a

Observation 0ef28c0e-b8b4-44ef-8681-188bea13fd59 · outbound

This paper cites Suzuki,Some results on the coefficients of the biquadratic theta series, J.

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

Resolution
verified fuzzy
raw_fallback, observed 2026-08-07T10:44:12.191120Z

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.

source=pdf_text observed=2026-08-07T10:44:10.787136Z digest=sha256:3c079fb4a96b6adf71a042fcf34f7b287556b422ae041638531e26b6013048b8

Observation 16b06a8a-9ace-4043-94ad-ac7c4542a9d0 · outbound

This paper cites an unresolved cited work.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work

Reference 46

Resolution
unresolved
raw_fallback, observed 2026-08-07T10:44:12.040047Z

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.

source=pdf_text observed=2026-08-07T10:44:10.865680Z digest=sha256:506607a9f9cfab0720b24a0727b171fd713546ceb9e47fc63ee7b6fc6c47cb85

Observation 6b874718-81af-444d-bb5b-d5881f39e98a · outbound

This paper cites an unresolved cited work.

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$ Unresolved cited work

Reference 47

Resolution
unresolved
raw_fallback, observed 2026-08-07T10:44:11.885407Z

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.

source=pdf_text observed=2026-08-07T10:44:10.939617Z digest=sha256:301e94f41343894d15efd0c8fbbea89e814f9ab72f6abe74fb1d226d91534a5f

Pith citing papers

No inbound Pith citation observations are available.