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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:b85361874d35cd4143351141fd3e98d2cc6ddfc41d56f8da39f7f536de598920

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:fb98b0d763820f557b9dc03e8f87e3ef461714cb2ff4e80c99d18cb8f020054a

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=sha256:18dc4bf29fa27fddc4cdd9fc402c56b187d96d1b674e07ecc248ac7a116ee3a4

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=sha256:1d5ee99c30b6d2d9d78325dd2a6e577b5d645b11d3e6c94429346b5ffdf9b1f1

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=sha256:5962ba2c8df18da52e49408d26d1e6516cb860a7b1876d40eaca506d1fb2ac8d

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:ba0cb03d19273695cf57932f68ccd4dd6c42877c7c7bc79cdf16c7c7f57674c8

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=sha256:7620fd128cbf9d5e7436b70d8353d43950b9953ea12bc3c5b5b44a46148e6787

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:d8b1f7ecfb20022613d9356b8ec4ef4b25be33e113ce8a815e56fd00596ecb21

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=sha256:42ca6f3a9c89597ac4aacff01d9edf5e80ac112e492eb589565bf94930686eb4

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:dad439fb4879c1ca3c6ed8c0d4896d5105c4eadb636c5dbdd9fd26ac9925774a

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=sha256:3f846537a610d50b2c9546fd91b443e82874c75be2132809f453fd1b8d5e908b

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=sha256:545e9e16602f5bb0d171368ae1873117f417fc1e60dfe0a5183b50cbdd701b2e

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=sha256:1fe21472d8dbd1b3e4a689f96779d341aab9fb1373f1e1135d312860de7ecb85

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:fa9ee9a96db91d227d16a9d22d67373b5e72ab8c7837683dcdad815dc5af177a

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=sha256:9d5d1640749e97104bfd705619c72696959b4f4d3a37c48449baf8768ddb2ba4

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:de5c009c68e3b66a3c0e8c169c5f7c6dcf73ac5d6a6fda5a862245da5a348978

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=sha256:37eaf5c3b75ac3961723d5935d36cf02f37d794e1c2ad02d87211f05015f2c1f

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=sha256:589c606c38e48221c8dd50eb1b59bb1e7c7b1ac8415603aa9994021079feef77

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:c2f56f21ed3ecf418fd9fd0496867a1e06120522bbefca77a59dff1385bbd757

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=sha256:22410501ebecfaa03aa37a367079efdb30adcaf62a600db1df4340b349eb38db

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=sha256:249757d35e82da6dc1dc639a0872361277fb793bc09b7253810fb0f591cf0ed1

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=sha256:781c5f3772d5a00b9950e3c08cc35c2a08f4019b2f1d3806049d2dc7c53d907d

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=sha256:4739dde7b0505105573b63092a15787370dc36db148e647dead4d7193eace4c7

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:b773bf05faf5ece31df29545aa77b087f915a343e862c4c6c28c101d5f226940

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:e12fb089694068ddd5d68fda1db64573b7b896b66d5c2f4d6ce184bec2ab9c15

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:dfcb6c4dc69cf0c5c62fc66be0ed9a2697e8bfb86e1f0892c88454eaa5b0bea7

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:f1e9c6582b7335993d9727937ad6527d2c6c6d6037b5859924f07d1a582ad480

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:e21c8ce3be98d3491f5e1f87bb1109f21ddead9e65aa03d637047710aaea9e07

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:e8b668a865195d836f0d2ce566d88b60786f4a25924bae959f605e43e896e078

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=sha256:16de50f3dc120b47019a3421b5fadc0cf0d448b756a9d2717d5a3727b8b1642d

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:bb5c9d62a8d33fee3626e504a2232d4abf53a82de9dec8a8e161f0a7d505cd17

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=sha256:6e9ff505f5cf1ec7117e84e62807020e67ca32a6485e392b433959e4abf31458

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=sha256:73d053f9ea44882a3cd440ba752531a0e543cb409ed3a3eb7037eb1e4e3b4003

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=sha256:6fda0d2890cc8fe04c6ec43dd054e1807a5ca14a01464d08377ba48e90eb4965

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=sha256:220d5defb32993bb1a20ab3f5fb7540a59372cc9f7e7758e90161d420e4169f3

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:d83a7b4b497d2375e241bb0c461d45d93bcdc6f13374414e84b7aec270fe11f9

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=sha256:4f1bb0300554b7180a75f6cfe858483969d974adcdd51135443bdb2b9611babd

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:e032c9a674be4496ea2ab9f77d04ec8bf7607ce1db117855f55f3416e287295f

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=sha256:4c36819aaa016efd89f622d15a1c110746608e08322f7cde77f5d9289672dd34

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=sha256:200c032d3f18650c7666802a2d8dd96c3124908471e07c574030fef749a74499

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=sha256:19d1c1407648e44211c9d6871df324ba535795a964ac8554ce328a0c7c556cfc

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=sha256:2881778fd52bca10ea58d87f5adf7e436461b3898226e869d3ea03ad76479c08

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:a2e7fb849963141d8e67a18c00ed8946e79ea35245bee2a5425e12f5a276dd48

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=sha256:8c9c69d191d33e5609c20c0438fffd172a09cd6a1b9c8337a6946cd586f37f0a

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:ed3317ddb8d94dacb7e1d5ea0f1e54d88c82b950daaebcd6f4f70fb0588e08a5

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=sha256:866d4d5591ccabb8c006b2e33c82e7cfd860779b9a524525e94711ec2dfde5d6

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:d3b45fc227b8545764a4d3407cfa88922d89a48f6efe4553bf557727443f27d3

Pith citing papers

No inbound Pith citation observations are available.