{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T15:13:01Z","timestamp":1776352381918,"version":"3.51.2"},"reference-count":14,"publisher":"World Scientific Pub Co Pte Ltd","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Data. Sc. Math. Sc."],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:p>As a continuation of our earlier paper [Z. Shi and L. Weng, Murmurations and Sato\u2013Tate conjectures for high rank zetas of elliptic curves, preprint, arXiv:2410.04952], we offer a new approach to murmurations and Sato\u2013Tate laws for higher rank zetas of elliptic curves. Our approach here does not depend on the Riemann hypothesis for the so-called [Formula: see text]-invariant [Formula: see text] in rank [Formula: see text] even for the Sato\u2013Tate law, rather, on a much refined structure, a similar version of which was already observed earlier by Zagier and the senior author of this paper in [L. Weng and D. Zagier, Higher rank zeta functions for elliptic curves, Proc. Natl. Acad. Sci USA 117(9) (2020) 4546\u20134558] when the rank [Formula: see text] Riemann hypothesis was established. Namely, instead of the rank [Formula: see text] Riemann hypothesis bounds [Formula: see text] on which our first paper is based, we use the asymptotic bounds [Formula: see text]. Accordingly, rank [Formula: see text] Sato\u2013Tate law can be established and rank [Formula: see text] murmurations can be formulated equally well, similar to the corresponding structures in the abelian framework for Artin zetas of elliptic curves.<\/jats:p>","DOI":"10.1142\/s2810939225400027","type":"journal-article","created":{"date-parts":[[2025,6,20]],"date-time":"2025-06-20T03:49:58Z","timestamp":1750391398000},"page":"45-61","source":"Crossref","is-referenced-by-count":1,"title":["Murmurations and Sato\u2013Tate conjecture for high rank zetas of elliptic curves II: Beyond Riemann hypothesis"],"prefix":"10.1142","volume":"03","author":[{"given":"Zhan","family":"Shi","sequence":"first","affiliation":[{"name":"Graduate Program of Mathematics for Innovation, Kyushu University, Fukuoka, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lin","family":"Weng","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics, Kyushu University, Fukuoka, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2025,7,17]]},"reference":[{"key":"S2810939225400027BIB001","doi-asserted-by":"publisher","DOI":"10.2977\/prims\/31"},{"key":"S2810939225400027BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/s10240-008-0016-1"},{"key":"S2810939225400027BIB003","doi-asserted-by":"publisher","DOI":"10.1007\/BF01106161"},{"key":"S2810939225400027BIB004","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2010.171.779"},{"key":"S2810939225400027BIB005","unstructured":"Y. H. He, K. H. Lee, T. Oliver and A. Pozdnyakov, Murmurations of elliptic curves, preprint (2022), arXiv:2204.10140."},{"key":"S2810939225400027BIB006","doi-asserted-by":"crossref","unstructured":"Z. Shi and L. Weng, Murmurations and Sato\u2013Tate conjectures for high rank zetas of elliptic curves, preprint (2024), arXiv:2410.04952.","DOI":"10.1142\/S2810939225400027"},{"key":"S2810939225400027BIB007","doi-asserted-by":"publisher","DOI":"10.1007\/s10240-008-0015-2"},{"key":"S2810939225400027BIB008","doi-asserted-by":"publisher","DOI":"10.1353\/ajm.2005.0035"},{"key":"S2810939225400027BIB009","unstructured":"L. Weng, Zeta functions for function fields, preprint (2012), arXiv:1202.3183."},{"key":"S2810939225400027BIB010","unstructured":"L. Weng, Higher rank zeta functions and Riemann hypothesis for elliptic curves, pdf file of the talk slides for the conference on Arithmetic and Algebraic Geometry at Tokyo University (2013), available at author\u2019s personal webpage."},{"key":"S2810939225400027BIB011","unstructured":"L. Weng, Riemann hypothesis for non-abelian zeta functions of curves over finite fields, preprint, arXiv:2201.03703."},{"key":"S2810939225400027BIB012","doi-asserted-by":"publisher","DOI":"10.1142\/10723"},{"key":"S2810939225400027BIB013","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1912023117"},{"key":"S2810939225400027BIB014","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1912501117"}],"container-title":["International Journal of Data Science in the Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S2810939225400027","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T03:58:33Z","timestamp":1776311913000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S2810939225400027"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,17]]},"references-count":14,"journal-issue":{"issue":"02","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["10.1142\/S2810939225400027"],"URL":"https:\/\/doi.org\/10.1142\/s2810939225400027","relation":{},"ISSN":["2810-9392","2810-9406"],"issn-type":[{"value":"2810-9392","type":"print"},{"value":"2810-9406","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,17]]}}}