{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:44:00Z","timestamp":1776833040156,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"252","license":[{"start":{"date-parts":[[2006,4,15]],"date-time":"2006-04-15T00:00:00Z","timestamp":1145059200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We describe numerical calculations which examine the Phillips-Sarnak conjecture concerning the disappearance of cusp forms on a noncompact finite volume Riemann surface\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    under deformation of the surface. Our calculations indicate that if the Teichm\u00fcller space of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is not trivial, then each cusp form has a set of deformations under which either the cusp form remains a cusp form or else it dissolves into a resonance whose constant term is uniformly a factor of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"10 Superscript 8\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mn>10<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>8<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">10^{8}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    smaller than a typical Fourier coefficient of the form. We give explicit examples of those deformations in several cases.\n                  <\/p>","DOI":"10.1090\/s0025-5718-05-01746-1","type":"journal-article","created":{"date-parts":[[2005,8,10]],"date-time":"2005-08-10T10:23:21Z","timestamp":1123669401000},"page":"1967-1982","source":"Crossref","is-referenced-by-count":14,"title":["Deformations of Maass forms"],"prefix":"10.1090","volume":"74","author":[{"given":"D.","family":"Farmer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S.","family":"Lemurell","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2005,4,15]]},"reference":[{"key":"1","unstructured":"[A] H. Avelin, Research Announcement on the deformation of cusp forms, U.U.D.M. Report 2002:26, Uppsala Univ."},{"key":"2","isbn-type":"print","first-page":"85","article-title":"A numerical survey of the reduction of modular curve genus by Fricke\u2019s involutions","author":"Cohn, Harvey","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387976701"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"xiii, 275--286","DOI":"10.5802\/aif.890","article-title":"Pseudo-laplaciens. I","volume":"32","author":"Colin de Verdi\u00e8re, Yves","year":"1982","journal-title":"Ann. Inst. Fourier (Grenoble)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-0956","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"87","DOI":"10.5802\/aif.917","article-title":"Pseudo-laplaciens. II","volume":"33","author":"Colin de Verdi\u00e8re, Yves","year":"1983","journal-title":"Ann. Inst. Fourier (Grenoble)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-0956","issn-type":"print"},{"key":"5","unstructured":"[FJ] D. Farmer and S. Lemurell, in preparation."},{"issue":"469","key":"6","doi-asserted-by":"publisher","first-page":"vi+165","DOI":"10.1090\/memo\/0469","article-title":"Eigenvalues of the Laplacian for Hecke triangle groups","volume":"97","author":"Hejhal, Dennis A.","year":"1992","journal-title":"Mem. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-9266","issn-type":"print"},{"key":"7","isbn-type":"print","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1007\/978-1-4612-1544-8_11","article-title":"On eigenfunctions of the Laplacian for Hecke triangle groups","author":"Hejhal, Dennis A.","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0387988246"},{"issue":"203","key":"8","doi-asserted-by":"publisher","first-page":"245","DOI":"10.2307\/2152951","article-title":"On Fourier coefficients of Maass waveforms for \ud835\udc43\ud835\udc46\ud835\udc3f(2,\ud835\udc19)","volume":"61","author":"Hejhal, D. A.","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"9","series-title":"Biblioteca de la Revista Matem\\'{a}tica Iberoamericana. [Library of the Revista Matem\\'{a}tica Iberoamericana]","volume-title":"Introduction to the spectral theory of automorphic forms","author":"Iwaniec, Henryk","year":"1995"},{"issue":"2","key":"10","doi-asserted-by":"publisher","first-page":"477","DOI":"10.2307\/3062104","article-title":"Nonvanishing of \ud835\udc3f-values and the Weyl law","volume":"154","author":"Luo, Wenzhi","year":"2001","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"1","key":"11","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1215\/S0012-7094-00-10316-X","article-title":"Perturbation of scattering poles for hyperbolic surfaces and central values of \ud835\udc3f-series","volume":"103","author":"Petridis, Yiannis N.","year":"2000","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1007\/BF01388610","article-title":"On cusp forms for co-finite subgroups of \ud835\udc43\ud835\udc46\ud835\udc3f(2,\ud835\udc45)","volume":"80","author":"Phillips, R. S.","year":"1985","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"issue":"6","key":"13","doi-asserted-by":"publisher","first-page":"853","DOI":"10.1002\/cpa.3160380614","article-title":"The Weyl theorem and the deformation of discrete groups","volume":"38","author":"Phillips, R. S.","year":"1985","journal-title":"Comm. Pure Appl. 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II","author":"Sarnak, P.","year":"1990"},{"issue":"4","key":"17","doi-asserted-by":"publisher","first-page":"600","DOI":"10.2969\/jmsj\/02740600","article-title":"A characterization of arithmetic Fuchsian groups","volume":"27","author":"Takeuchi, Kisao","year":"1975","journal-title":"J. Math. Soc. Japan","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5645","issn-type":"print"},{"issue":"2","key":"18","doi-asserted-by":"publisher","first-page":"239","DOI":"10.2307\/2946582","article-title":"Disappearance of cusp forms in special families","volume":"139","author":"Wolpert, Scott A.","year":"1994","journal-title":"Ann. of Math. 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