{"id":{"repo_id":"temple","oai_identifier":"oai:scholarshare.temple.edu:20.500.12613/738"},"canonical_url":"https://search.dev.ndltd.org/etd/temple/oai:scholarshare.temple.edu:20.500.12613/738","repository":{"repo_id":"temple","name":"Temple University","base_url":"https://scholarshare.temple.edu/server/oai/request"},"display":{"title":"Identities between Hecke Eigenforms","abstract":"In this dissertation, we study solutions to certain low degree polynomials in terms of Hecke eigenforms. We show that the number of solutions to the equation $h=af^2+bfg+g^2$ is finite for all $N$, where $f,g,h$ are Hecke newforms with respect to $\\Gamma_1(N)$ of weight $k>2$ and $a,b\\neq 0$. Using polynomial identities between Hecke eigenforms, we give another proof that the $j$-function is algebraic on zeros of Eisenstein series of weight $12k$. Assuming Maeda's conjecture, we prove that the Petersson inner product $\\langle f^2,g\\rangle$ is nonzero, where $f$ and $g$ are any nonzero cusp eigenforms for $SL_2(\\mathhbb{Z})$ of weight $k$ and $2k$, respectively. As a corollary, we obtain that, assuming Maeda's conjecture, identities between cusp eigenforms for $SL_2(\\mathbb{Z})$ of the form $X^2+\\sum_{i=1}^n \\alpha_iY_i=0$ all are forced by dimension considerations, i.e., a square of an eigenform for the full modular group is unbiased. We show by an example that this property does not hold in general for a congruence subgroup. Finally we attach our Sage code in the appendix.","abstract_html":"In this dissertation, we study solutions to certain low degree polynomials in terms of Hecke eigenforms. We show that the number of solutions to the equation <span class=\"etd-inline-math\">h=af<sup>2</sup>+bfg+g<sup>2</sup></span> is finite for all $N$, where $f,g,h$ are Hecke newforms with respect to <span class=\"etd-inline-math\">\\Gamma<sub>1</sub>(N)</span> of weight $k&gt;2$ and $a,b\\neq 0$. Using polynomial identities between Hecke eigenforms, we give another proof that the $j$-function is algebraic on zeros of Eisenstein series of weight $12k$. Assuming Maeda&#x27;s conjecture, we prove that the Petersson inner product <span class=\"etd-inline-math\">\\langle f<sup>2</sup>,g\\rangle</span> is nonzero, where $f$ and $g$ are any nonzero cusp eigenforms for <span class=\"etd-inline-math\">SL<sub>2</sub>(\\mathhbb{Z})</span> of weight $k$ and $2k$, respectively. As a corollary, we obtain that, assuming Maeda&#x27;s conjecture, identities between cusp eigenforms for <span class=\"etd-inline-math\">SL<sub>2</sub>(\\mathbb{Z})</span> of the form <span class=\"etd-inline-math\">X<sup>2</sup>+\\sum<sub>i=1</sub><sup>n</sup> &alpha;<sub>i</sub>Y<sub>i</sub>=0</span> all are forced by dimension considerations, i.e., a square of an eigenform for the full modular group is unbiased. We show by an example that this property does not hold in general for a congruence subgroup. Finally we attach our Sage code in the appendix.","abstract_has_math":true,"creators":["Bao, Dianbin"],"institution":"Temple University. Libraries","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Stover, Matthew"],"committee_chairs":[],"committee_members":["Stover, Matthew","Lorenz, Martin, 1951-","Linowitz, Benjamin","Dolgushev, Vasily"],"year":2017,"date_issued":"2017","date_published":"2017","updated_at":"2026-07-27T21:19:51Z","subjects":["Mathematics","Hecke eigenform","Maeda's conjecture"],"languages":["eng"],"rights":["IN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available."],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/20.500.12613/738","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Stover, Matthew"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Stover, Matthew","Lorenz, Martin, 1951-","Linowitz, Benjamin","Dolgushev, Vasily"]},{"key":"dc:creator","label":"Author","values":["Bao, Dianbin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2020-10-20T13:33:28Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2020-10-20T13:33:28Z"]},{"key":"dc:date.issued","label":"Date","values":["2017"]},{"key":"dc:publisher","label":"Institution","values":["Temple University. Libraries"]},{"key":"dc:type","label":"Dc Type","values":["Text"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Hecke eigenform","Maeda's conjecture"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["IN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/20.500.12613/738"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this dissertation, we study solutions to certain low degree polynomials in terms of Hecke eigenforms. We show that the number of solutions to the equation $h=af^2+bfg+g^2$ is finite for all $N$, where $f,g,h$ are Hecke newforms with respect to $\\Gamma_1(N)$ of weight $k>2$ and $a,b\\neq 0$. Using polynomial identities between Hecke eigenforms, we give another proof that the $j$-function is algebraic on zeros of Eisenstein series of weight $12k$. Assuming Maeda's conjecture, we prove that the Petersson inner product $\\langle f^2,g\\rangle$ is nonzero, where $f$ and $g$ are any nonzero cusp eigenforms for $SL_2(\\mathhbb{Z})$ of weight $k$ and $2k$, respectively. As a corollary, we obtain that, assuming Maeda's conjecture, identities between cusp eigenforms for $SL_2(\\mathbb{Z})$ of the form $X^2+\\sum_{i=1}^n \\alpha_iY_i=0$ all are forced by dimension considerations, i.e., a square of an eigenform for the full modular group is unbiased. We show by an example that this property does not hold in general for a congruence subgroup. Finally we attach our Sage code in the appendix."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Identities between Hecke Eigenforms"]}]}],"canonical_facts":{"dc:contributor.advisor":["Stover, Matthew"],"dc:contributor.committeemember":["Stover, Matthew","Lorenz, Martin, 1951-","Linowitz, Benjamin","Dolgushev, Vasily"],"dc:creator":["Bao, Dianbin"],"dc:date.accessioned":["2020-10-20T13:33:28Z"],"dc:date.available":["2020-10-20T13:33:28Z"],"dc:date.issued":["2017"],"dc:description.abstract":["In this dissertation, we study solutions to certain low degree polynomials in terms of Hecke eigenforms. We show that the number of solutions to the equation $h=af^2+bfg+g^2$ is finite for all $N$, where $f,g,h$ are Hecke newforms with respect to $\\Gamma_1(N)$ of weight $k>2$ and $a,b\\neq 0$. Using polynomial identities between Hecke eigenforms, we give another proof that the $j$-function is algebraic on zeros of Eisenstein series of weight $12k$. Assuming Maeda's conjecture, we prove that the Petersson inner product $\\langle f^2,g\\rangle$ is nonzero, where $f$ and $g$ are any nonzero cusp eigenforms for $SL_2(\\mathhbb{Z})$ of weight $k$ and $2k$, respectively. As a corollary, we obtain that, assuming Maeda's conjecture, identities between cusp eigenforms for $SL_2(\\mathbb{Z})$ of the form $X^2+\\sum_{i=1}^n \\alpha_iY_i=0$ all are forced by dimension considerations, i.e., a square of an eigenform for the full modular group is unbiased. We show by an example that this property does not hold in general for a congruence subgroup. Finally we attach our Sage code in the appendix."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["http://hdl.handle.net/20.500.12613/738"],"dc:language.iso":["eng"],"dc:publisher":["Temple University. Libraries"],"dc:rights":["IN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available."],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Mathematics","Hecke eigenform","Maeda's conjecture"],"dc:title":["Identities between Hecke Eigenforms"],"dc:type":["Text"]},"updated_at":"2026-07-27T21:19:51Z"}