Identities between Hecke Eigenforms

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=af²+bfg+g² is finite for all $N$, where $f,g,h$ are Hecke newforms with respect to \Gamma₁(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 f²,g\rangle is nonzero, where $f$ and $g$ are any nonzero cusp eigenforms for SL₂(\mathhbb{Z}) of weight $k$ and $2k$, respectively. As a corollary, we obtain that, assuming Maeda's conjecture, identities between cusp eigenforms for SL₂(\mathbb{Z}) of the form X²+\sum_i=1ⁿ α_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.

Grantor dc:publisher

Temple University. Libraries

Year dc:date.issued

2017

Author dc:creator

• Bao, Dianbin

Advisor dc:contributor.advisor

• Stover, Matthew

Committee members dc:contributor.committeemember

• Stover, Matthew
• Lorenz, Martin, 1951-
• Linowitz, Benjamin
• Dolgushev, Vasily

• Mathematics
• Hecke eigenform
• Maeda's conjecture

Statement 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.

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/20.500.12613/738

OAI identifier oai:identifier

oai:scholarshare.temple.edu:20.500.12613/738