The level 13 analogue of the Rogers–Ramanujan continued fraction and its modularity

Yoonjin Lee; Yoon Kyung Park

doi:10.1016/j.jnt.2016.04.009

The level 13 analogue of the Rogers–Ramanujan continued fraction and its modularity

Yoonjin Lee, Yoon Kyung Park

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We prove the modularity of the level 13 analogue r₁₃(τ) of the Rogers–Ramanujan continued fraction. We establish some properties of r₁₃(τ) using the modular function theory. We first prove that r₁₃(τ) is a generator of the function field on Γ₀(13). We then find modular equations of r₁₃(τ) of level n for every positive integer n by using affine models of modular curves; this is an extension of Cooper and Ye's results with levels n=2,3 and 7 to every level n. We further show that the value r₁₃(τ) is an algebraic unit for any τ∈K−Q, where K is an imaginary quadratic field.

Original language	English
Pages (from-to)	306-333
Number of pages	28
Journal	Journal of Number Theory
Volume	168
DOIs	https://doi.org/10.1016/j.jnt.2016.04.009
State	Published - 1 Nov 2016

Bibliographical note

Funding Information:
The authors are supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( 2009-0093827 ), and the first author is also supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) ( 2014-002731 ).

Publisher Copyright:
© 2016 Elsevier Inc.

Keywords

Modular function
Modular unit
Rogers–Ramanujan continued fraction

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10.1016/j.jnt.2016.04.009

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title = "The level 13 analogue of the Rogers–Ramanujan continued fraction and its modularity",

abstract = "We prove the modularity of the level 13 analogue r13(τ) of the Rogers–Ramanujan continued fraction. We establish some properties of r13(τ) using the modular function theory. We first prove that r13(τ) is a generator of the function field on Γ0(13). We then find modular equations of r13(τ) of level n for every positive integer n by using affine models of modular curves; this is an extension of Cooper and Ye's results with levels n=2,3 and 7 to every level n. We further show that the value r13(τ) is an algebraic unit for any τ∈K−Q, where K is an imaginary quadratic field.",

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N2 - We prove the modularity of the level 13 analogue r13(τ) of the Rogers–Ramanujan continued fraction. We establish some properties of r13(τ) using the modular function theory. We first prove that r13(τ) is a generator of the function field on Γ0(13). We then find modular equations of r13(τ) of level n for every positive integer n by using affine models of modular curves; this is an extension of Cooper and Ye's results with levels n=2,3 and 7 to every level n. We further show that the value r13(τ) is an algebraic unit for any τ∈K−Q, where K is an imaginary quadratic field.

AB - We prove the modularity of the level 13 analogue r13(τ) of the Rogers–Ramanujan continued fraction. We establish some properties of r13(τ) using the modular function theory. We first prove that r13(τ) is a generator of the function field on Γ0(13). We then find modular equations of r13(τ) of level n for every positive integer n by using affine models of modular curves; this is an extension of Cooper and Ye's results with levels n=2,3 and 7 to every level n. We further show that the value r13(τ) is an algebraic unit for any τ∈K−Q, where K is an imaginary quadratic field.

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The level 13 analogue of the Rogers–Ramanujan continued fraction and its modularity

Abstract

Bibliographical note

Keywords

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