This paper discusses several properties of the Weierstrass-℘ function, as defined on the fundamental parallelogram C/Γ, where C is the complex plane and Γ is the lattice generated by ω1 and ω2. Using the addition formula for ℘(z1 + z2), we develop a reccurence relation for ℘(nz) in terms of ℘(z). We then examine the degree of this expression, some coefficients, and patterns concerning the poles of this function. We also consider the geometric interpretation of taking an arbitrary z0 and adding it to itself, both in the fundamental parallelogram C/Γ and the elliptic curve generated by ℘(z) and ℘′(z).

Faculty Advisor Name

Goong Chen

Faculty Advisor Institution

Texas A&M University

Suggested Mathematics Subject Classification(s)

Primary 54C40, 14E20; Secondary 46E25, 20C20


The main results of this paper were obtained while both authors attended the REU Program at the Mathematics Department, Pennsylvania State University-University Park during Summer 2003. We thank Professor Misha Guysinsky for the helpful guidance and we thank Penn State’s Mathematics Department for the stimulating discussions and generous support.

Included in

Mathematics Commons



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