Glasnik Matematicki, Vol. 48, No. 2 (2013), 301-312.

BIRATIONAL MAPS OF X(1) INTO P2

Damir Mikoč and Goran Muić

Department of Mathematics, University of Rijeka, Omladinska 14, HR-51000 Rijeka, Croatia
e-mail: damir.mikoc@gmail.com

Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
e-mail: gmuic@math.hr


Abstract.   In this paper we study birational maps of modular curve X(1) attached to SL2(Z) into the projective plain P2. We prove that every curve of genus 0 and degree q in P2 can be uniformized by modular forms for SL2(Z) of weight 12q but not with modular forms of smaller weight, and that the corresponding uniformization can be chosen to be a birational equivalence. We study other regular maps X(1) → P2 and we compute the equation of obtained projective curve. We provide numerical examples in SAGE.

2010 Mathematics Subject Classification.   14H50, 11F11, 11F23.

Key words and phrases.   Modular forms, modular curves, birational equivalence.


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DOI: 10.3336/gm.48.2.06


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