TY - JOUR
T1 - Algebraic hypergeometric transformations of modular origin
AU - Maier, Robert S.
PY - 2007/8
Y1 - 2007/8
N2 - It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function 2F1 arises from a relation between modular curves, namely the covering of X0(3) by X0(9). In general, when 2 ≤ N ≤ 7, the N-fold cover of X0(N) by X0(N2) gives rise to an algebraic hypergeometric transformation. The N = 2, 3, 4 transformations are arithmetic-geometric mean iterations, but the N = 5, 6, 7 transformations are new. In the final two cases the change of variables is not parametrized by rational functions, since X0(6),X0(7) are of genus 1. Since their quotients X+0 (6),X+0 (7) under the Fricke involution (an Atkin-Lehner involution) are of genus 0, the parametrization is by two-valued algebraic functions. The resulting hypergeometric transformations are closely related to the two-valued modular equations of Fricke and H. Cohn.
AB - It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function 2F1 arises from a relation between modular curves, namely the covering of X0(3) by X0(9). In general, when 2 ≤ N ≤ 7, the N-fold cover of X0(N) by X0(N2) gives rise to an algebraic hypergeometric transformation. The N = 2, 3, 4 transformations are arithmetic-geometric mean iterations, but the N = 5, 6, 7 transformations are new. In the final two cases the change of variables is not parametrized by rational functions, since X0(6),X0(7) are of genus 1. Since their quotients X+0 (6),X+0 (7) under the Fricke involution (an Atkin-Lehner involution) are of genus 0, the parametrization is by two-valued algebraic functions. The resulting hypergeometric transformations are closely related to the two-valued modular equations of Fricke and H. Cohn.
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U2 - 10.1090/S0002-9947-07-04128-1
DO - 10.1090/S0002-9947-07-04128-1
M3 - Article
AN - SCOPUS:67651029530
SN - 0002-9947
VL - 359
SP - 3859
EP - 3885
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 8
ER -