##
**Modular functions and transcendence questions.**
*(English.
Russian original)*
Zbl 0898.11031

Sb. Math. 187, No. 9, 1319-1348 (1996); translation from Mat. Sb. 187, No. 9, 65-96 (1996).

In this important paper, the author proves a fundamental result on the algebraic independence of values of modular forms and functions. Let \(P\), \(Q\), \(R\) denote the normalized Eisenstein series of weights 2, 4, 6, respectively, \(J:= 1728Q^3/(Q^3- R^2)\) the elliptic modular function. Then for each \(q= \exp(2\pi i\tau)\in \mathbb C\), \(0<| q|< 1\), at least three of the numbers
\[
q,\;P(q),\;Q(q),\;R(q)
\]
are algebraically independent over \(\mathbb Q\) (Theorem 1). From this theorem, he derives among others

– a proof of a conjecture of D. Bertrand about the values of \(J(q)\) and its derivatives (Corollary 2),

– the algebraic independence of three of the numbers \(q\), \(J(q)\), \(J'(q)\), \(J''(q)\) if \(\tau\) is not an elliptic fixed point of the modular group (Corollary 1),

– new transcendence results for periods of elliptic curves (Corollary 3),

– and, in particular, the algebraic independence of the elements \[ \{\pi, e^\pi, \Gamma(1/4)\}, \quad \{\pi, e^{\pi\sqrt 3}, \Gamma(1/3)\},\quad \{\pi, e^{\pi\sqrt D}\} \] for any \(D\in\mathbb N\) (Corollaries 5 and 6).

The last consequence is obtained by specializing Theorem 1 to CM points and represents a remarkable progress in the direction of Schanuel’s conjecture. All results are effective (Theorem 2) and can be reduced to the following Theorem 3, which may be written (in a simplified version) as follows: If \(A\) is a nonzero complex polynomial in four variables, of degree \(\leq N\) in each variable, then the function \(A(z, P(z), Q(z), R(z))\) has a zero of order at most \(2\cdot 10^{45}N^4\) in the point \(z= 0\).

The proofs given in the paper are based on classical tools of transcendence such as Siegel’s lemma, Schwarz’ lemma, the use of differential equations and the estimation of heights, but also on classical algebraic geometry giving criteria for algebraic independence. Here one has to mention a series of former contributions of the author and the work of P. Philippon [Publ. Math., Inst. Hautes Etud. Sci. 64, 5–52 (1986; Zbl 0615.10044)]. The application of these techniques to elliptic modular forms and functions was greatly motivated by the recent solution of the Mahler-Manin conjecture on the transcendence of \(J(q)\) for algebraic \(q\) by K. Barré-Sirieix, G. Diaz, F. Gramain and G. Philibert [Invent. Math. 124, 1–9 (1996; Zbl 0853.11059)].

– a proof of a conjecture of D. Bertrand about the values of \(J(q)\) and its derivatives (Corollary 2),

– the algebraic independence of three of the numbers \(q\), \(J(q)\), \(J'(q)\), \(J''(q)\) if \(\tau\) is not an elliptic fixed point of the modular group (Corollary 1),

– new transcendence results for periods of elliptic curves (Corollary 3),

– and, in particular, the algebraic independence of the elements \[ \{\pi, e^\pi, \Gamma(1/4)\}, \quad \{\pi, e^{\pi\sqrt 3}, \Gamma(1/3)\},\quad \{\pi, e^{\pi\sqrt D}\} \] for any \(D\in\mathbb N\) (Corollaries 5 and 6).

The last consequence is obtained by specializing Theorem 1 to CM points and represents a remarkable progress in the direction of Schanuel’s conjecture. All results are effective (Theorem 2) and can be reduced to the following Theorem 3, which may be written (in a simplified version) as follows: If \(A\) is a nonzero complex polynomial in four variables, of degree \(\leq N\) in each variable, then the function \(A(z, P(z), Q(z), R(z))\) has a zero of order at most \(2\cdot 10^{45}N^4\) in the point \(z= 0\).

The proofs given in the paper are based on classical tools of transcendence such as Siegel’s lemma, Schwarz’ lemma, the use of differential equations and the estimation of heights, but also on classical algebraic geometry giving criteria for algebraic independence. Here one has to mention a series of former contributions of the author and the work of P. Philippon [Publ. Math., Inst. Hautes Etud. Sci. 64, 5–52 (1986; Zbl 0615.10044)]. The application of these techniques to elliptic modular forms and functions was greatly motivated by the recent solution of the Mahler-Manin conjecture on the transcendence of \(J(q)\) for algebraic \(q\) by K. Barré-Sirieix, G. Diaz, F. Gramain and G. Philibert [Invent. Math. 124, 1–9 (1996; Zbl 0853.11059)].

Reviewer: Jürgen Wolfart (Frankfurt am Main)

### MSC:

11J89 | Transcendence theory of elliptic and abelian functions |

11J85 | Algebraic independence; Gel’fond’s method |

11J91 | Transcendence theory of other special functions |

11F11 | Holomorphic modular forms of integral weight |

11F03 | Modular and automorphic functions |

### Keywords:

Bertrand conjecture; algebraic independence of values of modular forms; Eisenstein series; elliptic modular function; periods of elliptic curves; Schanuel’s conjecture; elliptic modular forms
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\textit{Yu. V. Nesterenko}, Sb. Math. 187, No. 9, 1319--1348 (1996; Zbl 0898.11031); translation from Mat. Sb. 187, No. 9, 65--96 (1996)

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### Online Encyclopedia of Integer Sequences:

Decimal expansion of exp(Pi).Decimal expansion of Gamma(1/3).