Ax–Schanuel for the $j$-function

@article{Pila2016AxSchanuelFT,
  title={Ax–Schanuel for the \$j\$-function},
  author={J. Pila and Jacob Tsimerman},
  journal={Duke Mathematical Journal},
  year={2016},
  volume={165},
  pages={2587-2605}
}
  • J. Pila, Jacob Tsimerman
  • Published 2016
  • Mathematics
  • Duke Mathematical Journal
  • In this paper we prove a functional transcendence statement for the j-function which is an analogue of the Ax-Schanuel theorem for the exponential function. It asserts, roughly, that atypical algebraic relations among functions and their compositions with the j-function are governed by modular relations. 
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    References

    SHOWING 1-10 OF 30 REFERENCES
    Modular Ax-Lindemann-Weierstrass with Derivatives
    • J. Pila
    • Computer Science, Mathematics
    • Notre Dame J. Formal Log.
    • 2013
    • 21
    • PDF
    Ax-Lindemann for Ag
    • 22
    • PDF
    Exponential Sums Equations and the Schanuel Conjecture
    • 118
    Ax-Lindemann for \mathcal{A}_g
    • 31
    • PDF
    On the transcendence degree of the differential field generated by Siegel modular forms
    • 24
    • PDF
    A Combination of the Conjectures of Mordell-Lang and André-Oort
    • 85
    • PDF
    Heights, Transcendence, and Linear Independence on Commutative Group Varieties
    • 18
    • Highly Influential
    The rational points of a definable set
    • 167
    • PDF