Elliptic Functions and Transcendence

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Algebraic independence of periods and logarithms of Drinfeld modules (with an appendix by Brian Conrad) (2010, background): “Now the Q-linear relations among the elliptic logarithms or general abelian logarithms of algebraic points have been studied extensively in the past several decades, but algebraic independence of these logarithms is still wide open (see [3, 24]).”
Diagonalization and Rationalization of algebraic Laurent series (2012, background): “• For nonzero algebraic numbers t in the open unit disc, f1(t) is an elliptic integral and πf1(t) is a period in the sense of Kontsevich and Zagier [23] which are both known to be transcendental (see for instance the complete survey [34]).”