Relevant Background
It would be impossible to give an exhaustive list of all the relevant literature (that would amount to almost all of maths), but below I give a list of articles that, for various reasons, I keep going back to.
Existential Closedness
For complex exponentiation
A remark on Zilber's pseudo-exponentiation, Dave Marker
J. Symbolic Logic 2006Polynomial exponential equations and Zilber's conjecture, Vincenzo Mantova (with an appendix by Vincenzo Mantova and Umberto Zannier)
Bull. Lond. Math. Soc. 2016Zero estimates with moving targets, W. D. Brownawell and David Masser
J. Lond. Math. Soc. 2017Generic solutions of equations with iterated exponentials, Paola D'Aquino, Antongiulio Fornasiero and Giuseppina Terzo
Trans. Amer. Math. Soc. 2018Blurred complex exponentiation, Jonathan Kirby
Selecta Math. 2019A weak version of the strong exponential closure, Paola D'Aquino, Antongiulio Fornasiero and Giuseppina Terzo
Israel J. Math. 2021Exponential sum equations and tropical geometry, Francesco Gallinaro
Selecta Math. (N.S.) 2023Polynomial exponential equations — some new cases of solvability, Vincenzo Mantova and David Masser
preprint 2023Quasiminimality of complex powers, Francesco Gallinaro and Jonathan Kirby
preprint 2023
For exponential maps of (semi) abelian varieties
Solving systems of equations of raising-to-powers type, Francesco Gallinaro
preprint 2021A geometric approach to some systems of exponential equations, Vahagn Aslanyan, Jonathan Kirby and Vincenzo Mantova
IMRN 2022On systems of equations in abelian varieties, Francesco Gallinaro
IMRN 2023
For the modular j-function
Solutions of equations involving the modular j-function, Sebastian Eterović and Sebastián Herrero
Trans. Amer. Math. Soc. 2021Solving systems of equations of raising-to-powers type, Francesco Gallinaro
preprint 2021Differential existential closedness for the j-function, Vahagn Aslanyan, Sebastian Eterović and Jonathan Kirby
Proc. Amer. Math. Soc. 2021Generic solutions of equations involving the modular j-function, Sebastian Eterović
preprint 2022Blurrings of the j-function, Vahagn Aslanyan and Jonathan Kirby
The Quarterly Journal of Mathematics, 2022A closure operator respecting the modular j-function, Vahagn Aslanyan, Sebastian Eterović and Jonathan Kirby
Israel J. Math. 2023Equations involving the modular j-function and its derivatives, Vahagn Aslanyan, Sebastian Eterović and Vincenzo Mantova
Preprint 2023
For automorphic functions
Algebraic varieties and automorphic functions, Sebastian Eterović and Roy Zhao
preprint 2021
For other functions
Some equations involving the Gamma function, Sebastian Eterović and Adele Padgett
preprint 2023
(Un)likely intersections and Zilber-Pink
Exponential maps
The theory of exponential equations of semi-abelian varieties, Jonathan Kirby
Selecta Math. (N.S.) 2009Exponentially closed fields and the conjecture on intersections with tori, Jonathan Kirby and Boris Zilber
Ann. Pure Appl. Logic 2014Some remarks on atypical intersections, Vahagn Aslanyan
Proc. Amer. Math. Soc. 2021
The modular j-function
O-minimality and the André-Oort conjecture for C^n, Jonathan Pila
Ann. of Math. 2011Some unlikely intersections beyond André-Oort, Philipp Habegger and Jonathan Pila
Compos. Math. 2012O-minimality and certain atypical intersections, Philipp Habegger and Jonathan Pila
Sci. Éc. Norm. Supér. 2016Some remarks on atypical intersections, Vahagn Aslanyan
Proc. Amer. Math. Soc. 2021Weak modular Zilber-Pink with derivatives, Vahagn Aslanyan
Math. Ann. 2022A note on unlikely intersections in Shimura varieties, Vahagn Aslanyan and Christopher Daw
preprint 2022Multiplicative relations among differences of singular moduli, Vahagn Aslanyan, Sebastian Eterović and Guy Fowler
preprint 2023
General settings
Applications of the hyperbolic Ax-Schanuel conjecture, Christopher Daw and Jinbo Ren
Compositio Math. 2018The André-Oort conjecture for A_g, Jacob Tsimerman
Ann. of Math. 2018Canonical heights on Shimura varieties and the André-Oort conjecture, Jonathan Pila, Ananth Shankar, Jacob Tsimerman, Hélène Esnault, Michael Groechenig
preprint 2021Effective transcendental Zilber-Pink for variations of Hodge-structure, Jonathan Pila and Thomas Scanlon
preprint 2021On the distribution of the Hodge loci, Gregorio Baldi, Bruno Klingler and Emmanuel Ullmo
Invent Math. 2024Likely intersections, Sebastian Eterović and Thomas Scanlon
preprint 2022Existence and density of typical Hodge loci, Nazim Khelifa and David Urbanik
preprint 2023Compact subvarieties of the moduli space of complex abelian varieties, Samuel Grushevsky, Gabriele Mondello, Riccardo Salvati Manni and Jacob Tsimerman
preprint 2024
Schanuel's Conjecture and Ax-Schanuel Theorems
For complex exponentiation
On Schanuel's conjecture, James Ax
Ann. of Math. 1971Périodes de 1-motifs et transcendance, Cristiana Bertolin
J. Number Theory 2002The uniform Schanuel conjecture of the real numbers, Jonathan Kirby and Boris Zilber
Bull. Lond. Math. Soc. 2006Exponential algebraicity in exponential fields, Jonathan Kirby
Bull. Lond. Math. Soc. 2010A Schanuel property for exponentially transcendental powers, Martin Bays, Jonathan Kirby and Alex Wilkie
Bull. Lond. Math. Soc. 2010
For the modular j-function
Ax-Schanuel for the j-function, Jonathan Pila and Jacob Tsimerman
Duke Math. J. 2015A Schanuel property for j, Sebastian Eterovic
Bull. Lond. Math. Soc. 2018Schanuel type conjectures and disjointness, Isaac Broudy and Sebastian Eterovic
Ramanujan J. 2023
General results
Some topics in differential algebraic geometry I: analytic subgroups of algebraic groups, James Ax
Am. J. Math. 1972Ax-Schanuel for Shimura varieties, Ngaiming Mok, Jonathan Pila and Jacob Tsimerman
Ann. of Math. 2019The Ax-Schanuel conjecture for variations of Hodge structures, Benjamin Bakker and Jacob Tsimerman
Invent. Math. 2019The Ax-Schanuel conjecture for variations of mixed Hodge structures, Ziyang Gao and Bruno Klingler
Math. Ann. 2023A differential approach to the Ax-Schanuel I, David Blasquez-Sans, Guy Casale, James Freitag and Ronnie Nagloo
preprint 2021Functional transcendence of periods and the geometric André-Grothendieck period conjecture, Benjamin Bakker and Jacob Tsimerman
preprint 2022Ax-Schanuel for variations of mixed Hodge structures, Kenneth Chiu
preprint 2022Mixed Ax-Schanuel with derivatives for mixed period mappings, Kenneth Chiu
preprint 2021
Categoricity
Categoricity of modular and Shimura curves, Christopher Daw and Adam Harris
J. Inst. Math. Jussieu, 2017Pseudo-exponential maps, variants and quasiminimality, Martin Bays and Jonathan Kirby
Algebra & Number Theory 2018Universal covers of commutative finite Morley rank groups, Martin Bays, Bradd Hart and Anand Pillay
J. Inst. Math. Jussieu 2020Categoricity of Shimura varieties, Sebastian Eterovic
preprint 2018Zilber's notion of logically perfect structures, John T. Baldwin and Andrés Villaveces
Model Theory 2024