Exponential Sums Equations and the Schanuel Conjecture

By B. Zilber. 2002

163 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology, Theoretical Computer Science

View Paper: https://www.semanticscholar.org/paper/c78bcfe7fdf817ae2fb2b51b0268e72720463f0c

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In Algebra and Number Theory

In Geometry and Topology

In Theoretical Computer Science

Citations in the corpus

Ax–Schanuel for the $j$-function (2014, unknown): “On the Zilber-Pink conjecture generally see [6, 24, 25, 29, 30].”
The theory of the exponential differential equations of semiabelian varieties (2007, unknown): “en there is J ∈ J U of codimension at least t and s ∈ S(C) such that X ⊆ s·J. This is a weak version of the Conjecture on the intersection of algebraic subgroups with subvarieties stated by Zilber in [Zil02], and is the natural generalization to semiabelian varieties of the version proved there for ”
Some unlikely intersections beyond André–Oort (2011, background): “There are variants of Pink’s Conjecture for semi-abelian varieties such as Zilber’s Conjecture 2 [45] and Pink’s Conjecture 5.”
Applications of the hyperbolic Ax–Schanuel conjecture (2017, unknown): “author [10, 22, 41]. Furthermore, Gao has generalized these proofs to all mixed Shimura varieties [16, 15]. In hiswork on Schanuel’s conjecture, Zilbermade hisown conjecture on unlikely intersections [47], which was closely related to the independent work of Bombieri, Masser, and Zannier [5]. To des”
Towards the Andr\'e-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points (2013, background): “1 is part of the more general Zilber-Pink Conjecture [54, 70, 56].”
O-minimality and the André-Oort conjecture for $\mathbb{C}^{n}$ (2011, result): “…by Zannier and implemented in three recent papers: a new proof of the Manin-Mumford conjecture by Pila-Zannier; a proof of a special (but new) case of Pink’s relative Manin-Mumford conjecture by Masser-Zannier; and new proofs of certain known results of André-OortManin-Mumford type by Pila.”
On the Bounded Height Conjecture (2008, background): “We proceed to show some finiteness results related to conjectures stated by Zilber and Pink, if the codimension of the subgroups is at least 1 + dim X.”
Torsion points on families of squares of elliptic curves (2012, background): “This is a variant of that stated by Pink [21] in 2005, which generalized the Zilber Conjectures [26] to schemes.”
O-minimality and certain atypical intersections (2014, background): “The Zilber-Pink Conjecture remains open for surfaces in abelian varieties defined over a number field.”