Editing Faltings's theorem (section)

==Proofs==
[[Igor Shafarevich]] conjectured that there are only finitely many isomorphism classes of [[abelian variety|abelian varieties]] of fixed dimension and fixed [[Abelian variety#Polarisations|polarization]] degree over a fixed number field with [[good reduction]] outside a fixed finite set of [[place (mathematics)|place]]s.{{sfn|Shafarevich|1963}} [[Aleksei Parshin]] showed that Shafarevich's finiteness conjecture would imply the Mordell conjecture, using what is now called Parshin's trick.{{sfn|Parshin|1968}} 

[[Gerd Faltings]] proved Shafarevich's finiteness conjecture using a known reduction to a case of the [[Tate conjecture]], together with tools from [[algebraic geometry]], including the theory of [[Néron model]]s.{{sfn|Faltings|1983}} The main idea of Faltings's proof is the comparison of [[Height function#Faltings height|Faltings heights]] and [[Height function#Naive height|naive heights]] via [[Siegel modular variety|Siegel modular varieties]].{{efn|"Faltings relates the two notions of height by means of the Siegel moduli space.... It is the main idea of the proof." {{cite journal |last=Bloch |first=Spencer |s2cid=306251 |author-link=Spencer Bloch |page=44 |title=The Proof of the Mordell Conjecture |journal=The Mathematical Intelligencer |volume=6 |issue=2 |year=1984|doi=10.1007/BF03024155 }}}}

===Later proofs===
* [[Paul Vojta]] gave a proof based on [[Diophantine approximation]].{{sfn|Vojta|1991}} [[Enrico Bombieri]] found a more elementary variant of Vojta's proof.{{sfn|Bombieri|1990}}
*Brian Lawrence  and [[Akshay Venkatesh]] gave a proof based on [[p-adic Hodge theory|{{mvar|p}}-adic Hodge theory]], borrowing also some of the easier ingredients of Faltings's original proof.{{sfn|Lawrence|Venkatesh|2020}}