Math Conjectures

Vojta’s Main Conjecture

OPENLandmarkConjectureProposed 1987 · Standard version

Canonical statement

Let XX be a smooth projective variety over a number field KK, let DD be a simple-normal-crossings divisor, AA a big divisor, SS a finite set of places of KK, r≥1r\ge1, and ε>0\varepsilon>0. There is a proper Zariski-closed Z⊂XZ\subset X containing the support of DD such that for every P∈X(Kˉ)∖ZP\in X(\bar K)\setminus Z with [K(P):K]≤r[K(P):K]\le r,
mD,S(P)+hKX(P)≤dK(P)+εhA(P)+O(1). m_{D,S}(P)+h_{K_X}(P) \le d_K(P)+\varepsilon h_A(P)+O(1).
Here mD,Sm_{D,S} is the proximity function, heights use fixed Weil-height choices, the O(1)O(1) constant may depend on the fixed data (X,D,A,K,S,r,ε)(X,D,A,K,S,r,\varepsilon) but is independent of PP, and
dK(P)=log⁡ ⁣∣NK/QDK(P)/K∣[K(P):K] d_K(P)= \frac{\log\!\left| N_{K/\mathbb Q}\mathfrak D_{K(P)/K}\right|} {[K(P):K]}
is the normalized logarithmic relative discriminant, with DK(P)/K\mathfrak D_{K(P)/K} the relative discriminant ideal.
View source LaTeX
Let \(X\) be a smooth projective variety over a number field \(K\), let \(D\) be a simple-normal-crossings divisor, \(A\) a big divisor, \(S\) a finite set of places of \(K\), \(r\ge1\), and \(\varepsilon>0\). There is a proper Zariski-closed \(Z\subset X\) containing the support of \(D\) such that for every \(P\in X(\bar K)\setminus Z\) with \([K(P):K]\le r\),
\[
  m_{D,S}(P)+h_{K_X}(P)
     \le d_K(P)+\varepsilon h_A(P)+O(1).
\] Here \(m_{D,S}\) is the proximity function, heights use fixed Weil-height choices, the \(O(1)\) constant may depend on the fixed data \((X,D,A,K,S,r,\varepsilon)\) but is independent of \(P\), and
\[
  d_K(P)=
    \frac{\log\!\left|
      N_{K/\mathbb Q}\mathfrak D_{K(P)/K}\right|}
         {[K(P):K]}
\] is the normalized logarithmic relative discriminant, with \(\mathfrak D_{K(P)/K}\) the relative discriminant ideal.
AG-DIO-007:[Vojta1987Diophantine][Vojta1998Integral][GuoSunWang2025VojtaTori]
Algebraic geometryNumber theory

Notes

Vojta's main conjecture arose from a systematic dictionary between value-distribution (Nevanlinna) theory and Diophantine approximation, set out in his 1987 monograph [Vojta1987Diophantine]. For a smooth projective variety XX over a number field with a simple-normal-crossings divisor DD, it predicts that for algebraic points PP of bounded degree outside some proper Zariski-closed subset, the proximity function mD,S(P)m_{D,S}(P) plus the height hKX(P)h_{K_X}(P) attached to the canonical divisor is bounded by the normalized discriminant term dK(P)d_K(P) plus εhA(P)+O(1)\varepsilon h_A(P)+O(1), for any big divisor AA and ε>0\varepsilon>0.

Much of the interest lies in its consequences: the inequality implies the abcabc conjecture, Roth-type theorems, and broad degeneracy statements for rational and integral points; Vojta himself later formulated a more general abcabc-type version [Vojta1998Integral]. Unconditionally the conjecture is known only for restricted classes of varieties, and the function-field setting is better understood, for instance in the case of algebraic tori [GuoSunWang2025VojtaTori].

The general height inequality over number fields remains open, and its resolution would reorganize a large part of Diophantine geometry.

References (3)

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