Math 2.0

Number theory Conjecture

Vojta's conjecture

Builds on Algebraic number fieldHeight functionCanonical bundleAmple line bundleWeil divisorZariski topologyRational point

Statement

Let FF be an algebraic number field and let XX be a smooth projective variety over FF. Let DD be an effective reduced divisor on XX with at most normal crossings. Let KXK_X be a canonical divisor and let AA be an ample divisor on XX. Let SS be a finite set of places of FF. Let hKXh_{K_X} and hAh_A be Weil heights for KXK_X and AA, and let λD,v\lambda_{D,v} be a local height function for DD at the place vv. Then for every ε>0\varepsilon > 0 there are a proper Zariski closed subset Z⊊XZ \subsetneq X and a constant CC such that ∑v∈SλD,v(P)+hKX(P)≤ε hA(P)+C\sum_{v \in S} \lambda_{D,v}(P) + h_{K_X}(P) \le \varepsilon \, h_A(P) + C for all P∈X(F)P \in X(F) with P∉ZP \notin Z.

Remarks

Vojta formulated the conjecture in 1987 by analogy with the second main theorem of Nevanlinna theory, through a dictionary between Diophantine approximation and value distribution. For X=ℙ1X = \mathbb{P}^1 and DD a sum of distinct points, the conjecture is Roth's theorem, which is known. For X=ℙnX = \mathbb{P}^n and DD a sum of hyperplanes in general position, it is Schmidt's subspace theorem, which is also known. Let XX be of general type and D=0D = 0. Outside a proper closed subset, the canonical height is at least δhA−C′\delta h_A - C' for some δ>0\delta > 0. Taking ε<δ\varepsilon < \delta then forces X(F)X(F) into a proper closed subset, which is the Bombieri–Lang conjecture. Generalized forms for algebraic points, with truncated local heights, are higher-dimensional analogues of the abc conjecture. The conjecture is open in general.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Algebraic number field, concept, no evidence levelAlgebraic number fieldconceptAmple line bundle, concept, no evidence levelAmple line bundleconceptCanonical bundle, concept, no evidence levelCanonical bundleconceptHeight function, concept, no evidence levelHeight functionconceptRational point, concept, no evidence levelRational pointconceptWeil divisor, concept, no evidence levelWeil divisorconceptZariski topology, concept, no evidence levelZariski topologyconceptVojta's conjecture, conjecture, no evidence levelVojta's conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic number field concept
  2. Ample line bundle concept
  3. Canonical bundle concept
  4. Height function concept
  5. Rational point concept
  6. Weil divisor concept
  7. Zariski topology concept
  8. Vojta's conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Vojta%27s_conjecture
About this page
TagQQ717VSFNN
URIm2://QQ717VSFNN
Kindconjecture
WritingW0 stub
MSC11J97, 11G50, 14G40
StatusThis copy of the site has no build data. Run m2 status QQ717VSFNN.
AboutHeight function, Rational point
UsesAlgebraic number field, Height function, Canonical bundle, Ample line bundle, Weil divisor, Zariski topology, Rational point
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