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A certain numerical function on the set of solutions of a Diophantine equation (cf. [[Diophantine equations|Diophantine equations]]). In the simplest case of a solution in integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h0468101.png" /> of a Diophantine equation, the height is a function of the solution, and equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h0468102.png" />. It is encountered in this form in Fermat's method of descent. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h0468103.png" /> be a projective algebraic variety defined over a global field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h0468104.png" />. The height is a class of real-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h0468105.png" /> defined on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h0468106.png" /> of rational points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h0468107.png" /> and depending on a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h0468108.png" /> of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h0468109.png" /> into the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681010.png" />. Each function in this class is also called a height. From the point of view of estimating the number of rational points there are no essential differences between the functions in this class: for any two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681012.png" /> there exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681014.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681015.png" />. Such functions are called equivalent, and this equivalence is denoted (here) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681016.png" />.
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A certain numerical function on the set of solutions of a Diophantine equation (cf. [[Diophantine equations]]). In the simplest case of a solution in integers $(x_0,\ldots,x_n)$ of a Diophantine equation, the height is a function of the solution, and equals $\max\{|x_i|\}$. It is encountered in this form in Fermat's method of descent. Let $X$ be a projective algebraic variety defined over a [[global field]] $K$. The height is a class of real-valued functions $h_L(P)$ defined on the set $X(K)$ of rational points $P$ and depending on a morphism $L:X\rightarrow P^n$ of the variety $X$ into the projective space $P^n$. Each function in this class is also called a height. From the point of view of estimating the number of rational points there are no essential differences between the functions in this class: for any two functions $h'$ and $h''$ there exist constants $c',c'' > 0$, such that $c' h' \le h'' \le c''h'$. Such functions are called equivalent, and this equivalence is denoted (here) as $\cong$.
  
Fundamental properties of the height. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681017.png" /> is functorial with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681018.png" />, i.e. for any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681019.png" /> and morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681020.png" />,
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Fundamental properties of the height. The function $h_L(P)$ is functorial with respect to $P$, i.e. for any morphism $f:X \rightarrow Y$ and morphism $L : Y \rightarrow P^n$,
 +
$$
 +
h_{f*L}(P) \cong h_L(f(P))\,\ \ P \in X(K) \ .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681021.png" /></td> </tr></table>
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If the morphisms $L$, $L_1$ and $L_2$ are defined by invertible sheaves $\mathcal{L}$, $\mathcal{L}_1$ and $\mathcal{L}_2$, and if $\mathcal{L} = \mathcal{L}_1 \otimes \mathcal{L}_2$, then $h_L \cong h_{L_1} h_{L_2}$. The set of points $P \in X(K)$ of bounded height is finite in the following sense: If the basic field $K$ is an [[algebraic number field]], the set is finite; if it is an algebraic function field with field of constants $k$, the elements of $X(K)$ depend on a finite number of parameters from the field $k$; in particular, $X(K)$ is finite if the field $k$ is finite. Let $|\cdot|_\nu$ run through the set of all norms of $K$. One may then define the height of a point $(x_0:\cdots:x_n)$ of the projective space $P^n$ with coordinates from $K$ as
 
 
If the morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681024.png" /> are defined by invertible sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681027.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681029.png" />. The set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681030.png" /> of bounded height is finite in the following sense: If the basic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681031.png" /> is an algebraic number field, the set is finite; if it is an algebraic function field with field of constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681032.png" />, the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681033.png" /> depend on a finite number of parameters from the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681034.png" />; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681035.png" /> is finite if the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681036.png" /> is finite. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681037.png" /> run through the set of all norms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681038.png" />. One may then define the height of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681039.png" /> of the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681040.png" /> with coordinates from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681041.png" /> as
 
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Weil, "Number theory and algebraic geometry" , ''Proc. Internat. Congress Mathematicians (Cambridge, 1950)'' , '''2''' , Amer. Math. Soc. (1952) pp. 90–100 {{MR|0045416}} {{ZBL|0049.02802}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lang, "Diophantine geometry" , Interscience (1962) {{MR|0142550}} {{ZBL|0115.38701}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974) (Appendix in Russian translation: Yu.I. Manin; The Mordell–Weil theorem (in Russian)) {{MR|2514037}} {{MR|1083353}} {{MR|0352106}} {{MR|0441983}} {{MR|0282985}} {{MR|0248146}} {{MR|0219542}} {{MR|0219541}} {{MR|0206003}} {{MR|0204427}} {{ZBL|0326.14012}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> Yu.I. Manin, "Height of theta points on an Abelian manifold, their variants and applications" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''28''' (1964) pp. 1363–1390 (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> D. Mumford, "A remark on Mordell's conjecture" ''Amer. J. Math.'' , '''87''' (1965) pp. 1007–1016 {{MR|186624}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A. Néron, "Quasi-fonctions et hauteurs sur les variétés abéliennes" ''Ann. of Math. (2)'' , '''82''' (1965) pp. 249–331 {{MR|0179173}} {{ZBL|0163.15205}} </TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> A. Weil, "Number theory and algebraic geometry" , ''Proc. Internat. Congress Mathematicians (Cambridge, 1950)'' , '''2''' , Amer. Math. Soc. (1952) pp. 90–100 {{MR|0045416}} {{ZBL|0049.02802}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lang, "Diophantine geometry" , Interscience (1962) {{MR|0142550}} {{ZBL|0115.38701}} </TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974) (Appendix in Russian translation: Yu.I. Manin; The Mordell–Weil theorem (in Russian)) {{MR|2514037}} {{MR|1083353}} {{MR|0352106}} {{MR|0441983}} {{MR|0282985}} {{MR|0248146}} {{MR|0219542}} {{MR|0219541}} {{MR|0206003}} {{MR|0204427}} {{ZBL|0326.14012}} </TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top"> Yu.I. Manin, "Height of theta points on an Abelian manifold, their variants and applications" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''28''' (1964) pp. 1363–1390 (In Russian)</TD></TR>
 +
<TR><TD valign="top">[5]</TD> <TD valign="top"> D. Mumford, "A remark on Mordell's conjecture" ''Amer. J. Math.'' , '''87''' (1965) pp. 1007–1016 {{MR|186624}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[6]</TD> <TD valign="top"> A. Néron, "Quasi-fonctions et hauteurs sur les variétés abéliennes" ''Ann. of Math. (2)'' , '''82''' (1965) pp. 249–331 {{MR|0179173}} {{ZBL|0163.15205}} </TD></TR>
 +
</table>
  
  
  
 
====Comments====
 
====Comments====
The notion of height is a major tool in arithmetic algebraic geometry. It plays an important role in Faltings' proof of the Tate conjecture on endomorphisms of Abelian varieties over number fields, the Shafarevich conjecture that there are only finitely many isomorphism classes of Abelian varieties over a number field over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681055.png" /> of given dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681056.png" /> with good reduction outside a finite set of places <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681057.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681058.png" />, and the Mordell conjecture on the finiteness of the set of rational points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681059.png" /> of a smooth curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681060.png" /> over a number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681061.png" />. Heights also play an important role in Arakelov intersection theory, which via moduli spaces of algebraic curves has also become important in string theory in mathematical physics.
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The notion of height is a major tool in arithmetic algebraic geometry. It plays an important role in Faltings' proof of the Tate conjecture on endomorphisms of Abelian varieties over number fields, the Shafarevich conjecture that there are only finitely many isomorphism classes of Abelian varieties over a number field over $K$ of given dimension $g\ge1$ with good reduction outside a finite set of places $S$ of $K$, and the Mordell conjecture on the finiteness of the set of rational points $X(K)$ of a smooth curve of genus $g \ge 2$ over a number field $K$. Heights also play an important role in Arakelov intersection theory, which via moduli spaces of algebraic curves has also become important in string theory in mathematical physics.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Faltings (ed.) G. Wüstholtz (ed.) , ''Rational points'' , Vieweg (1986) {{MR|0863887}} {{ZBL|0636.14019}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Faltings (ed.) G. Wüstholtz (ed.) , ''Rational points'' , Vieweg (1986) {{MR|0863887}} {{ZBL|0636.14019}} </TD></TR>
 +
</table>
 +
 
 +
{{TEX|part}}

Revision as of 11:50, 29 October 2017

A certain numerical function on the set of solutions of a Diophantine equation (cf. Diophantine equations). In the simplest case of a solution in integers $(x_0,\ldots,x_n)$ of a Diophantine equation, the height is a function of the solution, and equals $\max\{|x_i|\}$. It is encountered in this form in Fermat's method of descent. Let $X$ be a projective algebraic variety defined over a global field $K$. The height is a class of real-valued functions $h_L(P)$ defined on the set $X(K)$ of rational points $P$ and depending on a morphism $L:X\rightarrow P^n$ of the variety $X$ into the projective space $P^n$. Each function in this class is also called a height. From the point of view of estimating the number of rational points there are no essential differences between the functions in this class: for any two functions $h'$ and $h''$ there exist constants $c',c'' > 0$, such that $c' h' \le h'' \le c''h'$. Such functions are called equivalent, and this equivalence is denoted (here) as $\cong$.

Fundamental properties of the height. The function $h_L(P)$ is functorial with respect to $P$, i.e. for any morphism $f:X \rightarrow Y$ and morphism $L : Y \rightarrow P^n$, $$ h_{f*L}(P) \cong h_L(f(P))\,\ \ P \in X(K) \ . $$

If the morphisms $L$, $L_1$ and $L_2$ are defined by invertible sheaves $\mathcal{L}$, $\mathcal{L}_1$ and $\mathcal{L}_2$, and if $\mathcal{L} = \mathcal{L}_1 \otimes \mathcal{L}_2$, then $h_L \cong h_{L_1} h_{L_2}$. The set of points $P \in X(K)$ of bounded height is finite in the following sense: If the basic field $K$ is an algebraic number field, the set is finite; if it is an algebraic function field with field of constants $k$, the elements of $X(K)$ depend on a finite number of parameters from the field $k$; in particular, $X(K)$ is finite if the field $k$ is finite. Let $|\cdot|_\nu$ run through the set of all norms of $K$. One may then define the height of a point $(x_0:\cdots:x_n)$ of the projective space $P^n$ with coordinates from $K$ as

(*)

This is well defined because of the product formula , . Let be an arbitrary projective variety over and let be a closed imbedding of into the projective space; the height may then be obtained by transferring the function (*), using the imbedding, to the set . Various projective imbeddings, corresponding to the same sheaf , define equivalent functions on . A linear extension yields the desired function . The function is occasionally replaced by its logarithm — the so-called logarithmic height.

The above estimates may sometimes follow from exact equations [3], [4], [5]. There is a variant of the height function — the Néron–Tate height — which is defined on Abelian varieties and behaves as a functor with respect to the morphisms of Abelian varieties preserving the zero point. For the local aspect see [6]. The local components of a height constructed there play the role of intersection indices in arithmetic.

References

[1] A. Weil, "Number theory and algebraic geometry" , Proc. Internat. Congress Mathematicians (Cambridge, 1950) , 2 , Amer. Math. Soc. (1952) pp. 90–100 MR0045416 Zbl 0049.02802
[2] S. Lang, "Diophantine geometry" , Interscience (1962) MR0142550 Zbl 0115.38701
[3] D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974) (Appendix in Russian translation: Yu.I. Manin; The Mordell–Weil theorem (in Russian)) MR2514037 MR1083353 MR0352106 MR0441983 MR0282985 MR0248146 MR0219542 MR0219541 MR0206003 MR0204427 Zbl 0326.14012
[4] Yu.I. Manin, "Height of theta points on an Abelian manifold, their variants and applications" Izv. Akad. Nauk SSSR Ser. Mat. , 28 (1964) pp. 1363–1390 (In Russian)
[5] D. Mumford, "A remark on Mordell's conjecture" Amer. J. Math. , 87 (1965) pp. 1007–1016 MR186624
[6] A. Néron, "Quasi-fonctions et hauteurs sur les variétés abéliennes" Ann. of Math. (2) , 82 (1965) pp. 249–331 MR0179173 Zbl 0163.15205


Comments

The notion of height is a major tool in arithmetic algebraic geometry. It plays an important role in Faltings' proof of the Tate conjecture on endomorphisms of Abelian varieties over number fields, the Shafarevich conjecture that there are only finitely many isomorphism classes of Abelian varieties over a number field over $K$ of given dimension $g\ge1$ with good reduction outside a finite set of places $S$ of $K$, and the Mordell conjecture on the finiteness of the set of rational points $X(K)$ of a smooth curve of genus $g \ge 2$ over a number field $K$. Heights also play an important role in Arakelov intersection theory, which via moduli spaces of algebraic curves has also become important in string theory in mathematical physics.

References

[a1] G. Faltings (ed.) G. Wüstholtz (ed.) , Rational points , Vieweg (1986) MR0863887 Zbl 0636.14019
How to Cite This Entry:
Height, in Diophantine geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Height,_in_Diophantine_geometry&oldid=42211
This article was adapted from an original article by A.N. Parshin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article