View source for Height, in Diophantine geometry

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" />.

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" />,

<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>

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>

This is well defined because of the product formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681044.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681045.png" /> be an arbitrary projective variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681046.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681047.png" /> be a closed imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681048.png" /> into the projective space; the height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681049.png" /> may then be obtained by transferring the function (*), using the imbedding, to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681050.png" />. Various projective imbeddings, corresponding to the same sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681051.png" />, define equivalent functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681052.png" />. A linear extension yields the desired function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681053.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046810/h04681054.png" /> is occasionally replaced by its logarithm — the so-called logarithmic height.

The above estimates may sometimes follow from exact equations [[#References|[3]]], [[#References|[4]]], [[#References|[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 [[#References|[6]]]. The local components of a height constructed there play the role of intersection indices in arithmetic.

====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>



====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.

====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>