676 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxm 



can decide by rational operations whether or not /=0 is of genus zero and 

 if so we can find by rational operations n 1 linearly independent ternary 

 forms <f>i of degree n 2 with integral coefficients such that for arbitrary 

 parameters X the curve /=0 is cut by the curve 



in n 2 points varying with the parameters X t . Set 



$i = X;i0H ----- hXin-l<n-l (i=l, 2, 3), 



where the X t / are arbitrary parameters. Transform /=0 by 



y\ :y z :2/3 = $i : $2 : $3. 



The result is g(yi, y%, 2/3) =0, where g is an irreducible form of degree n2 

 in the y's with integral coefficients. Now give to the parameters X,-/ such 

 integral values that g remains irreducible. Since our transformation is 

 birational, every rational point on /=0 corresponds to a rational point on 

 g = and conversely. Hence the initial problem is reduced to the equation 

 gr = also of genus zero, but of lower degree by two units. Ultimately we 

 reach an equation of degree 1 or 2. For a linear equation l(u\, u z , u z ) =0, 

 we can evidently find three linear functions w of the homogeneous param- 

 eter ti/tz such that if i : u 2 : u 3 = coi : co 2 : o> 3 gives all rational solutions of 

 1 = when h, t 2 take all integral values. By applying the inverses of our 

 transformations, we get the initial/=0 and solutions x\ : o? 2 : 3 = p\ : p 2 : PS, 

 where the p f are forms of degree n in ti, t 2 . The only missing solutions are 

 those, finite in number and found rationally, which correspond to rational 

 singular points of /=0, where our transformations cease to be birational. 

 Second, if we reached a quadratic equation, it can be transformed rationally 

 into aiul+a z ul+azul = Q, the a's without square factors and relatively 

 prime in pairs. It has integral solutions if and only if the a's are not all 

 of like sign and if a 2 a 3 , a a ai, aid 2 are quadratic residues of a\, a z , a 3 , 

 respectively (papers 114, 116, 119 of Ch. XIII). When these conditions 

 are satisfied, the conic has rational points and can be transformed bira- 

 tionally into a straight fine; we proceed as before. 



M. Noether 14 had earlier proved that a rational curve can be trans- 

 formed birationally into a straight line or conic; a curve of order 2n with 

 a (2n l)-fold point is counted as curve of odd order. 



H. Poincare 15 proved the above result that any unicursal curve with 

 rational coefficients is equivalent to a conic or a straight line, two curves 

 being called equivalent if one can be transformed into the other by a 

 birational transformation with rational coefficients. A curve /=0 of genus 

 1 (bicursal curve) with rational coefficients is equivalent to a curve of 

 order p (p^3) if and only if /=0 has a rational group of p points, i. e., a 

 set of p points such that every elementary symmetric function of their 

 coordinates is rational. 



14 Math. Annalen, 3, 1871, 170. 



16 Jour, de Math., (5), 7, 1901, 161-233. For a special case, von Sz. Nagy 151 of Ch. XXI. 



