CHAP, xxni] EQUATIONS FORMED FROM LINEAR FUNCTIONS. 677 



J. von Sz. Nagy 16 proved that any curve of genus 2 with rational coeffi- 

 cients is equivalent in general to a quartic curve and contains an infinitude 

 of rational groups of two points. 



J. von Sz. Nagy 17 cited the known fact that a curve C p n of order n and 

 genus p>l has in general no birational automorphs besides identity, and 

 never more than S4(p 1), and concluded that we can derive at most a 

 finite number of rational points from one. The birational automorphs 

 of non-hyperelliptic and hyperelliptic curves are discussed. An example 

 shows that from a rational point we do not in general obtain all other rational 

 points by means of the birational automorphs of the curve. 



J. von Sz. Nagy 18 wrote Q n for the g.c.d. of n and 2p 2 and proved that 

 a curve C* of order n and genus p contains infinitely many rational groups of 

 hQ n points if h is an integer for which hQ n >p 1; it is equivalent to a 

 curve C p m for m>p-\-l if and only if it contains a rational group of m non- 

 singular points. In particular, they are equivalent if m is a multiple of Q n , 

 and hence if m = 2p 2, p>2, and the curves are not hyperelliptic. 



E. Maillet 180 considered a polynomial f(x, y) of degree n>2, irreducible, 

 with integral coefficients, and such that the curve /=0 is unicursal (of genus 

 0). If there are at least n 3 simple rational points, there is an infinitude 

 corresponding to the rational values of a parameter t, and x=fz(t)/fi(f), 

 y=Mi)lfi(t), where the/, are polynomials with integral coefficients having 

 no common divisor, of degrees n^n, one being of degree n (cf. papers 12, 

 15). The curve has an infinite number of points with integral coordinates 

 only when /i is a constant or of one of the forms a(Mt+N) n , with a, M, N 

 integers, or a(Mtf-\-Nt-\-P) nl ~, where n is even and N 2 4:MP is positive and 

 not a square, while a, M, N, P are integers. There are extensions to cer- 

 tain equations f(x, y) = of genus > and to certain unicursal surfaces. 



For cubic curves of genus unity, see Levi 307 and Hurwitz 312 of Ch. XXI. 



EQUATIONS FORMED FROM LINEAR FUNCTIONS. 



For related papers, see Lagrange, 142 Rados 194a ; papers 313-23 of Ch. 

 XXI; and Ch. XX. 



G. L. Dirichlet 19 stated a theorem, which he regarded as remarkable for 

 its simplicity and importance : if an equation 



(1) s n +as n ~ 1 -\ ----- \-gs+h = Q 



with integral coefficients has no rational divisor and if at least one of its 

 roots a, /3, , co is real, and if we set 



0(a) =X+ay-\ ----- \-a n ~ l z; 

 then the indeterminate equation 



(2) F(x, y, ...,g) = 0( 



16 Math. Naturw. Berichte aus Ungarn, 26, 1908 (1913), 186 (168-195). 



17 Jahresbericht d. Deutschen Math.-Vereinigung, 21, 1912, 183-191. 



18 Math. Annalen, 73, 1913, 230-240, 600. 



180 Comptes Rendus Paris, 168, 1919, 217-20; Jour. Ecole Polyt., (2), 20, 1919, 115-56. 



19 Comptes Rendus Paris, 10, 1840, 285-8; Werke, I, 619-623. 



