of points of K. If P is a point in V(K), w_p denotes the valuation corresponding to P. By a subring A of K we 

understand a ring A such that k  A  K and (A) = K. A subring A of K is a semilocal subring if, and only if, 

the set 
                        S (A) = (PV(K) : v_p(a)  0 for all a  A) 
is finite. If  is the integral closure of A in K then  / oa /A  is a finite dimensional vector space over k. We put d =  

dim A / A . 

                  Let A be the adel ring of the field K. The subring A₀ of  A is defined by  

 A₀ = A  : w_p((P)) =   for " PS (A) ). 

                 For any divisor D of K we associate the following subspaces: 

N ( D ) = ( A : w_p((P)) +  w_p ( D )  0 for " PV(K) ). 
L ( D ) = N ( N ) Ç K , L ( D ) = N ( D ) Ç A . 

PS (A)

                  Generalized Riemann-Roch Theorem of Rosenlicht can be stated as 
 
                            dim L ( D) = deg . D + 1 - ( g + d) + dim L ( D) 

where g is the genus of K. The proof of this theorem is given in the language of adels. We have also proved the 

following: Let D be a divisor of K prime to S(A). 0  dim L ( D + Q ) - dim L ( D)  1 for any QV(K)\ S(A). 
If deg . D  2( g + d ) - 1  then dim  ( D ) = 0. The main result is that dim L ( ( g + d )Q ) = 1 for almost all 

QV(K)\ S(A). A point QV(K)\ S(A) is called a Weierstrass point of K relative to the semi-local subring A if 

dim L ( ( g + d )Q ) > 1. There are at least 2( g + 1 ) - Card .S(A) - 4 d \ ( g + d )  and at most ( g + d - 1)( g + d) 

( g + d + 1 ) Weierstrass points of K relative to A. The Weierstrass points of K relative to A come out to be the 

zeroes of a certain determinant D(A). 

                   Let Q be an arbitrary point in V(K) and n a positive integer. It has been proved that there are only a 

finite number of points   PV(K) such that Q is a Weierstrass point of K relative to the local subring k+Pⁱ, i = 

2,3,...,n + 1 if and only if there exist elements y₁,y₂,...,y_nK such that 
           
                    w_p (y_i) = i,   w_Q(y_i) = - ( g + i ) 

for all  i = 1,2,...,n. This led us to introduce the concepts of (n, Q)-equivalence and Q-equivalence. 

                   Finally, we give an imbedding of K into a non-computative ring proposed by C.Arf.