ON THE NUMBER OF COINCIDENCES OF MORPHISMS BETWEEN CLOSED RIEMANN SURFACES A

Let M be a compact Riemann surface (complete complex algebraic curve) of genus g > 2, and 7: M ---> M an automorphism different from the identity . Then it is well known (see e.g. [F-K]) that T has at most 2g + 2 fixed points and that this bound is attained if and only if M is hyperelliptic and T is the hyperelliptic involution . With this in mind, we consider two distinct morphisms fi : MM' of degrees di (i = 1, 2) between compact Riemann surfaces of genera g and g' >_ 2 respectively, and look at the number of coincidentes, that is, the number of points at which fi and f2 agree . The result we obtain (Theorem 2 .9) is that fl and f2 have at most di+2g' dld2 +d2 coincidentes, and that this number (suitably counted) is attained if and only if M' is hyperelliptic and fi and f2 differ by composition with the hyperelliptic involution . When these morphisms are isomorphisms, Le . when dl = d2 = 1, then, of cotlrse, the coincidentes are the fixed points of the automorphism T = fi ' o f2 ; in this case our result agrees with the classical one .

Let M be a compact Riemann surface (complete complex algebraic curve) of genus g > 2, and 7-: M ---> M an automorphism different from the identity .Then it is well known (see e.g.[F-K]) that T has at most 2g + 2 fixed points and that this bound is attained if and only if M is hyperelliptic and T is the hyperelliptic involution.
With this in mind, we consider two distinct morphisms fi : M -M' of degrees di (i = 1, 2) between compact Riemann surfaces of genera g and g' >_ 2 respectively, and look at the number of coincidentes, that is, the number of points at which fi and f2 agree .
The result we obtain (Theorem 2 .9) is that fl and f2 have at most di+2g' dl d2 +d2 coincidentes, and that this number (suitably counted) is attained if and only if M' is hyperelliptic and fi and f2 differ by composition with the hyperelliptic involution.When these morphisms are isomorphisms, Le. when dl = d2 = 1, then, of cotlrse, the coincidentes are the fixed points of the automorphism T = fi ' o f2; in this case our result agrees with the classical one .
The proof uses a Lefschetz trace formula for the case oftwo morphisms, which is a straightforward generalization of the standard one and, no doubt, is well known to topologists .However, at least in the precise form we need it here, we have not been able to locate it in the literature (although see [Eich], [Lej, [K-L]) ; so we devote a preliminary section to establish it.
The main result is proved in Section 2; the work done there will allow us to obtain, as a byproduct, the well known theorem of de Franchis ( [Fra]) on morphisms between closed Riemann surfaces.
1 .Lefschetz's trace formula A) In this section we first recall the basasic facts in the proof of the standard Lefschetz formula for the number of fixed points of a self-mapping, and then we show how to derive, in a similar way, the Lefschetz formula for the number of coincidentes of two different mappings .
Let M be a compact oriented manifold of dimension n, let ns C M x M be the diagonal submanifold, and let rlo E HDR(M x M) be its Poincaré dual.For any self-mapping f : M --> M, the integral is called the Lefschetz number of f.The classical theorem of Lefschetz arises from evaluating this integral in two different ways corresponding to two different representatives of the de Rham cohomology class i70 .

P=1
On the other hand, the Poincaré dual of an oriented submanifold Z of X can always be represented by a form %, the Thom class, supported on an arbitrarily small tubular neighbourhood T of Z in X, diffeomorphic to the normal bundle Nz of Z in X, with the property that the integral of Pz along each fiber Tz , z E Z, is 1 ([B-T]) .In the case of our diagonal submanifold A C M x M, one sees that (f x id) * ( Do is supported only near the fixed point set of f, and hence, at least in the case in which f x id is transverse to 0, we have (see M where e(x) is the sign of the determinant of (Dfx -Id) .We recall that f x id being transverse to ns is equivalent to the matrix (Dfx -Id) being non-singular ([G-P]) .
More generally, let us only assume that f has a finite number of fixed points (not necessarily transverse to Ls) .We recall that L x (f ), the local Lefschetz number of f at an isolated fixed point x, is defined to be the degree of the map z H f (z) -z from the boundary of a small ball lf(z) -zi around x to the unit sphere S" -1 .
In this situation (see [G-P]) one can perturb f near the fixed points to obtain a map ft : M --> M enjoying the following properties (3.A) i) ft is homotopic to f ; ii) ft agrees with f outside compact balls B(x) around each fixed point x; iii) (ft x I) is transverse to Ls; iv) Lx(f) = E e( {YEB(x)/ft(y)=y1 Summing up, we obtain The above considerations translate word for word to the case in which one has two different mappings fi : M --> M'(¡ = 1, 2) between (in general, distinct) compact oriented manifolds of the same dimension .
Definition 1.1.The Lefschetz number of two mappings fi : M -4 M'(¡ = 1, 2) between two compact oriented manifolds of equal dimension is defined to be L(fi, f2) = fm (fi x f2) *?7o, where 91o is the Poincaré dual of the diagonal submanifold Ps C M' x M' .
Poincaré duality between HD R (M') and HDR (M') allows us to make the following Definition 1.2.Let f : M -M' be a mapping between compact oriented manifolds of the same dimension n.Then we shall denote by f* the linear map the previous formula (1 .A) being obtained by letting the second mapping be the identity.Again the form (f1 x f2) *, Dp is non zero only near the coincidentes of fl and f2 .In case f1 x f2 is transverse te the submanifold Ls C M' x M', which again means that the matrix (Dfl .-Df2,x) is non-singular at any such point x, each of these points contributes to the integral fm (fl x f2) * 4)o with ±1 according to whether the determinant of (Dfl,x -Df2,x) is positive or negative .Thus, the analogue te (2.A) is where E(x) is the sign of the determinant of (Dfl,x -Df2,x) .
If fl, f2 satisfy the weaker condition of having a finite number of coincidences, then by perturbing f1 in the way indicated above ([G-P]), we obtain a map ft enjoying the following properties i) ft is homotopic to fl ; ii) ft agrees with f1 outside compact balls B(x) around each of these finite number of points ; iii) (ft x f2) is transverse te Ls ; We can now write (3.B) Summmarizing we have Definition 1.3.Let f1, f2 be as in Definition 1 .1 and let x be an isolated point of coincidente, then the local Lefschetz number of f1, f2 at x, Lx(f1, f2), is defined to be the degree of the map z ~---~I fi (z) -f2(z) I from the boundary of a small ball around x to the unit sphere S'-1.
Let us assume that the set F of coincidentes is finite, then Lx(f1, f2) .
In the formula (1 .B) the linear maps fi , f2 .are composed in different order.The change is valid because of the well known fact that for any two matrices A, B the traces of A -B and B -A agree, whenever the two products make sense.
Of the above sequence of traces, the first and the last ones are the easiest to work out .Let us denote by d2 the degree of the map fi ; then we have i) fi o f2.: H°(M) H H°(M) is multiplication by d2 ii) fi o f2.: Hn(M) H Hn(M) is multiplication by d1 .
From this, i) and ii) follow easily.

A bound for the number of coincidentes
1.In what follows we will concentrate in the case in which the manifolds M and M' are compact Riemann surfaces of genera g and g', and the mappings fi : M --> M' are holomorphic and non constant.In this situation the Lefschetz formula of our previous section reads Moreover, it is well known that the first de Rham cohomology group (with complex coefficients) splits into the direct sum of the vector space of holomorphic 1-forms and its conjugate ; namely we have the following result L(f,, .f2)= dl -trace fi o f2, IH1(M) + d2 .HDR(M, C) = F(M, 9) ® F(M, Q) ; Lemma 2.1.
Let f : M -~M' be a holomorphic map between compact Riemann surfaces, and let f* : HLR(M, C) ' HD R (M', C) be the C-linear map obtained by extending the R-linear map f* introduced in the preceding section; then we have i)  in this way we obtain a globally well defined holomorphic form w' on M' (see [Spr,p. 276]) .
We claim that w' = f* w.Indeed, for any 1-form rl on M' a standard partition of unity argument shows that The rest of the statements in the lemma follow from this fact .
Notation.At this point it is convenient to introduce a change in our notation.From now on, given a holomorphic map fi : M -+ M', we shall denote by fi* the restriction of the C-linear operator fi* of the lemma above to F(M, S2) .Accordingly fi o f2* will always denote a C-linear endomorphism of F(M, S2) .
When fi : M -> M' (i = 1, 2) are isomorphisms, then we have dl = d2 = 1, and fi o f2 .= (fa 1 o fi) * ; thus, in this case, our formula is just the usual Lefschetz's formula for the automorphism (fa 1 o fi) (see 2 .It is well known that the vector space F(M, 9) carries a hermitian structure given by We have the following result Proposition 2.4.
and f2 o fi. are adjoint of each other.
ii) f* o f* is self adjoint.iii) There is an orthogonal basis 0 = {wi, . . ., w9 } of F(M, 9) with respect to which f2 o f2 and f2 o fl .o fi o f2 are represented by the following diagonal g x g matrices of rank g' Proof.. We have < fi o f2 v, w > = i f fi o f2 .v n w = i f v n f2 o fi .w =< v, f2 o fi .w > which preves i) and ii) .
In order to prove iii) we make the observation that the action of f* o f en I'(M', 9) is just multiplication by d = deg(f) ; this can be deduced either from the explicit construction of f*w carried out in the proof of lemma 2.1, or from the definition given in Section 1 .Indeed, for any two This observation shows that f2 o fi * o fi o f2 .= di f2 o f2 ., and therefore it is enough to prove the statement concerning f2 o f2* .Now, since f2 o f2 . is self adjoint, there is an orthogonal basis 0 with respect to which its matrix is diagonal.Clearly this matrix has rank at most g', but on the other hand the observation above also shows that the forms in f2 (F(M', SZ)) are all eigenvectors of f2 o f2, with eigenvalue d2 .This completes the proof.We have the bound L(fi, f2 ) <_ dl + 2g' dld2 + d2 .Equality holds if and only if the matrix of fi o f2, with respect to the basis ,0 above is and only if This means that Proof. .Let A = (a2j ) be the matrix of fi o f2 with respect to the orthogonal basis above.Then, by part i) of the Proposition, tA will be the matrix of f2 o fl, ; thus, by part iii) we have Definition 2 .6.
Let P E M be a coincidente of fl and f2 ; and let .fl (z) -.f2 (z) = ckzk + ck+lzk+1 + . . .; ck 7~o be the Taylor expansion of fi -f2 with respect to small parametric discs D of P and D' of fi(P) .We define the multiplicity of fi, f2 at P te be Proposition 2.7.
Let P E M be a coincidente of fi and f2 ; then LP(fi, f2) = mp(fi, f2) .Because of this proposition, in the rest of the paper we will refer to the global Lefschetz number L(fi, f2) as the number of coincidentes counted with multiplicities (or appropriately counted) .
In any case, this number is always greater than or equal te the actual number of coincidentes, so we have.Corollary 2.8.i) #{P E M/fi(P) _ f2(P)} c di + 2g' dld2 + d2 .We observe that when fl, f2 are isomorphisms then we have dl = d2 = 1, g = g' ; and our bounds all equal 2g + 2 = 2g' + 2, as it should be.
4 .We now address the question of whether our bound is sharp.By Corollary .2.5 ., the number of coincidentes L(fi, f2) attains this bound if and only if for the first g' forms wl , . . ., w9, of the orthogonal basis ,(3, we have It follows that the inclusions f0 : C(M') y C(M) between the function fields of M' and M induced by the maps fi (i = 1, 2) agree on the subfield K C C(M') generated by quotients of 1-forms on M'.
Let us now assume that g' >_ 2; then, if M' is not hyperelliptic we have K = C(M') and, by the well known equivalente between compact Riemann surfaces and their function fields, it follows that fi = f2, which is in contradiction with fi o f2 .wi= -dl d2 wi .
If on the other hand M' is hyperelliptic, then K = C(x) is the subfield of degree 2 generated by the hyperelliptic function x : M' ~Pl and we see that in this case either fl = f2 (which again is impossible), or f2 = J o fi, where J is the hyperelliptic involution of M' (see [F-K]) .
Summarizing we have proved the following result Theorem 2.9.
In this case, we can make everything explicit .We have -deg(fij) = n.

Remark 3.2.
There is a more direct approach to estimate #{P/fi(P) = f2(P)} .Let cP be a meromorphic function on M', then we can write This computation makes sense whenever ep is such that Wofi and coo f2 are distinct; in order to guarantee that this property is satisfied, we must allow cp to have degree g' + 1 .(We recall that, by the Riemann-Roch theorem, for any P' E M' there exists a function of degree g' + 1 that takes the value oo only at the point P' .) This gives us the bound (g' + 1) (dl + d2), which is satisfactory for the automorphism case (dl = d2 = 1) where we obtain the correct number 2g + 2. However, in the general case this bound exceeds ours by g'( dl -d2)2 .
Remark 3.3.We note that the morphisms between the surfaces M and M' cannot be replaced simply by continuous (surjective) mapsl .Already when M = M', one has a family of homeomorphisms f z : M --> M whose action on the first homology group is represented, with respect to a canonical basis, by the family of 29 x 2g matrices A,n = ~o . . .o -1 o . . . . . .oJ this is because An is symplectic.
The corresponding family of Lefschetz numbers is L(fn ) = 2 + 2gn, which is not bounded with g.
Remark 3 .4. (de Franchis theorem) .The work done in Section 2 also allows us to obtain the following result of de Franchis.Let M, M' have genes > 2; then i) The number of possible maps fi : M --> M' is finite .ii) The number of possible targets M' for fixed M is finite .
We describe the proof briefly ; it naturally falls into two parts : 1) First, one proves that the linear endomorphism fi o fi.(resp.fi* o fi ., fl kept fixed) of 1'(M, 9) determines fi up to postcomposition with an automorphism of M' (resp.determines fi completely).
2) Then, one shows that there can only be finitely many such linear endomorphisms .
The proof of statement 1) is contained in the discussion of Section 2 .4 that precedes Theorem 2 .9.Indeed, if fi o fi.= fj* o fj .(resp.fi o fl .= fj * o fl* ) then the induced inclusions between function fields fi*, fi : C(M') y C(M) would have the same image (resp .would coincide) .Therefore, from the well known equivalente between Riemann surfaces and therr function fields, we deduce that fi and fj differ by postcomposition with an automorphism of M' (resp.fi and fj agree) .Again, the case in which M' is hyperelliptic will have to be treated separately.
In order to prove 2), we work with the cohomology group with integer coefficients H1 (M, Z); this way we represent fz ofi.= Ti (resp .fz ofl.= Til) by a matrix with integer entries.Then, we use Proposition 2.4.iii) to obtain that its Euclidean norm 11 Ti 112 := trace (Ti -Ti) is 2d?g' (resp .2dldig'), where T* stands for the adjoint of the operator T. From this, we deduce that there is a finite number of operators Ti (resp .Til) .
In conclusion, the finiteness-of the operators Ti (resp.Til) proves part ii) (resp.part i)) of de Franchis theorem.
It should be said that this proof is very similar to that of H. Martens ([Mal) (see also [Ta], [H-SI) .The only difference is that in our proof jacobians do not appear ; instead we let function fields play the main role.

Added on Proof.
We have recently learnt (W.Fulton, "Intersection theory", Springer-Verlag, 1984, p. 312) that the bound given in our Theorem 2 .9.i)can also be obtained by means of the Intersection theory of algebraic surfaces.Not so (as far as we can see), the identification of the case in which this bound is attained (Theorem 2.9.ii) .
S2) ; in fact, for any holomorphic 1-form w, f* w = f*w .Proof.Let U' be an open set of M' well covered by f .This means that f -'(U') is the disjoint union of open sets Ui (i = 1, . . ., d = deg(f)) such that the restriction of f to each of them is an isomorphism.Now, given a holomorphic form w on M, we assign to each such open set U' the form d wíu' = 1:(f1-l)*w . i=1 , f2) = dl -(trace fi o f2 * + trace fi o f2 * ) + d2 .
3. Final remarks and examplesLet us take as M the Fermat Riemann surface of algebraic equation x2n + yen = 1 , and as M' the hyperelliptic surface of equation y2

'
We are grateful to C .Earle for bringing this question to our attention .