aa r X i v : . [ m a t h . AG ] A ug Remark on Weil’s conjectures
Igor V. Nikolaev
Abstract
We introduce a cohomology theory for a class of projective varietiesover a finite field coming from the canonical trace on a C ∗ -algebra at-tached to the variety. Using the cohomology, we prove the rationality,functional equation and the Betti numbers conjectures for the zetafunction of the variety. Key words and phrases: Weil conjectures, Serre C ∗ -algebrasMSC: 14F42 (motives); 46L85 (noncommutative topology) The aim of our note is a cohomology theory for projective varieties over thefield with q = p r elements. Such a cohomology comes from the canonicaltrace on a C ∗ -algebra attached to the variety V ; this theory will be calleda trace cohomology and denoted by H ∗ tr ( V ). The trace cohomology is muchparallel to the ℓ -adic cohomology H ∗ et ( V ; Q ℓ ), see [Grothendieck 1968] [3] and[Hartshorne 1977, pp. 453-457] [4]. Unlike the ℓ -adic cohomology, it does notdepend on a prime ℓ and the endomorphisms of H ∗ tr ( V ) always have an in-teger trace. While the ℓ -adic cohomology counts isolated fixed points of theFrobenius endomorphism geometrically, the trace cohomology does it alge-braically, i.e. taking into account the index ± H ∗ et ( V ; Q ℓ ) are complex alge-braic numbers of the absolute value q i , yet such eigenvalues are real algebraicon the trace cohomology H ∗ tr ( V ). The cohomology groups H itr ( V ) are trulyconcrete and simple; they can be found explicitly in many important special1ases, e.g. when V is an algebraic curve, see Section 4. We shall pass to adetailed construction.Denote by V C an n -dimensional projective variety over the field of com-plex numbers, such that the reduction of V C modulo the prime ideal over p is isomorphic to the variety V := V ( F q ). (In other words, we assume that V can be lifted to the characteristic zero and the reduction modulo p is functo-rial; the corresponding category is described in [Hartshorne 2010, Theorem22.1] [5].) Let B ( V C , L , σ ) be the twisted homogeneous coordinate ring ofprojective variety V C , where L is the invertible sheaf of linear forms on V C and σ an automorphism of V C , see [Stafford & van den Bergh 2001, p.180] [14] for the details. The norm-closure of a self-adjoint representationof the ring B ( V C , L , σ ) by the bounded linear operators on a Hilbert space H is a C ∗ -algebra, see e.g. [Murphy 1990] [7] for an introduction; we callit a Serre C ∗ -algebra of V C and denote by A V . Let K be the C ∗ -algebra ofall compact operators on H . We shall write τ : A V ⊗ K → R to denotethe canonical normalized trace on A V ⊗ K , i.e. a positive linear functionalof norm 1 such that τ ( yx ) = τ ( xy ) for all x, y ∈ A V ⊗ K , see [Blackadar1986, p. 31] [2]. Because A V is a crossed product C ∗ -algebra of the form A V ∼ = C ( V C ) ⋊ Z , one can use the Pimsner-Voiculescu six term exact se-quence for the crossed products, see e.g. [Blackadar 1986, p. 83] [2] for thedetails. Thus one gets the short exact sequence of the algebraic K -groups:0 → K ( C ( V C )) i ∗ → K ( A V ) → K ( C ( V C )) →
0, where map i ∗ is induced bythe natural embedding of C ( V C ) into A V . We have K ( C ( V C )) ∼ = K ( V C ) and K ( C ( V C )) ∼ = K − ( V C ), where K and K − are the topological K -groups of V C , see [Blackadar 1986, p. 80] [2]. By the Chern character formula, one gets K ( V C ) ⊗ Q ∼ = H even ( V C ; Q ) and K − ( V C ) ⊗ Q ∼ = H odd ( V C ; Q ), where H even ( H odd ) is the direct sum of even (odd, resp.) cohomology groups of V C . No-tice that K ( A V ⊗ K ) ∼ = K ( A V ) because of a stability of the K -group withrespect to tensor products by the algebra K , see e.g. [Blackadar 1986, p.32] [2]. One gets the commutative diagram in Fig. 1, where τ ∗ denotes ahomomorphism induced on K by the canonical trace τ on the C ∗ -algebra A V ⊗ K . Since H even ( V C ) := ⊕ ni =0 H i ( V C ) and H odd ( V ) := ⊕ ni =1 H i − ( V C ),one gets for each 0 ≤ i ≤ n an injective homomorphism τ ∗ : H i ( V C ) −→ R . Definition 1
By an i -th trace cohomology group H itr ( V ) of variety V oneunderstands the abelian subgroup of R defined by the map τ ∗ . Notice that each endomorphism of H itr ( V ) is given by a real number ω , such2 ❍❍❍❍❍❥ ✟✟✟✟✟✙ H even ( V C ) ⊗ Q i ∗ −→ K ( A V ⊗ K ) ⊗ Q −→ H odd ( V C ) ⊗ Q τ ∗ R Figure 1: The trace cohomology.that ωH itr ( V ) ⊆ H itr ( V ); thus the ring End ( H itr ( V )) of all endomorphisms of H itr ( V ) is commutative. The End ( H itr ( V )) is a commutative subring of thering End ( H i ( V C )) of all endomorphisms of the cohomology group H i ( V C ).Moreover, each regular map f : V → V corresponds to an algebraic map f C : V C → V C and, therefore, to an endomorphism ω ∈ End ( H itr ( V )). Onthe other hand, it is easy to see that End ( H itr ( V )) ∼ = Z or End ( H itr ( V )) ⊗ Q is an algebraic number field. In the latter case H itr ( V ) ⊂ End ( H itr ( V )) ⊗ Q ,see [Manin 2004, Lemma 1.1.1] [6] for the case of quadratic fields. We shallwrite tr ( ω ) to denote the trace of an algebraic number ω ∈ End ( H itr ( V )).Our main results are as follows. Theorem 1
The cardinality of variety V ( F q ) is given by the formula: | V ( F q ) | = 1 + q n + n − X i =1 ( − i tr ( ω i ) , (1) where ω i ∈ End ( H itr ( V )) is generated by the Frobenius map of V ( F q ) . Theorem 2
The zeta function Z V ( t ) := exp (cid:16)P ∞ r =1 | V ( F qr ) | r t r (cid:17) of V ( F q ) hasthe following properties:(i) Z V ( t ) = P ( t ) ...P n − ( t ) P ( t ) ...P n ( t ) is a rational function;(ii) Z V ( t ) satisfies the functional equation Z V (cid:16) q n t (cid:17) = ± q n χ ( V C )2 t χ ( V C ) Z V ( t ) ,where χ ( V C ) is the Euler-Poincar´e characteristic of V C ;(iii) deg P i ( t ) = dim H i ( V C ) . Remark 1
Roughly speaking, theorem 2 says that the standard propertiesof the trace cohomology imply all Weil’s conjectures, except for an analog ofthe Riemann hypothesis | α ij | = q i for the roots α ij of polynomials P i ( t ), see[Weil 1949, p. 507] [15]; the latter property is proved in [10].3he article is organized as follows. Section 2 contains some useful definitionsand notation. Theorems 1 and 2 are proved in Section 3. We calculatethe trace cohomology for the algebraic (and elliptic, in particular) curves inSection 4. In this section we briefly review the twisted homogeneous coordinate ringsand the Serre C ∗ -algebras associated to projective varieties, see [Artin & vanden Bergh 1990] [1]) and [Stafford & van den Bergh 2001] [14] for a de-tailed account. The C ∗ -algebras and their K -theory are covered in [Murphy1990] [7] and [Blackadar 1986] [2], respectively. The Serre C ∗ -algebras wereintroduced in [9]. Let V be a projective scheme over a field k , and let L be the invertible sheaf O V (1) of linear forms on V . Recall, that the homogeneous coordinate ringof V is a graded k -algebra, which is isomorphic to the algebra B ( V, L ) = M n ≥ H ( V, L ⊗ n ) . (2)Denote by Coh the category of quasi-coherent sheaves on a scheme V and by Mod the category of graded left modules over a graded ring B . If M = ⊕ M n and M n = 0 for n >>
0, then the graded module M is called right bounded.The direct limit M = lim M α is called a torsion, if each M α is a right boundedgraded module. Denote by Tors the full subcategory of
Mod of the torsionmodules. The following result is basic about the graded ring B = B ( V, L ). Lemma 1 ([Serre 1955] [11]) Mod ( B ) / Tors ∼ = Coh ( V ) . Let σ be an automorphism of V . The pullback of sheaf L along σ will bedenoted by L σ , i.e. L σ ( U ) := L ( σU ) for every U ⊂ V . The graded k -algebra B ( V, L , σ ) = M n ≥ H ( V, L ⊗ L σ ⊗ . . . ⊗ L σ n − ) . (3)is called a twisted homogeneous coordinate ring of scheme V ; notice thatsuch a ring is non-commutative, unless σ is the trivial automorphism. The4ultiplication of sections is defined by the rule ab = a ⊗ b σ m , whenever a ∈ B m and b ∈ B n . Given a pair ( V, σ ) consisting of a Noetherian scheme V and anautomorphism σ of V , an invertible sheaf L on V is called σ -ample, if for everycoherent sheaf F on V , the cohomology group H q ( V, L⊗L σ ⊗ . . . ⊗L σ n − ⊗F )vanishes for q > n >>
0. Notice, that if σ is trivial, this definition isequivalent to the usual definition of ample invertible sheaf [Serre 1955] [11].A non-commutative generalization of the Serre theorem is as follows. Lemma 2 ([Artin & van den Bergh 1990] [1])
Let σ : V → V bean automorphism of a projective scheme V over k and let L be a σ -ampleinvertible sheaf on V . If B ( V, L , σ ) is the ring (3), then Mod ( B ( V, L , σ )) / Tors ∼ = Coh ( V ) . (4) C ∗ -algebras Let V be a projective scheme and B ( V, L , σ ) its twisted homogeneous coor-dinate ring. Let R be a commutative graded ring, such that V = Spec ( R ).Denote by R [ t, t − ; σ ] the ring of skew Laurent polynomials defined by thecommutation relation b σ t = tb for all b ∈ R , where b σ is the image of b underautomorphism σ : V → V . Lemma 3 ([Artin & van den Bergh 1990] [1]) R [ t, t − ; σ ] ∼ = B ( V, L , σ ) . Let H be a Hilbert space and B ( H ) the algebra of all bounded linear operatorson H . For a ring of skew Laurent polynomials R [ t, t − ; σ ], we shall considera homomorphism ρ : R [ t, t − ; σ ] −→ B ( H ) . (5)Recall that algebra B ( H ) is endowed with a ∗ -involution; the involutioncomes from the scalar product on the Hilbert space H . We shall call rep-resentation (5) ∗ -coherent, if (i) ρ ( t ) and ρ ( t − ) are unitary operators, suchthat ρ ∗ ( t ) = ρ ( t − ) and (ii) for all b ∈ R it holds ( ρ ∗ ( b )) σ ( ρ ) = ρ ∗ ( b σ ), where σ ( ρ ) is an automorphism of ρ ( R ) induced by σ . Whenever B = R [ t, t − ; σ ]admits a ∗ -coherent representation, ρ ( B ) is a ∗ -algebra; the norm-closure of ρ ( B ) yields a C ∗ -algebra, see e.g. [Murphy 1990, Section 2.1] [7]. We shallrefer to such as the Serre C ∗ -algebra and denote it by A V .Recall that if A is a C ∗ -algebra and σ : G → Aut ( A ) is a continuous ho-momorphism of the locally compact group G group, then the triple ( A , G, σ )defines a C ∗ -algebra called a crossed product and denoted by A ⋊ σ G ; we refer5he reader to [Williams 2007, pp. 47-54] [16] for the details. It is not hard tosee, that A V is a crossed product C ∗ -algebra of the form A V ∼ = C ( V ) ⋊ σ Z ,where C ( V ) is the C ∗ -algebra of all continuous complex-valued functions on V and σ is a ∗ -coherent automorphism of V . We shall prove a stronger result contained in the following lemma.
Lemma 4
The Lefschetz number of the Frobenius map f C : V C → V C is givenby the formula: L ( f C ) = 1 − q n + n − X i =1 ( − i tr ( ω i ) . (6) Proof.
Recall that the Lefschetz number of a continuous map g C : V C → V C is defined as L ( g C ) = n X i =0 ( − i tr ( g ∗ i ) , (7)where g ∗ i : H i ( V C ) → H i ( V C ) is an induced linear map of the cohomol-ogy. Because f ∗ i is nothing but the matrix form of an endomorphism ω i ∈ End ( H itr ( V )), one gets L ( f C ) = n X i =0 ( − i tr ( ω i ) . (8)We shall write equation (8) in the form L ( f C ) = tr ( ω ) + tr ( ω n ) + n − X i =1 ( − i tr ( ω i ) . (9)It is known, that H ( V C ) ∼ = Z and ω = 1 is the trivial endomorphism; thus tr ( ω ) = 1. Likewise, H n ∼ = Z , but ω n = sgn [ N ( ω )] q n , (10)6here N ( • ) is the norm of an algebraic number. It is known, that the endo-morphism ω ∈ End ( H tr ( V )) has the following matrix form q (cid:18) A II (cid:19) , (11)where A is a positive symmetric and I is the identity matrix, see ([9], Lemma3). Thus sgn [ N ( ω )] = sgn det (cid:18) A II (cid:19) == sgn [ − det ( I )] = − sgn det ( I ) = − . (12)Therefore, from (10) one obtains ω n = − q n ; in other words, the Frobeniusendomorphism acts on H ntr ( V ) ∼ = Z by multiplication on the negative integer − q n . Clearly, tr ( ω n ) = − q n and the substitution of these data in (9) givesus L ( f C ) = 1 − q n + n − X i =1 ( − i tr ( ω i ) . (13)Lemma 4 follows. (cid:3) Corollary 1
The total number of the index − fixed points of the Frobeniusmap f C is equal to q n .Proof. It is known, that | V ( F q ) | = 1 + q n + n − X i =1 ( − i tr ( F r ∗ i ) , (14)where F r ∗ i : H iet ( V ; Q ℓ ) → H iet ( V ; Q ℓ ) is a linear map on the i -th ℓ -adiccohomology induced by the Frobenius endomorphism of V , see [Hartshorne1977, pp. 453-457] [4]. But according to [9], it holds tr ( F r ∗ i ) = tr ( ω i ) , ≤ i ≤ n − . (15)Since | F ix ( f C ) | = | V ( F q ) | , one concludes from lemma 4 that the algebraiccount L ( f C ) of the fixed points of f C differs from its geometric count | F ix ( f C ) | by exactly q n points of the index −
1. Corollary 1 is proved. (cid:3)
Theorem 1 follows formally from the equations (14) and (15). (cid:3) .2 Proof of theorem 2 For the sake of clarity, let us outline the main idea. Since the trace cohomol-ogy accounts for the fixed points of the Frobenius map f C algebraically (seecorollary 1), we shall deal with the corresponding Lefschetz zeta function Z LV ( t ) := exp ∞ X r =1 L ( f r C ) r t r ! (16)and prove items (i)-(iii) for the Z LV ( t ). Because f C : V C → V C is the Anosov-type map, one can use Smale’s formulas linking Z LV ( t ) and Z V ( t ), see [Smale1967, Proposition 4.14] [13]; it will follow that items (i)-(iii) are true for thefunction Z V ( t ) as well. We shall pass to a detailed argument; the followinggeneral lemma will be helpful. Lemma 5 If f : V → V is a regular map, then all eigenvalues λ ij of thecorresponding endomorphisms ω i ∈ End ( H itr ( V )) of the trace cohomologyare real algebraic numbers.Proof. Since the endomorphisms of H itr ( V ) commute with each other, thereexists a basis of H itr ( V ), such that each endomorphism is given in this basisby a symmetric integer matrix [9]. But the spectrum of a real symmetricmatrix is known to be totally real and the eigenvalues of an integer matrixare algebraic numbers. Lemma 5 follows. (cid:3) (i) Let us prove rationality of the function Z LV ( t ) given by formula (16).Using lemma 4, one getslog Z LV ( t ) = P ∞ r =1 h − q n ) r + P n − i =1 ( − i tr ( ω ri ) i t r r == ∞ X r =1 t r r + P ∞ r =1 ( − q n t ) r r + P ∞ r =1 (cid:16)P n − i =1 ( − i tr ( ω ri ) (cid:17) t r r . (17)Taking into account the well-known summation formulas P ∞ r =1 t r r = − log(1 − t ) and P ∞ r =1 ( − q n t ) r r = − log(1 + q n t ), one can bring equation (17) to the formlog Z LV ( t ) = − log(1 − t )(1 + q n t ) + n − X i =1 ( − i ∞ X r =1 tr ( ω ri ) t r r . (18)On the other hand, it easy to see that tr ( ω ri ) = λ r + . . . + λ rb i , (19)8here λ j are the eigenvalues of the Frobenius endomorphism ω i ∈ End ( H itr ( V ))and b i is the i -th Betti number of V C . Thus one can bring (18) to the formlog Z LV ( t ) = − log(1 − t )(1 + q n t )++ n − X i =1 ( − i P ∞ r =1 h ( λ t ) r r + . . . + ( λ bi t ) r r i . (20)Using the summation formula P ∞ r =1 ( λ j t ) r r = − log(1 − λ j t ), one gets from (20)log Z LV ( t ) = − log(1 − t )(1 + q n t )+ n − X i =1 ( − i +1 log [(1 − λ t ) . . . (1 − λ b i t )] . (21)Notice that the product (1 − λ t ) . . . (1 − λ b i t ) is nothing but the characteristicpolynomial P i ( t ) of the Frobenius endomorphism on the trace cohomology H itr ( V ); thus one can write (21) in the formlog Z LV ( t ) = log P ( t ) . . . P n − ( t )(1 − t ) P ( t ) . . . P n − ( t )(1 + q n t ) . (22)Taking exponents in the last equation, one obtains Z LV ( t ) = P ( t ) . . . P n − ( t ) P ( t ) . . . P n ( t ) , (23)where P ( t ) = 1 − t and P n ( t ) = 1 + q n t . Thus Z LV ( t ) is a rational function.To prove rationality of Z V ( t ), recall that a map f C : V C → V C is called Anosov-type , if there exist a (possibly singular) pair of orthogonal foliations F u and F s of V C preserved by f C . (Note that our definition is more gen-eral than the standard and includes all continuous maps f C .) Considerthe trace cohomology H tr ( V ) endowed with the Frobenius endomorphism ω ∈ End ( H tr ( V )). Let F s be a foliation of V C , whose holonomy (Plante)group is isomorphic to H tr ( V ). Because ω H tr ( V ) ⊂ H tr ( V ), one concludesthat F s is an invariant stable foliation of the map f C . The unstable folia-tion F u can be constructed likewise. Thus f C is the Anosov-type map of themanifold V C . One can apply now (an extension of) [Smale 1967, Proposition9.14] [13], which says that one of the following formulas must hold: Z V ( t ) = Z LV ( t ) ,Z V ( t ) = Z LV ( − t ) ,Z V ( t ) = Z LV ( − t ) . (24)Since Z LV ( t ) is known to be a rational function (23), it follows from Smale’sformulas (24) that Z V ( t ) is rational as well. Item (i) is proved.(ii) Recall that the cohomology H ∗ ( V C ) satisfies the Poincar´e dulality; theduality can be given by a pairing H i ( V C ) × H n − i ( V C ) −→ H n ( V C ) (25)obtained from the cup-product on H ∗ ( V C ).Let f : V → V be the Frobenius endomorphism and f C : V C → V C thecorresponding algebraic map of V C and consider the action ( f n C ) ∗ on thepairing h• , •i given by (25). Since H n ( V C ) ∼ = Z and the linear map ( f n C ) ∗ multiplies H n ( V C ) by the constant q n , one gets h ( f i C ) ∗ x, ( f n − i C ) ∗ y i = q n h x, y i , (26)for all x ∈ H i ( V C ) and all y ∈ H n − i ( V C ). Recall the linear algebra identities,given e.g. in [Hartshorne 1977, Lemma 4.3, p. 456] [4]; then (26) implies thefollowing formulas det ( I − ( f i C ) ∗ t ) = ( − bi ( q n ) bi t bi det ( f n − i C ) ∗ det h I − q n t ( f n − i C ) ∗ i det ( f i C ) ∗ = ( q n ) bi det ( f n − i C ) ∗ , (27)where b i = dim H i ( V C ) are the i -th Betti numbers. But det ( I − ( f i C ) ∗ t ) := P i ( t ) and det h I − q n t ( f n − i C ) ∗ i := P n − i (cid:16) q n t (cid:17) ; therefore, the first equationof (27) yields the identity P i ( t ) = ( − b i ( q n ) b i det ( f n − i C ) ∗ t b i P n − i q n t ! . (28)10et us calculate Z LV (cid:16) q n t (cid:17) using (28); one gets the following expression Z LV q n t ! = P ( qnt ) ...P n − ( qnt ) P ( qnt ) ...P n ( qnt ) == P ( t ) . . . P n − ( t ) P ( t ) . . . P n ( t ) t ( b − b + ... ) ( − ( b − b + ... ) det ( f C ) ∗ ... det ( f n C ) ∗ det ( f C ) ∗ ... det ( f n − C ) ∗ . (29)Note that b − b + . . . = χ ( V C ) is the Euler-Poincar´e characteristic of V C . Fromthe second equation of (27) one obtains the identity det ( f i C ) ∗ det ( f n − i C ) ∗ =( q n ) b i . Thus (29) can be written in the form Z LV q n t ! = t χ ( V C ) ( − χ ( V C ) ( q n )
12 ( b ...b n ) ( q n )
12 ( b ... + b n − Z LV V ( t ) == t χ ( V C ) ( − χ ( V C ) ( q n ) χ ( V C ) Z LV ( t ) . (30)Taking into account ( − − χ ( V C ) = ±
1, one gets a functional equation for Z LV ( t ). We encourage the reader to verify using formulas (24), that the sameequation holds for the function Z V ( t ). Item (ii) of theorem 2 is proved.(iii) To prove the Betti numbers conjecture, notice that equality (19) im-plies that deg P i ( t ) = dim H itr ( V ). But dim H itr ( V ) = dim H i ( V C ) by thedefinition of trace cohomoogy; thus deg P i ( t ) = dim H i ( V C ) for the polyno-mials P i ( t ) in formula (23). Again, the reader can verify using (24), that thesame relationship holds for the polynomials representing the rational function Z V ( t ). Item (iii) is proved.This argument completes the proof of theorem 2 (cid:3) The groups H itr ( V ) are truly concrete and simple; in this section we calculatethe trace cohomology for n = 1, i.e. when V is a smooth algebraic curve. Inparticular, we find the cardinality of the set E ( F q ) obtained by the reductionmodulo q of an elliptic curve with complex multiplication. The reader canverify, that the lifting condition for V is satisfied, see footnote 1.11 xample 1 The trace cohomology of smooth algebraic curve C ( F q ) of genus g ≥ H tr ( C ) ∼ = Z ,H tr ( C ) ∼ = Z + Z θ + . . . + Z θ g − ,H tr ( C ) ∼ = Z , (31)where θ i ∈ R are algebraically independent integers of a number field ofdegree 2 g . Proof.
It is known that the Serre C ∗ -algebra of the (generic) complex alge-braic curve C is isomorphic to a toric AF -algebra A θ , see [8] for the notationand details. Moreover, up to a scaling constant µ >
0, it holds τ ∗ ( K ( A θ ⊗ K )) = (cid:26) Z + Z θ if g = 1 Z + Z θ + . . . + Z θ g − if g >
1, (32)where constants θ i ∈ R parametrize the moduli (Teichm¨uller) space of curve C , ibid. If C is defined over a number field k , then each θ i is algebraic andtheir total number is equal to 2 g −
1. (Indeed, since
Gal (¯ k | k ) acts on thetorsion points of C ( k ), it is easy to see that the endomorphism ring of C ( k )is non-trivial. Because such a ring is isomorphic to the endomorphism ringof jacobian J ac C and dim C J ac C = g , one concludes that End C ( k ) is a Z -module of rank 2 g and each θ i is an algebraic number.) After scaling by aconstant µ >
0, one gets H tr ( C ) := τ ∗ ( K ( A θ ⊗ K )) = Z + Z θ + . . . + Z θ g − (33)Because H ( C ) ∼ = H ( C ) ∼ = Z , one obtains the rest of formulas (31). (cid:3) Remark 2
Using theorem 1, one gets the formula |C ( F q ) | = 1 + q − tr ( ω ) = 1 + q − g X i =1 λ i , (34)where λ i are real eigenvalues of the Frobenius endomorphism ω ∈ End ( H tr ( C )).Note that λ + . . . + λ g = α + . . . + α g , (35)12here α i are the eigenvalues of the Frobenius endomorphism of H et ( C ; Q ℓ ).However, there is no trace cohomology analog of the classical formula |C ( F q r ) | = 1 + q r − g X i =1 α ri , (36)unless r = 1; this difference is due to an algebraic count of the fixed pointsby the trace cohomology. Example 2
The case g = 1 is particularly instructive; for the sake of clarity,we shall consider elliptic curves having complex multiplication. Let E ( F q ) bethe reduction modulo q of an elliptic with complex multiplication by the ringof integers of an imaginary quadratic field Q ( √− d ), see e.g. [Silverman 1994,Chapter 2] [12]. It is known, that in this case the trace cohomology formulas(31) take the form H tr ( E ( F q )) ∼ = Z ,H tr ( E ( F q )) ∼ = Z + Z √ d,H tr ( E ( F q )) ∼ = Z . (37)We shall denote by ψ ( P ) ∈ Q ( √− d ) the Gr¨ossencharacter of the prime ideal P over p , see [Silverman 1994, p. 174] [12]. It is easy to see, that in this casethe Frobenius endomorphism ω ∈ End ( H tr ( E ( F q ))) is given by the formula ω = 12 h ψ ( P ) + ψ ( P ) i + 12 r(cid:16) ψ ( P ) + ψ ( P ) (cid:17) + 4 q (38)and the corresponding eigenvalues λ = ω = h ψ ( P ) + ψ ( P ) i + r(cid:16) ψ ( P ) + ψ ( P ) (cid:17) + 4 q,λ = ¯ ω = h ψ ( P ) + ψ ( P ) i − r(cid:16) ψ ( P ) + ψ ( P ) (cid:17) + 4 q. (39)Using formula (34), one gets the following equation |E ( F q ) | = 1 − ( λ + λ ) + q = 1 − ψ ( P ) − ψ ( P ) + q, (40)which coincides with the well-known expression for |E ( F q ) | in terms of theGr¨ossencharacter, see e.g. [Silverman 1994, p. 175] [12].13 eferences [1] M. Artin and M. van den Bergh, Twisted homogeneous coordinaterings, J. of Algebra 133 (1990), 249-271.[2] B. Blackadar, K -Theory for Operator Algebras, MSRI Publications,Springer, 1986[3] A. Grothendieck, Standard conjectures on algebraic cycles, in: Alge-braic Geometry, Internat. Colloq. Tata Inst. Fund. Res., Bombay, 1968.[4] R. Hartshorne, Algebraic Geometry, GTM 52, Springer, 1977.[5] R. Hartshorne, Deformation Theory, GTM 257, Springer, 2010.[6] Yu. I. Manin, Real multiplication and noncommutative geometry, in“Legacy of Niels Hendrik Abel”, 685-727, Springer, 2004.[7] G. J. Murphy, C ∗ -Algebras and Operator Theory, Academic Press,1990.[8] I. Nikolaev, Noncommutative geometry of algebraic curves, Proc.Amer. Math. Soc. 137 (2009), 3283-3290.[9] I. Nikolaev, On traces of Frobenius endomorphisms, Finite Fields andtheir Applications 25 (2014), 270-279.[10] I. Nikolaev, On trace cohomology, arXiv:1407.3982 [11] J. P. Serre, Fasceaux alg´ebriques coh´erents, Ann. of Math. 61 (1955),197-278.[12] J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves,GTM 151, Springer 1994.[13] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73(1967), 747-817.[14] J. T. Stafford and M. van den Bergh, Noncommutative curves andnoncommutative surfaces, Bull. Amer. Math. Soc. 38 (2001), 171-216.[15] A. Weil, Numbers of solutions of equations in finite fields, Bull. Amer.Math. Soc. 55 (1949), 497-508.1416] D. P. Williams, Crossed Products of C ∗ -Algebras, Math. Surveys andMonographs, Vol. 134, Amer. Math. Soc. 2007. Department of Mathematics and Computer Science, St. John’sUniversity, 8000 Utopia Parkway, New York, NY 11439, UnitedStates; E-mail: [email protected]@gmail.com