Remark on Weil’s conjectures Igor V. Nikolaev Abstract We introduce a cohomology theory for a class of projective varieties over 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 zeta function of the variety. Key words and phrases: Weil conjectures, Serre C∗-algebras MSC: 14F42 (motives); 46L85 (noncommutative topology) 1 Introduction The aim of our note is a cohomology theory for projective varieties over the field with q = pr elements. Such a cohomology comes from the canonical trace on a C∗-algebra attached to the variety V ; this theory will be called ∗ a trace cohomology and denoted by Htr(V ). The trace cohomology is much ∗ parallel to the ℓ-adic cohomology Het(V ; Qℓ), see [Grothendieck 1968] [3] and arXiv:1309.2551v4 [math.AG] 4 Aug 2016 [Hartshorne 1977, pp. 453-457] [4]. Unlike the ℓ-adic cohomology, it does not ∗ depend on a prime ℓ and the endomorphisms of Htr(V ) always have an in- teger trace. While the ℓ-adic cohomology counts isolated fixed points of the Frobenius endomorphism geometrically, the trace cohomology does it alge- braically, i.e. taking into account the index 1 of a fixed point. Moreover, ± ∗ the eigenvalues of Frobenius endomorphism on Het(V ; Qℓ) are complex alge- i braic numbers of the absolute value q 2 , yet such eigenvalues are real algebraic ∗ i on the trace cohomology Htr(V ). The cohomology groups Htr(V ) are truly concrete and simple; they can be found explicitly in many important special 1 cases, e.g. when V is an algebraic curve, see Section 4. We shall pass to a detailed construction. Denote by VC an n-dimensional projective variety over the field of com- plex numbers, such that the reduction of VC modulo the prime ideal over p is isomorphic to the variety V := V (Fq). (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, Theorem 22.1] [5].) Let B(VC, , σ) be the twisted homogeneous coordinate ring of L projective variety VC, where is the invertible sheaf of linear forms on VC L and σ an automorphism of VC, see [Stafford & van den Bergh 2001, p. 180] [14] for the details. The norm-closure of a self-adjoint representation of the ring B(VC, , σ) by the bounded linear operators on a Hilbert space L is a C∗-algebra, see e.g. [Murphy 1990] [7] for an introduction; we call H ∗ ∗ it a Serre C -algebra of VC and denote by . Let be the C -algebra of AV K all compact operators on . We shall write τ : R to denote H AV ⊗K → the canonical normalized trace on , i.e. a positive linear functional AV ⊗K of norm 1 such that τ(yx) = τ(xy) for all x, y V , see [Blackadar 1986, p. 31] [2]. Because is a crossed product∈ AC∗-algebra⊗K of the form AV V ∼= C(VC) ⋊ Z, one can use the Pimsner-Voiculescu six term exact se- quenceA for the crossed products, see e.g. [Blackadar 1986, p. 83] [2] for the details. Thus one gets the short exact sequence of the algebraic K-groups: i∗ 0 K0(C(VC)) K0( V ) K1(C(VC)) 0, where map i∗ is induced by → → A → → 0 the natural embedding of C(VC) into V . We have K0(C(VC)) = K (VC) and −1 0 A −1 ∼ K1(C(VC)) ∼= K (VC), where K and K are the topological K-groups of VC, see [Blackadar 1986, p. 80] [2]. By the Chern character formula, one gets 0 even −1 odd even K (VC) Q = H (VC; Q) and K (VC) Q = H (VC; Q), where H odd ⊗ ∼ ⊗ ∼ (H ) is the direct sum of even (odd, resp.) cohomology groups of VC. No- tice that K0( V ) ∼= K0( V ) because of a stability of the K0-group with respect to tensorA ⊗K products byA the algebra , see e.g. [Blackadar 1986, p. K 32] [2]. One gets the commutative diagram in Fig. 1, where τ∗ denotes a ∗ homomorphism induced on K0 by the canonical trace τ on the C -algebra even n 2i odd n 2i−1 V . Since H (VC) := i=0H (VC) and H (V ) := i=1H (VC), A ⊗K ⊕ ⊕ i one gets for each 0 i 2n an injective homomorphism τ : H (VC) R. ≤ ≤ ∗ −→ i Definition 1 By an i-th trace cohomology group Htr(V ) of variety V one understands the abelian subgroup of R defined by the map τ∗. i Notice that each endomorphism of Htr(V ) is given by a real number ω, such 2 even i∗ odd H (VC) Q K ( ) Q H (VC) Q ⊗ −→ 0 AV ⊗K ⊗ −→ ⊗ ❍ ✟ ❍ ✟ ❍ τ∗ ✟ ❍❍❥ ✟✙✟ ❄ R Figure 1: The trace cohomology. i i i that ωHtr(V ) Htr(V ); thus the ring End (Htr(V )) of all endomorphisms of i ⊆ i Htr(V ) is commutative. The End (Htr(V )) is a commutative subring of the i i ring End (H (VC)) of all endomorphisms of the cohomology group H (VC). Moreover, each regular map f : V V corresponds to an algebraic map → i fC : VC VC and, therefore, to an endomorphism ω End (Htr(V )). On → i ∈ i the other hand, it is easy to see that End (Htr(V )) = Z or End (Htr(V )) Q i ∼ i ⊗ is an algebraic number field. In the latter case Htr(V ) End (Htr(V )) Q, see [Manin 2004, Lemma 1.1.1] [6] for the case of quadratic⊂ fields. We s⊗hall write tr (ω) to denote the trace of an algebraic number ω End (Hi (V )). ∈ tr Our main results are as follows. Theorem 1 The cardinality of variety V (Fq) is given by the formula: 2n−1 V (F ) =1+ qn + ( 1)i tr (ω ), (1) | q | − i Xi=1 where ω End (Hi (V )) is generated by the Frobenius map of V (F ). i ∈ tr q |V (F r )| ∞ q r F Theorem 2 The zeta function ZV (t) := exp r=1 r t of V ( q) has the following properties: P P1(t)...P2n−1(t) (i) ZV (t)= is a rational function; P0(t)...P2n(t) χ(VC) 1 n χ(VC) (ii) Z (t) satisfies the functional equation Z = q 2 t Z (t), V V qnt ± V where χ(VC) is the Euler-Poincar´echaracteristic of VC; i (iii) deg Pi(t) = dim H (VC). Remark 1 Roughly speaking, theorem 2 says that the standard properties of the trace cohomology imply all Weil’s conjectures, except for an analog of i the Riemann hypothesis αij = q 2 for the roots αij of polynomials Pi(t), see [Weil 1949, p. 507] [15]; the| | latter property is proved in [10]. 3 The article is organized as follows. Section 2 contains some useful definitions and notation. Theorems 1 and 2 are proved in Section 3. We calculate the trace cohomology for the algebraic (and elliptic, in particular) curves in Section 4. 2 Preliminaries In this section we briefly review the twisted homogeneous coordinate rings and the Serre C∗-algebras associated to projective varieties, see [Artin & van den 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 [Murphy 1990] [7] and [Blackadar 1986] [2], respectively. The Serre C∗-algebras were introduced in [9]. 2.1 Twisted homogeneous coordinate rings Let V be a projective scheme over a field k, and let be the invertible sheaf (1) of linear forms on V . Recall, that the homogeneousL coordinate ring OV of V is a graded k-algebra, which is isomorphic to the algebra B(V, )= H0(V, ⊗n). (2) L L nM≥0 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 Mn = 0 for n >> 0, then the graded module M is called right bounded. The direct limit M = lim Mα is called a torsion, if each Mα is a right bounded graded module. Denote by Tors the full subcategory of Mod of the torsion modules. 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 along σ will be denoted by σ, i.e. σ(U) := (σU) for every U V . TheL graded k-algebra L L L ⊂ n−1 B(V, , σ)= H0(V, σ ... σ ). (3) L L⊗L ⊗ ⊗L nM≥0 is called a twisted homogeneous coordinate ring of scheme V ; notice that such a ring is non-commutative, unless σ is the trivial automorphism. The 4 σm multiplication of sections is defined by the rule ab = a b , whenever a Bm and b B . Given a pair (V, σ) consisting of a Noetherian⊗ scheme V and∈ an ∈ n automorphism σ of V , an invertible sheaf on V is called σ-ample, if for every n− coherent sheaf on V , the cohomology groupL Hq(V, σ ... σ 1 ) vanishes for q >F 0 and n >> 0. Notice, that if σ is trivial,L⊗L ⊗ this⊗L definition⊗F is equivalent to the usual definition of ample invertible sheaf [Serre 1955] [11]. A non-commutative generalization of the Serre theorem is as follows.