aa r X i v : . [ m a t h . OA ] O c t On traces of Frobenius endomorphisms
Igor Nikolaev ∗ Abstract
We compute the number of points of projective variety V over afinite field in terms of invariants of the so-called Serre C ∗ -algebra of V . Key words and phrases: projective variety, C ∗ -algebraMSC: 14G15 (finite fields); 46L85 (noncommutative topology) The number of solutions of a system of polynomial equations over a finitefield is an important invariant of the system and an old problem dating backto Gauss. Recall that if F q is a field with q = p r elements and V ( F q ) a smooth n -dimensional projective variety over F q , then one can define a zeta function Z ( V ; t ) := exp (cid:16)P ∞ r =1 | V ( F q r ) | t r r (cid:17) ; the function is rational, i.e. Z ( V ; t ) = P ( t ) P ( t ) . . . P n − ( t ) P ( t ) P ( t ) . . . P n ( t ) , (1)where P ( t ) = 1 − t , P n ( t ) = 1 − q n t and for each 1 ≤ i ≤ n − P i ( t ) ∈ Z [ t ] can be written as P i ( t ) = Q deg P i ( t ) j =1 (1 − α ij t ) so that α ij are algebraic integers with | α ij | = q i , see e.g. [Hartshorne 1977] [7],pp. 454-457. The P i ( t ) can be viewed as characteristic polynomial of theFrobenius endomorphism F r iq of the i -th ℓ -adic cohomology group H i ( V ); ∗ Partially supported by NSERC. V ( F q ) according to the formula ( a , . . . , a n ) ( a q , . . . , a qn ). (We assumethroughout the Standard Conjectures, see [Grothendieck 1968] [5].) If V ( F q )is defined by a system of polynomial equations, then the number of solutionsof the system is given by the formula: | V ( F q ) | = n X i =0 ( − i tr ( F r iq ) , (2)where tr is the trace of Frobenius endomorphism loc. cit .Let B ( V, L , σ ) be the twisted homogeneous coordinate ring of an n -dimensional projective variety V over a field k , where L is the invertiblesheaf of linear forms on V and σ an automorphism of V , see [Stafford & vanden Bergh 2001] [14], p. 180 for the notation and details. Denote by A V the Serre C ∗ -algebra of V , i.e. the norm-closure of a self-adjoint representationof B ( V, L , σ ) by linear operators on a Hilbert space H . Consider a stable C ∗ -algebra of A V , i.e. the C ∗ -algebra A V ⊗ K , where K is the C ∗ -algebraof compact operators on H . Let τ : A V ⊗ K → R be the unique normalizedtrace (tracial state) on A V ⊗ K , i.e. a positive linear functional of norm 1such that τ ( yx ) = τ ( xy ) for all x, y ∈ A V ⊗ K , see [Blackadar 1986] [1], p.31. Recall that A V is the crossed product C ∗ -algebra of the form A V ∼ = C ( V ) ⋊ Z , where C ( V ) is the commutative C ∗ -algebra of complex valuedfunctions on V and the product is taken by an automorphism of algebra C ( V )induced by the map σ : V → V [Nikolaev 2012] [arXiv:1208.2049]. From thePimsner-Voiculescu six term exact sequence for crossed products, one gets theshort exact sequence of algebraic K -groups: 0 → K ( C ( V )) i ∗ → K ( A V ) → K ( C ( V )) →
0, where map i ∗ is induced by an embedding of C ( V ) into A V ,see [Blackadar 1986] [1], p. 83 for the details. We have K ( C ( V )) ∼ = K ( V )and K ( C ( V )) ∼ = K − ( V ), where K and K − are the topological K -groupsof variety V , see [Blackadar 1986] [1], p. 80. By the Chern character formula,one gets K ( V ) ⊗ Q ∼ = H even ( V ; Q ) and K − ( V ) ⊗ Q ∼ = H odd ( V ; Q ), where H even ( H odd ) is the direct sum of even (odd, resp.) cohomology groups of V . (Notice that K ( A V ⊗ K ) ∼ = K ( A V ) because of stability of the K -groupwith respect to tensor products by the algebra K , see e.g. [Blackadar 1986][1], p. 32.) Thus one gets the following commutative diagram:2 ❍❍❍❍❍❥ ✟✟✟✟✟✙ H even ( V ) ⊗ Q i ∗ −→ K ( A V ⊗ K ) ⊗ Q −→ H odd ( V ) ⊗ Q τ ∗ R where τ ∗ is a homomorphism induced on K by the canonical trace τ onthe C ∗ -algebra A V ⊗ K . Because H even ( V ) := ⊕ ni =0 H i ( V ) and H odd ( V ) := ⊕ ni =1 H i − ( V ), one gets for each 0 ≤ i ≤ n an injective homomorphism H i ( V ) → R (3)and we shall denote by Λ i an additive abelian subgroup of real numbersdefined by the homomorphism. The Λ i is known as a pseudo-lattice , see[Manin 2004] [10], Section 1.Recall that endomorphisms of a pseudo-lattice are given as multiplicationof points of Λ i by the real numbers α such that α Λ i ⊆ Λ i . It is known that End (Λ i ) ∼ = Z or End (Λ i ) ⊗ Q is a real algebraic number field such thatΛ i ⊂ End (Λ i ) ⊗ Q , see e.g. [Manin 2004] [10], Lemma 1.1.1 for the case ofquadratic fields. We shall write ε i to denote the unit of the order in the field K i := End (Λ i ) ⊗ Q , which induces the shift automorphism of Λ i , see [Effros1981] [4], p. 38 for the details and terminology.Let p be a good prime and V ( F q ) a reduction of V modulo q = p r . Con-sider a sub-lattice Λ qi of Λ i of the index q ; by an index of the sub-lattice weunderstand its index as an abelian subgroup of Λ i . We shall write π i ( q ) todenote an integer, such that multiplication by ε π i ( q ) i induces the shift auto-morphism of Λ qi . The trace of an algebraic number will be written as tr ( • ).Our main result relates invariants ε i and π i ( q ) of the C ∗ -algebra A V to thecardinality of the set V ( F q ). Theorem 1 | V ( F q ) | = P ni =0 ( − i tr (cid:16) ε π i ( q ) i (cid:17) . The article is organized as follows. Theorem 1 is proved in Section 2. Someexamples can be found in Section 3. 3
Proof of theorem 1
We shall split the proof in a series of lemmas starting with the followingwell-known
Lemma 1
There exists a symplectic unitary matrix Θ iq ∈ Sp ( deg P i ; R ) ,such that: F r iq = q i Θ iq . (4) Proof.
Recall that the eigenvalues of
F r iq have absolute value q i ; they come inthe complex conjugate pairs. On the other hand, symplectic unitary matricesin group Sp ( deg P i ; R ) are known to have eigenvalues of absolute value 1coming in complex conjugate pairs. Since the spectrum of a matrix definesthe similarity class of matrix, one can write the characteristic polynomial of F r iq in the form: P i ( t ) = det ( I − q i Θ iq t ) , (5)where matrix Θ iq ∈ Sp ( deg P i ; Z ) and its eigenvalues have absolute value 1.It remains to compare (5) with the formula: P i ( t ) = det ( I − F r iq t ) , (6)i.e. F r iq = q i Θ iq . Lemma 1 follows. (cid:3) Lemma 2
Using a symplectic transformation one can bring matrix Θ iq tothe block form: Θ iq = (cid:18) A I − I (cid:19) , (7) where A is a positive symmetric and I the identity matrix.Proof. Let us write Θ iq in the block form:Θ iq = (cid:18) A BC D (cid:19) , (8)where matrices A, B, C, D are invertible and their transpose A T , B T , C T , D T satisfy the symplectic equations: A T D − C T B = I,A T C − C T A = 0 ,B T D − D T B = 0 . (9)4ecall that symplectic matrices correspond to the linear fractional transfor-mations τ Aτ + BCτ + D of the Siegel half-space H n = { τ = ( τ j ) ∈ C n ( n +1)2 | ℑ ( τ j ) > } consisting of symmetric n × n matrices, see e.g. [Mumford 1983] [11], p.173. One can always multiply the nominator and denominator of such atransformation by B − without affecting the transformation; thus with noloss of generality, we can assume that B = I .We shall consider the symplectic matrix T and its inverse T − given bythe formulas: T = (cid:18) I D I (cid:19) and T − = (cid:18) I − D I (cid:19) . (10)It is verified directly, that T − Θ iq T = (cid:18) I − D I (cid:19) (cid:18)
A IC D (cid:19) (cid:18) I D I (cid:19) = (cid:18) A + D IC − DA (cid:19) . (11)The system of equations (9) with B = I implies the following two equa-tions: A T D − C T = I and D = D T . (12)Applying transposition to the both parts of the first equation of (12), onegets ( A T D − C T ) T = I T and, therefore, D T A − C = I . But the second of(12) says that D T = D ; thus one arrives at the equation DA − C = I . Thelatter gives us C − DA = − I , which we substitute in (11) and get (in a newnotation) the conclusion of lemma 2.Finally, the middle of equations (9) with C = − I implies A = A T , i.e. A is a symmetric matrix. Since the eigenvalues of symmetric matrix are alwaysreal and in view of tr ( A ) > tr ( F r iq ) > A is similar to a positive matrix, see e.g. [Handelman 1981] [6], Theorem 1.Lemma 2 follows. (cid:3) Lemma 3
The symplectic unitary transformation Θ iq of H i ( V ; Z ) descendsto an automorphism of Λ i given by the matrix: M iq = (cid:18) A II (cid:19) . (13) Proof.
Since Λ i ⊂ K i there exists a basis of Λ i consisting of algebraic num-bers; denote by ( µ , . . . , µ k ; ν , . . . , ν k ) a basis of Λ i consisting of positivealgebraic numbers µ i > ν i >
0. Using the injective homomorphism τ ∗ iq given by (7) to an automorphism of Λ i sothat (cid:18) µ ′ ν ′ (cid:19) = (cid:18) A I − I (cid:19) (cid:18) µν (cid:19) = (cid:18) Aµ + ν − µ (cid:19) , (14)where µ = ( µ , . . . , µ k ) and ν = ( ν , . . . , ν k ). Because vectors µ and ν consistof positive entries and A is a positive matrix, it is immediate that µ ′ = Aµ + ν > ν ′ = − µ < µ , . . . , µ k ; ν , . . . , ν k ) of Λ i byan algebraic unit λ > K i ; in particular, any such an automorphismmust be given by a non-negative matrix, whose Perron-Frobenius eigenvaluecoincides with λ . Thus for any automorphism of Λ i it must hold µ ′ > ν ′ > i given by matrix M iq = ( A, I, I, M iq it holds µ ′ = Aµ + ν > ν ′ = µ > M iq is a non-negative matrix satisfying the necessary condition tobelong to the Markov category. It is also a sufficient one, because the similar-ity class of M iq contains a representative whose Perron-Frobenius eigenvectorcan be taken for a basis ( µ, ν ) of Λ i . This argument finishes the proof oflemma 3. (cid:3) Corollary 1 tr ( M iq ) = tr (Θ iq ) .Proof. This fact is an implication of formulas (7) and (13) and a directcomputation tr ( M iq ) = tr ( A ) = tr (Θ iq ). (cid:3) Definition 1
We shall call q i M iq a Markov endomorphism of Λ i and denoteit by M k iq . Lemma 4 tr ( M k iq ) = tr ( F r iq ) .Proof. Corollary 1 says that tr ( M iq ) = tr (Θ iq ), and therefore: tr ( M k iq ) = tr ( q i M iq ) = q i tr ( M iq ) == q i tr (Θ iq ) = tr ( q i Θ iq ) = tr ( F r iq ) . (15)In words, Frobenius and Markov endomorphisms have the same trace, i.e. tr ( M k iq ) = tr ( F r iq ). (cid:3) emark 1 Notice that, unless i or r are even, neither Θ iq nor M iq are integermatrices; yet F r iq and M k iq are always integer matrices. Lemma 5
There exists an algebraic unit ω i ∈ K i such that:(i) ω i corresponds to the shift automorphism of an index q sub-lattice ofpseudo-lattice Λ i ;(ii) tr ( ω i ) = tr ( M k iq ) .Proof. To prove lemma 5, we shall use the notion of a stationary dimensiongroup and the corresponding shift automorphism; we refer the reader to[Effros 1981] [4], p. 37 and [Handelman 1981] [6], p.57 for the notation anddetails on stationary dimension groups and a survey of [Wagoner 1999] [15]for the general theory of subshifts of finite type.Consider a stationary dimension group, G ( M k iq ), generated by the Markovendomorphism M k iq : Z b i Mk iq → Z b i Mk iq → Z b i Mk iq → . . . , (16)where b i = deg P i ( t ). Let λ M be the Perron-Frobenius eigenvalue of matrix M iq . It is known, that G ( M k iq ) is order-isomorphic to a dense additive abeliansubgroup Z [ λ M ] of R ; here Z [ x ] is the set of all polynomials in one variablewith the integer coefficients.Let d M k iq be a shift automorphism of G ( M k iq ) [Effros 1981] [4], p. 37.To calculate the automorphism, notice that multiplication of Z [ λ M ] by λ M induces an automorphism of dimension group Z [ λ M ]. Since the determinantof matrix M iq (i.e. the degree of Markov endomorphism) is equal to q n , oneconcludes that such an automorphism corresponds to a unit of the endomor-phism ring of a sub-lattice of Λ i of index q n . We shall denote such a unitby ω i . Clearly, ω i generates the required shift automorphism d M k iq throughmultiplication of dimension group Z [ λ M ] by the algebraic number ω i . Item(i) of lemma 5 follows.Consider the Artin-Mazur zeta function of M k iq : ζ Mk iq ( t ) = exp ∞ X k =1 tr h ( M k iq ) k i k t k (17)7nd such of d M k iq : ζ d Mk iq ( t ) = exp ∞ X k =1 tr h ( d M k iq ) k i k t k . (18)Since M k iq and d M k iq are shift equivalent matrices, one concludes that ζ Mk iq ( t ) ≡ ζ d Mk iq ( t ), see [Wagoner 1999] [15], p. 273. In particular, tr ( M k iq ) = tr ( d M k iq ) . (19)But tr ( d M k iq ) = tr ( ω i ), where on the right hand side is the trace of analgebraic number. In view of (19) one gets the conclusion of item (ii) oflemma 5. (cid:3) Lemma 6
There exists a positive integer π i ( q ) , such that: ω i = ε π i ( q ) i , (20) where ε i ∈ End (Λ i ) is the fundamental unit corresponding to the shift auto-morphism of pseudo-lattice Λ i .Proof. Given an automorphism ω i of a finite-index sub-lattice of Λ i onecan extend ω i to an automorphism of entire Λ i , since ω i Λ i = Λ i . Thereforeeach unit of (endomorphism ring of) a sub-lattice is also a unit of the hostpseudo-lattice. Notice that the converse statement is false in general.By virtue of the Dirichlet Unit Theorem each unit of End (Λ i ) is a prod-uct of a finite number of (powers of) fundamental units of End (Λ i ). Weshall denote by π i ( q ) the least positive integer, such that ε π i ( q ) i is the shiftautomorphism of a sub-lattice of index q of pseudo-lattice Λ i . The number π i ( q ) exists and uniquely defined, albeit no general formula for its calculationis known, see remark 2. It is clear from construction, that π i ( q ) satisfies theclaim of lemma 6. (cid:3) Remark 2
No general formula for the number π i ( q ) as a function of q isknown; however, if the rank of Λ i is two (i.e. n = 1), then there are classicalresults recorded in e.g. [Hasse 1950] [8], p.298. See also examples section ofthis paper.Theorem 1 follows from formula (2) and lemmas 4-6. (cid:3) Examples
We shall consider two sets of examples illustrating theorem 1; both dealwith non-singular elliptic curves defined over the field of complex numbers.We shall assume that E τ is such a curve isomorphic to a complex torus, i.e. E τ = C / ( Z + Z τ ), where τ ∈ H = { x + iy ∈ C | y > } is a complex modulus,see e.g. [Hartshorne 1977] [7], p. 326. The Serre C ∗ -algebra A E τ of ellipticcurve E τ is known to be isomorphic to the so-called noncommutative torus A θ with the unit scaled by a constant 0 < log µ < ∞ , where θ is irrational and µ a positive real number [Nikolaev 2011] [arXiv:1109.6688]; we refer the readerto [Rieffel 1990] [12] for the definition and properties of noncommutative tori.It is known that K ( A θ ) = K ( A θ ) ∼ = Z so that the canonical trace τ on A θ provides us with the following useful formula: τ ∗ ( K ( A E τ ⊗ K )) = µ ( Z + Z θ ) . (21)Because H ( E τ ; Z ) = H ( E τ ; Z ) ∼ = Z while H ( E τ ; Z ) ∼ = Z , one gets frominjective homomorphism (3) the following collection of pseudo-lattices:Λ = Λ ∼ = Z and Λ ∼ = µ ( Z + Z θ ) . (22) To get an arithmetic variety, we shall assume that E τ has complex multi-plication; the multiplication is characterized by the endomorphism ring of E τ which is an order of conductor f ≥ Q ( √− D ), see e.g. [Hartshorne 1977] [7], p. 330. In this case τ ∈ Q ( √− D ) E τ ∼ = E ( k ), where k is the Hilbert class field (i.e. maximal abelian extension)of the field Q ( √− D ). For such a curve formulas (22) will depend on f and D : Λ = Λ ∼ = Z and Λ = ε [ Z + ( f ω ) Z ] , (23)where ω = (1+ √ D ) if D ≡ mod D = 1 or ω = √ D if D ≡ , mod ε > Z + ( f ω ) Z . As expected, Λ ⊂ K ,where K is the real quadratic field Q ( √ D ).Let p be a good prime. Consider a localization E ( F p ) of curve E ( k ) atthe prime ideal P over p . It is well known, that the Frobenius endomor-phism of elliptic curve with complex multiplication is defined by the so-calledGr¨ossencharacter, which is essentially a complex number α P ∈ Q ( √− D ) of9bsolute value √ p ; multiplication of the lattice L CM = Z + Z τ by α P inducesthe Frobenius endomorphism F r p on H ( E ( k ); Z ), see e.g. [Silverman 1994][13], p. 174. Thus one arrives at the following matrix form for the Frobenius& Markov endomorphisms and the shift automorphism, respectively: F r p = (cid:18) tr ( α P ) p − (cid:19) ,M k p = (cid:18) tr ( α P ) p (cid:19) , \ M k p = (cid:18) tr ( α P ) 11 0 (cid:19) . (24)To calculate positive integer π ( p ) appearing in theorem 1, denote by (cid:16) Dp (cid:17) the Legendre symbol of D and p . A classical result of the theory of realquadratic fields asserts that π ( p ) must be one of the divisors of the integernumber: p − Dp ! , (25)see e.g. [Hasse 1950] [8], p. 298. Thus the trace of Frobenius endomorphismon H ( E ( p ); Z ) is given by the formula: tr ( α P ) = tr ( ε π ( p ) ) , (26)where ε is taken from (23). The right hand side of (26) can be furthersimplified, since tr ( ε π ( p ) ) = 2 T π ( p ) (cid:20) tr ( ε ) (cid:21) , (27)where T π ( p ) ( x ) is the Chebyshev polynomial (of the first kind) of degree π ( p ). Thus one obtains a formula for the number of (projective) solutionsof a cubic equation over field F p in terms of invariants of pseudo-lattice Λ : |E ( F p ) | = 1 + p − T π ( p ) (cid:20) tr ( ε ) (cid:21) . (28) Suppose that b ≥ y z = x ( x − z ) x − b − b + 2 z ! . (29)10he Serre C ∗ -algebra of projective variety (29) is isomorphic (modulo anideal) to the so-called Cuntz-Krieger algebra O B , where B = (cid:18) b − b − (cid:19) (30)is a positive integer matrix [Nikolaev 2012] [arXiv:1201.1047]; for the defini-tion and properties of algebra O B we refer the reader to [Cuntz & Krieger1980] [3].Recall that O B ⊗ K is the crossed product C ∗ -algebra of a stationary AF C ∗ -algebra by its shift automorphism, see [Blackadar 1986] [1], p. 104; theAF C ∗ -algebra has the following dimension group: Z B T → Z B T → Z B T → . . . , (31)where B T is the transpose of matrix B . Because µ in formula (22) must bea positive eigenvalue of matrix B T , one gets: µ = 2 − b + √ b − . (32)Likewise, since θ in formula (22) must be the corresponding positive eigen-vector (1 , θ ) of the same matrix, one gets: θ = 12 s b + 2 b − − . (33)Therefore, pseudo-lattices Λ i are Λ = Λ ∼ = Z andΛ ∼ = 2 − b + √ b − Z + 12 s b + 2 b − − Z . (34)As expected, pseudo-lattice Λ ⊂ K , where K = Q ( √ b −
4) is a realquadratic field.Let p be a good prime and let E ( F p ) be the reduction of curve (29) modulo p . Using the same argument as in Section 3.1, we determine number π ( p )as one of the divisors of integer number: p − b − p ! . (35)11nlike the case of complex multiplication, the Gr¨ossencharacter is no longeravailable for elliptic curve (29); yet the trace of Frobenius endomorphism canbe computed using theorem 1: tr ( F r p ) = tr h ( B T ) π ( p ) i . (36)Using the Chebyshev polynomials, one can write (36) in the form: tr ( F r p ) = 2 T π ( p ) (cid:20) tr ( B T ) (cid:21) . (37)In view of (30) one gets tr ( B T ) = b , so that (37) takes the form: tr ( F r p ) = 2 T π ( p ) b ! . (38)Thus one obtains a formula for the number of solutions of equation (29) overfield F p in terms of invariants of pseudo-lattice Λ : |E ( F p ) | = 1 + p − T π ( p ) b ! . (39)We conclude by an example comparing formula (39) with the known resultsfor a rational elliptic curve in the Legendre form, see e.g. [Hartshorne 1977][7], p. 333 and [Kirwan 1992] [9], pp. 49-50. Example 1
Suppose that b ≡ mod
4. Recall that the j -invariant takesthe same value on λ , 1 − λ and λ , see e.g. [Hartshorne 1977] [7], p. 320.Therefore, one can bring (29) to the form: y z = x ( x − z )( x − λz ) , (40)where λ = ( b + 2) = 2 , , , . . . Notice that for curve (40): tr ( B T ) = b = 2(2 λ − . (41)To calculate (39) for elliptic curve (40), recall that in view of (38) one gets: tr ( F r p ) = 2 T π ( p ) (2 λ − . (42)12t will be useful to express Chebyshev polynomial T π ( p ) (2 λ −
1) in terms ofthe hypergeometric function F ( a, b ; c ; z ); the standard formula brings (42)to the form: tr ( F r p ) = 2 F ( − π ( p ) , π ( p ); 12 ; 1 − λ ) . (43)We leave to the reader to prove the identity:2 F ( − π ( p ) , π ( p ); ; 1 − λ ) == ( − π ( p ) 2 F ( π ( p ) + 1 , π ( p ) + 1; 1; λ ) . (44)In the last formula: F ( π ( p ) + 1 , π ( p ) + 1; 1; λ ) = π ( p ) X r =0 (cid:18) π ( p ) r (cid:19) λ r , (45)see [Carlitz 1966] [2], p.328.Recall that π ( p ) is a divisor of (35), which in our case takes the value p − . Bringing together formulas (39), (43)-(45) one gets: |E ( F p ) | = 1 + p + ( − p − p − X r =0 (cid:18) p − r (cid:19) λ r , (46)compare with [Kirwan 1992] [9], pp. 49-50 and [Hartshorne 1977] [7], p. 333for a relation with the Hasse invariant. References [1] B. Blackadar, K -Theory for Operator Algebras, MSRI Publications,Springer, 1986[2] L. Carlitz, Some binomial coefficient identities, Fibonacci Quart. 4(1966), 323-331.[3] J. Cuntz and W. Krieger, A class of C ∗ -algebras and topologicalMarkov chains, Invent. Math. 56 (1980), 251-268.[4] E. G. Effros, Dimensions and C ∗ -Algebras, in: Conf. Board of theMath. Sciences, Regional conference series in Math., No.46, AMS, 1981.135] A. Grothendieck, Standard conjectures on algebraic cycles, in: Alge-braic Geometry, Internat. Colloq. Tata Inst. Fund. Res., Bombay, 1968.[6] D. Handelman, Positive matrices and dimension groups affiliated to C ∗ -algebras and topological Markov chains, J. Operator Theory 6 (1981),55-74.[7] R. Hartshorne, Algebraic Geometry, GTM 52, Springer, 1977.[8] H. Hasse, Vorlesungen ¨uber Zahlentheorie, Springer, 1950.[9] F. Kirwan, Complex Algebraic Curves, LMS Student Texts 23, Cam-bridge, 1992.[10] Yu. I. Manin, Real multiplication and noncommutative geometry, in“Legacy of Niels Hendrik Abel”, 685-727, Springer, 2004.[11] D. Mumford, Tata Lectures on Theta I, Birkh¨auser, 1983.[12] M. A. Rieffel, Non-commutative tori – a case study of non-commutativedifferentiable manifolds, Contemp. Math. 105 (1990), 191-211. Avail-able http://math.berkeley.edu/ ∼ rieffel/ [13] J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves,GTM 151, Springer 1994.[14] J. T. Stafford and M. van den Bergh, Noncommutative curves andnoncommutative surfaces, Bull. Amer. Math. Soc. 38 (2001), 171-216.[15] J. B. Wagoner, Strong shift equivalence theory and the shift equivalenceproblem, Bull. Amer. Math. Soc. 36 (1999), 271-296. The Fields Institute for Research in Mathematical Sciences,Toronto, ON, Canada, E-mail: [email protected]