ALGEBRAIC FLOWS ON ABELIAN VARIETIES.

Transcription

1 ALGEBRAIC FLOWS ON ABELIAN VARIETIES. E. ULLMO, A. YAFAEV Table of contents. 1. Introduction. 1 Acknowledgements Mumford-Tate and Asymptotic Mumford-Tate Tori Mumford-Tate Tori Asymptotic Mumford-Tate tori for curves Weyl criterion and Mumford-Tate tori Weyl Criterion Characters of the Mumford-Tate torus Characters of the asymptotic Mumford-Tate Tori of the branches of a curve Local computations for oscillatory integrals in the complex plane Proof of the Theorem Examples Complex geodesic flows on abelian varieties The curve Z 1 Z 2 = Hyperelliptic curves in C Bibliography Introduction. The purpose of this paper is to study algebraic flows on abelian varieties. We formulate two (interrelated) conjectures, prove a rather general statement concerning these conjectures and provide examples where our statement applies. Let A be a complex abelian variety of dimension g. Let Γ be a lattice in C g such that A C g /Γ and π : C g A C g /Γ be the uniformizing map. A complex algebraic subvariety Z of A is said to be weakly special if Z = P + B for an abelian subvariety B and 1

2 2 E. ULLMO, A. YAFAEV a point P of A. If moreover P is a torsion point then Z = B + P is said to be special. The following abelian version of Ax-Lindemann theorem is due to Ax (see [1] and [2]) and plays an important role in the new proof by Pila and Zannier of the Manin-Mumford conjecture [6]. Note that the paper [6] provides a different proof of the abelian Ax-Lindemann theorem. Yet another proof was given by Orr (see [5]). Theorem 1.1. (Abelian Ax-Lindemann) Let Θ be a complex algebraic subvariety of C g. Then any component of the Zariski closure of π(θ) is weakly special. In this paper, instead of looking at the Zariski closure of π(θ), we attempt to characterise the usual topological closure π(θ) of π(θ). Let W C g be a R-vector space such that Γ W := Γ W is a lattice in W. Then W/Γ W is a real torus and is a closed real analytic subset of A. A real analytic subvariety Z of A is said to be real weakly special if Z = P + W/Γ W for a point P and a real subtorus W/Γ W of A. A real affine subspace of C g R 2g of the form W + Q where Q is a point in C g and W is a R-vector subspace of C g R 2g is said to be real weakly pre-special if π(q) + π(w ) is real weakly special. By definition Q + W is real weakly pre-special if and only if W Γ is a lattice in W. We will study the following conjecture. Conjecture 1.2. Let A be a abelian variety of dimension g. Let Θ be a complex irreducible algebraic subvariety of C g. Then there exists a finite number Z 1,..., Z r of real weakly special subvarieties of A such that r π(θ) = π(θ) Z k. Let Z = W/Γ W be a real subtorus of A. Then the normalized Lebesgue measure on W induces a canonical probability measure µ Z on A. The support Supp(µ Z ) of µ Z is Z. Using the description of a real weakly special subvariety Z of A of the form Z = P + W/Γ W we see that we have also a canonical probability measure µ Z on A such that Supp(µ Z ) = Z. Let ω be the canonical translation invariant positive (1, 1)-form on C g. We fix a C-basis of C g with associated complex coordinates (z 1,..., z g ) with z k = x k + iy k for 1 k g. Then we may choose ω = i g g dz k dz k = dx k dy k. 2 k=1 k=1 k=1

3 ALGEBRAIC FLOWS ON ABELIAN VARIETIES. 3 For R > 0, we let B(0, R) = {(z 1,..., z g ) C g, z k < R}. By slight abuse of notations we will often identify functions on A with Γ-invariant functions on C g. Assume that Θ is an algebraic subvariety of C g of dimension d. Define, for all R big enough, the probability measure µ Θ,R on C g such that for any continuous function f on C g, µ Θ,R (f) = 1 V R Θ B(0,R) fω d where V R = Θ B(0,R) ωd. The pushforward of µ Θ,R (that we will still, by abuse of notation denote µ Θ,R ) to A induces a probability measure on A with support contained in π(θ). The measure theoretical version of the topological conjecture 1.2 is Conjecture 1.3. Let A be a abelian variety of dimension g. Let Θ be an irreducible complex algebraic subvariety of C g. Then there exists a finite number of real weakly special subvarieties Z 1,..., Z r and some positive real numbers c 1,..., c r such that µ Θ,R converges weakly to the measure r k=1 c kµ Zk. This means that for any continuous function f on A we have r µ Θ,R (f) c k µ Zk (f) as R. k=1 The main result of this paper is the proof of these conjectures for the image π(c) of a curve C in C g. In this case the real weakly special subvarieties Z 1,..., Z r appearing in the statements of the conjectures 1.2 and 1.3 are described simply in terms of Γ and the geometry at infinity of C. The precise version of this result is given in the statement of the theorem 2.4. In the sections 2 and 3 we define the Mumford-Tate torus of an algebraic subvariety of C g. We also define the asymptotic Mumford-Tate torus of an infinite branch of a curve contained in C g. We prove a certain number of properties of these objects that we use later on. We then review charaters of these Mumford-Tate tori and the Weyl criterion. The section 4, which is the main technical part of the proof of the theorem 2.4, concerns itself with the estimation of some oscillatori integrals. After proving the main result in section 5, we provide enlightening examples where the conclusion of the conjecture 1.2 holds. The hyperbolic version of the Ax-Lindemann theorem for Shimura varieties had been proven (see [9], [7] and[4]). The conjectures 1.2 and

4 4 E. ULLMO, A. YAFAEV 1.3 have natural and interesting formulations in this context. These questions will be studied in further works. Acknowledgements. We are extremely grateful to James Wright who provided very detailed answers to our questions regarding oscillatory intergrals. We would like to extend our thanks to the referee for his careful reading and his detailed comments which helped us improve the presentation of the paper. The second author is very grateful to the ERC for financial support. 2. Mumford-Tate and Asymptotic Mumford-Tate Tori Mumford-Tate Tori. Let Θ be an irreducible algebraic subvariety of C g containing the origin O of C g. The Mumford-Tate group MT (Θ) of Θ is defined as the smallest Q-vector subspace W of Γ Q such that Θ W R. More generally, let Θ be an irreducible algebraic subvariety of C g and P Θ. Then we define MT (Θ) as MT (Θ P ). One can check that the definition is independent of the choice of P Θ. Let W Θ := MT (Θ) R and T Θ = W Θ /(W Θ Γ). We denote by T Θ the real weakly-special subvariety of A T Θ = π(p ) + T Θ. Then T Θ is independent of P and T Θ is the smallest real weakly special subvariety of A containing π(θ). Then we have the first natural upper bound for the topological closure of π(θ) in A (1) π(θ) T Θ Asymptotic Mumford-Tate tori for curves. Let C be a curve in C g. Let C be the Zariski closure of C in P 1 (C) g. Then C C is a finite set of points {P 1,..., P s }. For each i {1,..., s} we can fix a neighborhood U i of P i such that the connected components of C U i have a parametrization by power series. A infinite branch C α of C near a point at infinity P i is a complex analytic connected component of C U i. We assume that for all i and j we have U i U j =. Let C α be such a branch. Let (z 1,..., z g ) be the coordinates on C g. Then there exists at least one coordinate z = z i which is unbounded when Z = (z 1,..., z g ) varies in C α. We say that a real weakly pre-special subvariety Q+W is asymptotic to C α if the Euclidean distance from a point Z = (z 1,..., z g ) of C α to Q + W tends to 0 as z = z i tends to. Note that by definition if P is

5 ALGEBRAIC FLOWS ON ABELIAN VARIETIES. 5 a point of C then P + MT (C) R is asymptotic to C α for all infinite branches C α of C. Lemma 2.1. An intersection of two real weakly pre-special subvarieties which are asymptotic to C α is real weakly pre-special and asymptotic to C α. Proof. Let P +W and Q+W be two real weakly pre-special subvarieties which are asymptotic to C α. Then the distance between P + W and Q + W is zero. Therefore (P + W ) (Q + W ) is not empty. Let R (P + W ) (Q + W ). Then P + W = R + W, Q + W = R + W. As a consequence (P + W ) (Q + W ) = R + W W. The lattice Γ defines a Q-structure on C g R 2g. By definition W and W are defined over Q for this Q-structure. Therefore W W is defined over Q and R + W W is real weakly pre-special. Let Z n be a sequence of points of C α tending to the point at infinity. Then there exists P n W, Q n W and two sequences ɛ n and ɛ n of points in C g tending to O such that Z n = R + P n + ɛ n = R + Q n + ɛ n. Therefore we have P n and d(p n, W ) 0 as n. It follows that d(z n, R + W W ) 0 as n. To see this we may write {f i = 0}, i = 1,..., s and {g i = 0}, i = 1,... r the linear equations defining W and W respectively. Then f i (P n ) = 0 and g i (P n ) 0. By extracting from the system {f i = 0, g i = 0} a maximal free subsystem that we call {h i }, i = 1,..., t, we obtain a system of equations defining W W. By the above, h i (P n ) 0. We can make a change of variables (z 1,..., z 2g ) (which does not affect the property of convergence of distances to zero) so that z i = h i for i {1,..., t}. This shows that d(p n, W W ) 0 and therefore d(z n, R + W W ) 0. As the property d(z n, R+W W ) 0 is true for all sequences (Z n ) of points of C α tending to the point at infinity, we get that R +W W is asymptotic to C α. Let P α + W α be the smallest real weakly pre-special subvariety of C g which is asymptotic to C α. Let T α = W α /W α Γ and T α = π(p α )+T α. We say that T α is the asymptotic Mumford-Tate torus of C α. By definition for all α we have the inclusion T α T C. We remark that even if the infinite branches of C depend on the choice of some small enough neighborhood of some point at infinity of the curve, the associated asymptotic Mumford-Tate tori does not depend of this choice.

6 6 E. ULLMO, A. YAFAEV Lemma 2.2. We have π(c α ) π(c α ) T α. Proof. Let P be a point of π(c α ) π(c α ). There exists a sequence Q n of points of C α tending to the point at infinity of C α such that P n := π(q n ) tends to P. By hypothesis d(q n, P α + W α ) tends to 0 as n tends to. Therefore d(p n, T α) tends to 0. As T α is a closed subset of A, we find that P T α. The following proposition summarizes what we have obtained so far. Proposition 2.3. Let C be a curve in C g. Let C 1,..., C r be the set of all branches of C through all points at infinity. Let T 1,..., T r be the associated asymptotic Mumford-Tate tori. Then r π(c) π(c) T α T C. α=1 The main result of the paper is the following theorem. Theorem 2.4. Let C be a curve in C g. Let C 1,..., C r be the set of all branches of C through all points at infinity. Let T 1,..., T r be the associated asymptotic Mumford-Tate tori. (i) We have r π(c) = π(c) T α. (ii) Let µ α be the canonical probability measure on T α. There exists positive real numbers c 1,..., c r such that µ C,R converges weakly to r α=1 c αµ α. The first part of the theorem is a consequence of the proposition 2.3 and the second part of the theorem. In the rest of the paper we will give a proof of the second part of the theorem using some tools from harmonic analysis. Note also that very often at least one of the asymptotic Mumford-Tate tori of C is just the Mumford-Tate torus of C. In this case we have π(c) = T C. General examples of this type will be given in the section 6. α=1 3. Weyl criterion and Mumford-Tate tori Weyl Criterion. Let, be the canonical Hermitian scalar product on C g. For any vectors z, z of C g, we denote by (z, z ) := Re( z, z ) the real part of z, z. Then (, ) is the usual Euclidean product on C g R 2g. For any subset E of R 2g we write E for the orthogonal complement of E with respect to the Euclidean product (, ).

7 ALGEBRAIC FLOWS ON ABELIAN VARIETIES. 7 For any θ C g, we denote by χ θ the function on C g such that χ θ (z) = exp(2πi(θ, z)). Let Γ be the dual lattice of Γ defined as Γ = {θ C g such that for all γ Γ, (θ, γ) Z}. The group of characters X (A) of the abelian variety A viewed as a real torus of dimension 2g is a free Z-module of rank 2g. We denote by 1 A the trivial character of A. We have an isomorphism Γ X (A) given by θ χ θ. The Weyl criterion says that a sequence of probability measures µ n on A converges weakly to a probability measure µ if and only if for all θ Γ we have µ n (χ θ ) µ(χ θ ) as n. Let T = W T /(Γ W T ) be a real subtorus of A. Let X (T) be the set of characters of T. We denote by 1 T the trivial character of T. Let θ Γ. The restriction (χ θ ) T of χ θ to T is a character of T. Let N(T) be the free Z-submodule of X (A) defined by We have an exact sequence N(T) := {χ θ X (A) (χ θ ) T = 1 T }. 0 N(T) X (A) X (T) 0. If we identify Γ and X (A), we can realize N(T) as a subset of C g R 2g. Then N(T) is a lattice in N(T) R W T R 2g and N(T) = Γ (N(T) R). Let T be the real torus A/T. Then T N(T) R/N(T) and we have an identification N(T) = X (T ). Let P be a point of C g. Let T = π(p ) + T. Let µ T be the canonical probability measure on T. Then for all θ in Γ, we have (2) χ θ dµ T = exp(2πi(θ, z))dµ T = exp(2iπ(p, θ)) if θ N(T) and (3) A A A χ θ dµ T = A exp(2πi(θ, z))dµ T = 0 if θ / N(T). As the proof of theorem 2.4 is an application of the Weyl criterion, we will need to understand the group of characters of the Mumford-Tate torus of an algebraic subvariety of C g and of the asymptotic Mumford- Tate tori of a branch of a curve in C g. The relevant results are given in the next two sections.

8 8 E. ULLMO, A. YAFAEV 3.2. Characters of the Mumford-Tate torus. Let Θ be an irreducible subvariety of C g. Let P be a point of Θ. Let W Θ = MT (Θ P ) R and T Θ = W Θ /(Γ W Θ ). The Mumford-Tate torus of Θ is T Θ = π(p ) + T Θ. By the discussion of the previous section we have N(T Θ ) = X (A/T Θ ) = W Θ Γ. We will need the following lemma. Lemma 3.1. We have N(T Θ ) = {χ θ, θ Γ (Θ P ) } Γ (Θ P ). Proof. Let N (T Θ ) = Γ (Θ P ). As Θ P W Θ we get N(T Θ ) N (T Θ ). Moreover N (T Θ ) is a saturated Z-submodule of Γ in the sense that if there exists θ Γ and a non zero integer n such that nθ N (T Θ ) then θ N (T Θ ). As a consequence there is a real subtorus T Θ of A and an exact sequence 0 N (T Θ ) Γ X (A) X (T Θ) 0. By definition π(θ P ) T Θ and as N(T Θ) N (T Θ ) we have T Θ T Θ. By the definition of the Mumford-Tate tori of Θ P we find that T Θ = T Θ and N (T Θ ) = N(T Θ ) Characters of the asymptotic Mumford-Tate Tori of the branches of a curve. Let C be an irreducible algebraic curve in C g. Let C α be an infinite branch of C. Let P α + W α be the smallest real weakly pre-special subvariety which is asymptotic to C α and T α = π(p α ) + T α be the associated asymptotic torus. Let Z = (z 1,..., z g ) be a system of coordinates on C α. We fix a coordinate z = z i which is unbounded as Z varies in C α. For all j {1,..., g} we have a Puiseux expansion of the form z j = n 0 a n,j z α j n e j for some α j Q and some e j N (see [10] ch. 4). The sequence of complex numbers a n,j is such that the power series n 0 a n,jx n has a radius of convergence ρ j > 0. Let a 0 (z j ) be the constant term of z j in this expansion. Note that a 0 (z j ) = 0 if and only if α j n e j 0 for all n N, or if there exists

9 ALGEBRAIC FLOWS ON ABELIAN VARIETIES. 9 n 0 N such that α j n 0 e j = 0 and a n0,j = 0. Let Q α be the point of C g such that Q α = (a 0 (z 1 ),..., a 0 (z g )). Let N α be the Z-submodule of Γ consisting of the θ = (θ 1,..., θ g ) Γ such that the function g ψ θ (Z) = (Z, θ) = Re( θ k z k ) is bounded on C α. Then N α is a saturated submodule of Γ. There exists therefore a real subtorus S α of A and an exact sequence k=1 0 N α Γ X (A) X (S α ) 0. Lemma 3.2. We have T α = S α, N(T α ) = N α and T α = π(q α ) + S α = π(p α ) + S α. Proof. Let θ = (θ 1,..., θ g ) Γ. Let H θ be the real hyperplane of C g R 2g with equation g Re( θ k z k ) = 0. k=1 Then Γ θ := Γ H θ is a lattice in H θ. If there exists a point Q such that C α is asymptotic Q + H θ then the distance d(z, H θ ) is bounded as Z varies in C α. By the computation of the distance of a point to a hyperplane this implies that Re( g k=1 θ kz k ) is bounded when Z varies in C α. If Re( g k=1 θ kz k ) is bounded when Z varies in C α, the coefficients of the positive powers of z in the Puiseux expansion of ψ θ (Z) = Re( are vanishing. Moreover the function g θ k z k ) k=1 ψ θ (Z Q α ) tends to zero as z tends to. As a consequence C α is asymptotic to Q α + H θ and P α + W α Q α + H θ = P α + H θ. If W α C g, then W α is the intersection of the hyperplanes H θ with θ Γ such that C α is asymptotic to P α +H θ. By the previous discussion we have W α = θ N α H θ.

10 10 E. ULLMO, A. YAFAEV This shows that T α = S α = W α /(Γ W α ) and that T α = π(q α ) + S α = π(p α ) + S α. If W α = C g, then there is no hyperplane H θ such that C α is asymptotic to P α + H θ for some point P α. Therefore N α is trivial and S α = C g = T α. 4. Local computations for oscillatory integrals in the complex plane. In this section we give the computations for oscillatory integrals which are relevant for the application of the Weyl criterion in the proof of the theorem 2.4. We consider a function φ of a complex variable z defined for z > A for a suitable real number A. We assume moreover that we have an expansion of the form φ(z) = n 0 a n z α n e where e is a positive integer, 0 α is a rational number and (a n ) n N is a sequence of complex numbers such that the power series a n T n has a positive radius of convergence ρ. If α 0 we assume that a 0 0. By making A larger we may and do assume that A > 1 and if ρ that 1 (4) < ρ 2. A 1 e For a complex number z = r exp(iθ) with θ [0, 2π[ and a positive integer k we define z 1 k as z 1 k = r 1 k exp( iθ). k The purpose of this part is to study the behaviour for large R of the integral (5) J φ (R) = A< z <R idz dz exp(ire(φ(z))). 2R 2 Theorem 4.1. For any function φ satisfying the previous properties (i) If α > 0 then J φ (R) 0 as R. (ii) If α = 0 then J φ (R) π exp(ire(a 0 ))

11 as R. with ALGEBRAIC FLOWS ON ABELIAN VARIETIES. 11 We write z = r exp(iθ), a n = A n exp(iγ n ). Then ψ(r, θ) := Re(φ(z)) = n 0 A n r α n e bn (θ) b n (θ) = cos((α n e )θ + γ n). With these notations we find that J φ (R) = 2π 1 0 A R exp(iψ(rt, θ)) t dt dθ. We fix θ [0, 2π]. For t [ A, 1] we define R f θ,r (t) = ψ(rt, θ) = n 0 A n R α n e t α n e bn (θ). The main input in the proof of the first part of the theorem 4.1 is the Van der Corput lemma that can be stated in the following way. Lemma 4.2. (Van der Corput Lemma.) Let k 2 be an integer, there exists a constant C = C k such that for any real numbers a < b, any functions f C k ([a, b]) such that f (k) (x) > λ > 0 on [a, b] and any function ψ C 1 ([a, b]) (6) b a exp(if(x))ψ(x)dx C k ( ψ L ([a,b]) + ψ L 1 ([a,b]))λ 1 k. If k = 1 and f is monotone the conclusion also holds. A proof of this result is given in [8]. The next lemma summarizes what we need in order to be able to apply the Van der Corput lemma. Lemma 4.3. (i) We have f θ,r(t) = n 0(α n e )A nr α n e t α 1 n e bn (θ) and f θ,r(t) = n 0(α n e )(α 1 n e )A nr α n e t α 2 n e bn (θ). (ii) Assume that α = 1. Let η be a positive real number such that η < 1. There exists a positive constant c 2e 1 = c 1 (η) such that the following holds. For R big enough, for all θ [0, 2π] such that A 0 b 0 (θ) > R η and for t [ A, 1] we have R 1 2 f θ,r(t) > c 1 R 1 η.

12 12 E. ULLMO, A. YAFAEV (iii) Assume α > 0. If α = 1 assume that the smallest positive integer n 0 such that A n0 0 verifies n 0 < e. There exist real numbers β > 0, 1 > δ > 0 and µ > 0 such that for all t [ A, 1] and all 2 R δ θ [0, 2π] such that α(α 1)A 0 b 0 (θ) > R β if α 1 and such that n 0 e (1 n 0 e )A n0 b n0 (θ) > R β if α = 1 we have f θ,r(t) > c 2 R µ for a constant c 2 > 0. (iv) Assume α = 0 or α = 1 and for all k > 0 such that 0 < k < e we have A k = 0. Let n 0 be the smaller positive integer k such that a k 0. There exist real numbers β > 0 and δ > 0 such that for t [ A, 1] and R δ all θ [0, 2π] such that A n0 b n0 (θ) > R β the function f θ,r (t) is non vanishing. Proof. To prove (i) we remark that for t [ A, 1] and n big enough we R have (α n e )A nr α n e t α 1 n e bn (θ) 2n A n RA α 1 ( 1 ) n. The series with general term B n = 2n A n RA α 1 ( 1 A 1 e A 1 e ) n is convergent as the power series A n T n has a convergence radius ρ > 0 and as 1 by hypothesis. This shows the validity of the expression for f θ,r A 1 e < ρ 2 (t). The proof for f θ,r (t) is done in the same way. To prove (ii) we assume that α = 1. We use for t [ A, 1] the upper R 1 2 bound for all n > 0 (1 n e )A nr 1 n e t n e bn (θ) (n + 1) A n R e ( ) n. Let c n := (n + 1) A n. The power series c n T n has a radius of convergence ρ and is therefore bounded for T ρ. By hypothesis 1 < ρ. 2 2 Therefore if R is large enough, as η < 1, if A 2e 0b 0 (θ) > R η and t [ A, 1] we get R 1 2 f θ,r(t) > c 1 (η)r 1 η. For the part (iii) we assume first that 0 < α 1. Let β = min( α 2, 1 4e ). Let δ > 0. If α > 2 assume that (7) δ(α 2) 1 4e < 1. A 1 e A 1 e Assume that α(α 1)A 0 b 0 (θ) > R β.

13 ALGEBRAIC FLOWS ON ABELIAN VARIETIES. 13 Then for all t [ A R δ, 1] we have (8) α(α 1)R α A 0 b 0 (θ)t α 2 R α β if α 2 and (9) α(α 1)R α A 0 b 0 (θ)t α 2 A 2 R α β δ(α 2) if α > 2 Let µ = max( α, α 1 ). Remark that if α 2, then β = 1 and 2 2e 4e µ = α 1. Using the upper bound (7) for δ when α > 2 shows that 2e for all α > 0 with α 1 we have (10) α(α 1)R α A 0 b 0 (θ)t α 2 A 2 R µ. Assume moreover that δ > 0 satisfies 1 (11) 2e < αδ + 1 (1 δ) 2δ, e and δ < 1. Remark that such a δ exists as the limit when δ tends to 2 zero of the righthand side of the equation is 1 > 1. e 2e For all t [ A, 1] and all n 1 such that α 2 n < 0 we have R δ e (α n e )(α 1 n e )A nr α n e t α 2 n e bn (θ) R λ 1 (n+1) 2 A n A α 2 ( 1 ) n. with λ 1 = α(1 δ) 1 (1 δ) + 2δ < µ e by our choice of δ. For all t [ A, 1] and all n 1 such that α 2 n R δ e A 1 e 0 we have (α n e )(α 1 n e )A nr α n e t α 2 n e bn (θ) R λ 2 (n + 1) 2 A n. with λ 2 = α n < µ. Let λ = max(λ e 1, λ 2 ). Then λ < µ. Let d n := (n + 1) 2 A n A α 2. The power series d n T n has a radius of convergence ρ and is therefore bounded for T ρ. By hypothesis 2 1 < ρ. Therefore there exists a positive constant c such that for all A 1 e 2 R big enough, n 1(α n e )(α 1 n e )A nr α n e t α 2 n e bn (θ) cr λ. In view of equation (10) and of the expression for f θ,r (t) obtained in the part (i) of the lemma this finishes the proof of the part (iii) when α 1. The proof of part (iii) when α = 1 is obtained by considering the function g θ,r (t) = f θ,r (t) A 0 Rtb 0 (θ).

14 14 E. ULLMO, A. YAFAEV We have g θ,r (t) = f θ,r (t) and g θ,r(t) has an expansion of the form g θ,r (t) = n 0 A n+n0 R α 1 n e t α 1 n e bn+n0 (θ) with α 1 = 1 n 0 e > 0 and A n0 0. Therefore g θ,r (t) satisfies the hypothesis of the part (iii) with α 1 1. This finishes the proof of the part (iii) of the lemma. The part (iv) of the lemma is proven similarly. We may assume as previously that α = 0 by considering the function g θ,r (t) = f θ,r (t) A 0 Rtb 0 (θ). In the case α = 0 we choose β = 1. Assume that 2e Then for all t [ A, 1] we have R n 0 e (n 0 e + 1)A n 0 b n0 (θ) > R β. (12) n 0 e (n 0 e + 1)A n 0 b n0 (θ)r n 0 e t 2 n 0 e R n 0 β e = R n e. Let δ > 0 such that ( n )δ < β = 1. For all t [ A, 1] and all e 2e R δ n n we have n e (1 + n e )A nr n e t 2 n e bn (θ) R λ (n + 1) 2 A n A 2 ( 1 with λ = n 0+1 (1 δ) + 2δ < n β = n by our choice of δ. e e e Let d n := (n + 1) 2 A n A 2. The power series d n T n has a radius of convergence ρ and is therefore bounded for T ρ. By hypothesis 2 1 < ρ. Therefore there exists a positive constant c such that for all A 1 e 2 R big enough, n e (1 + n e )A nr n e t 2 n e bn (θ) cr λ. This shows that n n 0 +1 f θ,r(t) = n e (1 + n e )A nr n e t 2 n e bn (θ) n n 0 is non vanishing if t [ A R δ, 1]. This finishes the proof of the lemma. We first prove the first part of the theorem 4.1. Let β be the constant defined in part (iii) of the lemma if α 1 and β be the constant defined in part (iii) (resp. part (iv)) if α = 1 and the hypothesis of part (iii) (resp. part (iv)) hold. A 1 e ) n.

15 ALGEBRAIC FLOWS ON ABELIAN VARIETIES. 15 If α 1, let E 1 be the subset of θ [0, 2π[ such that α(α 1)A 0 b 0 (θ) > R β. If α = 1 let n 0 be the integer defined in part (iii) or in part (iv) of the lemma. Let E 1 be the subset of θ [0, 2π[ such that A 0 b 0 (θ) > R β and such that n 0 e (1 n 0 e )A n 0 b n0 (θ) > R β if the hypothesis of part (iii) hold and such that A n0 b n0 (θ) > R β if the hypothesis of part (iv) hold. Let E 2 such that [0, 2π[ is a disjoint union of E 1 and E 2. Then there exists ɛ 1 > 0 such that E 2 is of Lebesgue measure bounded by R ɛ 1 for R big enough. We have with and Jφ(R) 1 = Jφ(R) 2 = J φ (R) = J 1 φ(r) + J 2 φ(r) θ E 1 θ E 2 1 A R 1 A R exp(i(ψ(rt, θ)) t dt dθ. exp(i(ψ(rt, θ)) t dt dθ. Then for all R big enough Jφ 2(R) R ɛ 1 by the previous discussion. Let δ be the constant defined in part (iii) if α 1 or if the hypothesis of part (iii) hold. If α = 1 and the hypothesis of part (iv) hold let δ be the constant defined in part (iv). We then write Jφ(R) 1 = Jφ(R) 3 + Jφ(R) 4 with and Jφ(R) 3 = Jφ(R) 4 = θ E 1 θ E 1 A R δ A R 1 exp(i(ψ(rt, θ)) t dt dθ. exp(i(ψ(rt, θ)) t dt dθ. A R δ Then Jφ,γ 3 A2 (R) < 2π. R 2δ If α 1 or α = 1 and the hypothesis of part (iii) hold, then using the result of part (iii) of the lemma and the Van der Corput lemma with k = 2 and λ = c 2 R µ we find that that for all R big enough J 4 φ(r) R µ 2.

16 16 E. ULLMO, A. YAFAEV Assume finally that α = 1 and the hypothesis of part (iv) hold. Using the result of part (iv), of part (ii) and the Van der Corput lemma with k = 1 and dividing into t > A and t A in the computation of J 4 φ (R), shows that there exists ɛ 2 > 0 such that R 1 2 J 4 φ(r) R ɛ 2. This finishes the proof to the first part of the theorem 4.1 For the proof of the second part of the theorem 4.1, we assume α = 0. Therefore we have an expansion of φ(z) of the form As previously we write and R 1 2 φ(z) = a 0 + n 1 a n z n e. f θ,r (t) = n 0 A n R n e t n e bn (θ) J φ (R) = 2π 1 0 A R exp(if θ,r (t))t dt dθ. Lemma 4.4. There exists a positive constant c such that for all t, 1] and all θ [0, 2π] [ A R 1 2 (13) n 1 A n R n e t n e bn (θ) cr 1 2e Proof. For all n 1 and all t [ A R 1 2, 1] we have A n R n e t n e bn (θ) R 1 2e An ( 1 This finishes the proof of the lemma as the power series A n T n has radius ρ > 0 and 1 ρ. A 1 e 2 Lemma 4.5. There exists a positive constant c such that for all t, 1] and all θ [0, 2π] [ A R 1 2 A 1 e ) n. exp(i n 1 A n R n e t n e bn (θ)) 1 c R 1 2e. Proof. Let ψ(t, R, θ) = n 1 A nr n e t n e b n (θ). We have exp(iψ(t, R, θ)) 1 = (cos(ψ(t, R, θ)) 1) 2 + sin(ψ(t, R, θ)) 2.

17 ALGEBRAIC FLOWS ON ABELIAN VARIETIES. 17 This finishes the proof using the inequality of the previous lemma for t [ A, 1] and θ [0, 2π]. R 1 2 We can now give the proof of the second part of the theorem 4.1. Let J φ(r) = 2π exp(if θ,r (t))t dt dθ. Then J φ (R) J φ (R) 0 as R. We just need to prove that as R. We have J φ(r) π exp(ire(a 0 )) J φ(r) π exp(ire(a 0 )) = with and J 1 = J 2 = 2π 0 0 A R 1 2 2π 1 0 A R 1 2 2π exp(ire(a 0 ))(exp(if θ,r (t)) 1) t dt dθ = J 1 +J 2 exp(ire(a 0 ))(exp(if θ,r (t)) 1) t dt dθ exp(ire(a 0 ))(exp(if θ,r (t)) 1) t dt dθ. Then J 1 2πA2 R and J 2 0 as R by lemma Proof of the Theorem 2.4 Let C be a curve in C g. Let C 1,..., C r be the branches at infinity of C. We recall that we defined ω := i g dz k dz k 2 and the hypercube k=1 B(0, R) = {(z 1,..., z g ) C g, such that for all k {1,..., g}, z k < R}. Lemma 5.1. There exists a positive constant a such that ω R a 2 as R. C B(0,R)

18 18 E. ULLMO, A. YAFAEV Proof. The question only depends on the behaviour of C near the points at infinity. We therefore just need to show that for each branch C β of C there exists a positive constant a β such that (14) C β B(0,R) ω R 2 a β. We fix a branch C β of C and to simplify notations we write D = C β. We just need to show that for all j {1,..., g} there exists a non negative constant a j such that ψ j (R) := D B(0,R) idz j dz j 2R 2 a j and that there exists a j {1,..., g} such that a j > 0. If z j is bounded when Z = (z 1,..., z g ) varies on D then ψ j (R) 0. Let j be such that z j is unbounded. Writing z = z j as in the previous section and using the theory of Puiseux expansions (see [10] chapter 4) we have (15) z k = n 0 a k,n z α k n e k, with a k,0 0, α k Q and e k N. If there exists k {1,..., g} such that α k > 1 = α j then for z tending to infinity z = z k /a k,0 1 α k (1 + o(1)). Then for Z = (z 1,..., z g ) D B(0, R), we have z R 1 α k and ψ j (R) R 2 α k 2. In this case ψ j (R) 0. If α j = max k (α k ), we define E j = {k {1,..., g} such that α k = 1 = α j }. When Z = (z 1,..., z g ) varies in D B(0, R) the function z = z j varies in 1 (16) {z C, A j < z < R(1 + o(1))} max k Ej ( a k,0 ) for some positive constant A j. In this situation there exists a positive constant d j such that (17) ψ j (R) = πd j (1 + o(1)). max k Ej ( a k,0 )

19 ALGEBRAIC FLOWS ON ABELIAN VARIETIES. 19 Let C β be a branch of C and T β = π(p β) + T β be the associated asymptotic Mumford-Tate torus. Let µ β be the normalized Haar measure on T β. Let µ β R and µ β R be the measures on A such that for all continuous Γ-invariant functions f on C g, µ β,r (f) = C β B(0,R) fω C β B(0,R) ω and µ C β,r(f) = fω β B(0,R). R 2 The theorem 2.4 is now a consequence of the following proposition. Proposition 5.2. The measures µ β,r converge weakly to µ β as R. There is a positive constant c β such that µ β,r converges weakly to c β µ β as R. Proof. (of theorem 2.4) Let us first verify that the proposition implies the theorem 2.4. Let f be a continuous Γ-invariant function on C g and C B(0,R) I f (R) = fω C B(0,R) ω. By the lemma 5.1 I f (R) = C B(0,R) fω ar 2 + o(1) = r β=1 C β B(0,R) fω ar 2 + o(1). Using the proof of the lemma 5.1 we get the existence of positive constants a β for β {1,..., r} such that r I f (R) = a β µ β,r (f) + o(1). β=1 Then by the proposition 5.2 we see that r I f (R) a β µ β (f). β=1 This finishes the proof of the theorem 2.4 assuming the proposition 5.2. Proof. (of proposition 5.2) We can start proving proposition 5.2 By the equation (14) in proof of the lemma 5.1, the two statements of the proposition 5.2 are equivalent.

20 20 E. ULLMO, A. YAFAEV Let C β be a branch of C. Let T β = π(p β) + T β be the associated asymptotic torus. Let N(T β ) be as previously the subset of Γ X (A) consisting of characters of A which have a trivial restriction to T β. Let θ Γ, we recall that and µ β (χ θ ) = 0 if θ / N(T β ) µ β (χ θ ) = exp(2iπ(p β, θ)) if θ N(T β ). By lemma 3.2, N(T α ) is the subset of Γ consisting of the θ = (θ 1,..., θ g ) such the function ψ θ (z) := Re(φ θ (z)) (where φ θ (z) = g k=1 θ kz k ) is bounded as Z = (z 1,..., z g ) varies in C β. By the Weyl criterion we must check that there is a positive constant c β independent of θ such that and µ β,r(χ θ ) 0 if θ / N(T β ) µ β,r(χ θ ) c β exp(2iπ(p β, θ)) if θ N(T β ). For k {1,..., g}, let µ β,k,r be the measure on A such that for all continuous Γ-invariant function f we have µ C β,k,r(f) = f idz β B(0,R) k dz k 2R 2 By definition µ β,r = r k=1 µ β,k,r. By the proof of the lemma 5.1 if z j is a coordinate which is bounded on C β for all θ Γ µ β,j,r(χ θ ) 0. If z = z j is unbounded on C β we use the notations of the Puiseux expansions given in the equation (15). Then if there is k {1,..., g} such that α k > 1 = α j once more as in the proof of the lemma 5.1, for all θ Γ µ β,j,r(χ θ ) 0. Let z = z j be unbounded on C β and assume that α j = 1 = max k (α k ). Then φ θ (z) admits a Puiseux expansion of the form considered in section 4 φ θ (z) = n 0 a n z α n e

21 ALGEBRAIC FLOWS ON ABELIAN VARIETIES. 21 with α 1. Then φ θ (z) is unbounded if α > 0 and bounded if α 0. By the proof of lemma 5.1 we have µ idz dz β,j,r(χ θ ) = exp(ire(φ θ (z)) + o(1) 2R 2 A< z <cr for some positive constants A = A β,k and c = c β,k. By the first part of the theorem 4.1, if θ / N(T β ) then µ β,j,r(χ θ ) 0. Let a 0 (z k ) be the constant term in the Puiseux expansion of z k given by the equation (15). Let Q β be the point of C g with coordinates Q β = (a 0 (z 1 ),..., a 0 (z g )). Let a 0 (ψ θ ) be the real part of the constant term in the Puiseux expansion of φ θ (z). Then g a 0 (ψ θ ) = Re( θ k a 0 (z k )) = (θ, Q β ) i=1 By the second part of theorem 4.1, if θ N(T β ), using the equation (17) we obtain that there exists a positive constant d β,j such that µ β,j,r(χ θ ) d β,j exp(ia 0 (ψ θ )) = d β,j exp(i(q β, θ)). By the lemma 3.2, we have (Q β, θ) = (P β, θ). This finishes the proof of the proposition 5.2 and of the theorem Examples We give in this part three examples illustrating the results of this paper Complex geodesic flows on abelian varieties. As in the previous section, A is a complex abelian variety of dimension g. We fix a lattice Γ such that A C g /Γ and we denote by π : C g A the uniformizing map. The conjectures 1.2 and 1.3 for Θ a complex linear subspace of C g are simple applications of Weyl s criterion. In this case Θ is a totally geodesic subvariety of C g. Let V be a C-vector space of dimension 2 and (e 1, e 2 ) be a C-basis of V. Let Γ be the lattice Γ := Ze 1 Z 1e 1 Ze 2 Z 5e 2

22 22 E. ULLMO, A. YAFAEV Then A := A/Γ is an abelian variety of dimension 2. In fact A is a product of 2 elliptic curves. Let W be a sub-vector space C(e 1 + e 2 ) of V. Then the Mumford-Tate group of W is MT (W ) = Q(e 1 + e 2 ) + Q 1e 1 + Q 5e 2. and MT (W ) R = R(e 1 + e 2 ) + R 1e 1 + R 5e 2. As a consequence MT (W ) R/Γ MT (W ) R is a real torus of real dimension 3. This shows that we can not expect that in the conjecture 1.2 the analytic closure of π(θ) has a complex structure The curve Z 1 Z 2 = 1. Let A be a complex abelian surface and π : C 2 A = C 2 /Γ be the uniformizing map. Let C be the curve in C 2 with equation Z 1 Z 2 = 1. There are two branches 1 C 1 = {(Z 1, ), Z 1 > 1} Z 1 and C 2 = {( 1, Z 2 ), Z 2 > 1}. Z 2 Let T 1 = π(p 1 ) +T 1 and T 2 = π(p 2 )+T 2 be the asymptotic Mumford- Tate tori associated to C 1 and C 2. Suppose A is a product of two elliptic curves E 1 E 2. Then there are some lattices Γ 1 and Γ 2 in C such that E 1 = C/Γ 1 and E 2 = C/Γ 2 and Γ = Γ 1 Γ 2. Then Γ = Γ 1 Γ 2. We know by lemma 3.2 that a character χ θ with θ = (θ 1, θ 2 ) Γ = X (A) is trivial on T 1 if and only if Re(θ 1 Z 1 + θ 2 Z 2 ) is bounded on C 1. This is equivalent to saying that θ = (0, θ 2 ) {0} Γ 2. Then we have the exact sequence 0 {0} Γ 2 X (A) = Γ 1 Γ 2 X (T 1 ) = Γ 1 {0} = X (E 1 {0}) 0 As a consequence the asymptotic torus T 1 associated to C 1 is E 1 {0} and the same proof shows that the one associated to C 2 is {0} E 2. In this case π(c) = π(c) E 1 {0} {0} E 2. If Γ has no vector of the form θ = (0, θ 2 ) with θ 2 0, then the only character of A which is trivial on T 1 is 1 A. In this situation π(c) = A. This is the generic situation.

23 ALGEBRAIC FLOWS ON ABELIAN VARIETIES. 23 It is also possible to construct examples of lattices Γ C 2 where the set of θ Γ of the form θ = (0, θ 2 ) is a rank one Z-submodule of Γ. In this situation T 1 is a real torus of dimension 3. If there is no element θ Γ of the form θ = (θ 1, 0) with θ 1 0, then T 2 = A and π(c) = A but the sequence µ C,R does not converge to µ A. In fact by the theorem 2.4, µ C,R aµ A + bµ T1 for some positive real constants a and b. If there is some θ Γ of the form θ = (θ 1, 0) with θ 1 0, then T 2 will be also a real torus of dimension 3 and π(c) = π(c) T 1 T Hyperelliptic curves in C 2. Let A be a complex abelian surface and π : C 2 A = C 2 /Γ be the uniformizing map. Let C be an hyperelliptic curve in C 2 with equation Z 2 2 = Z r 1 + a r 1 Z r a 0 for some integer r 3 and some complex numbers a 0,..., a r 1. Proposition 6.1. We have π(c) = A and µ C,R converges weakly to the normalized Haar measure µ A of A. Proof. For any branch C α of C we have Z 2 Z 1 r 2. Let T α = π(p ) + T α be the asymptotic torus associated to C α. We know by lemma 3.2 that a character χ θ with θ = (θ 1, θ 2 ) Γ = X (A) is trivial on T α if and only if Re(θ 1 Z 1 + θ 2 Z 2 ) is bounded on C α. This is only possible if θ 1 = θ 2 = 0. Therefore T α = A and the result is a consequence of the theorem 2.4. Bibliography. [1] J. Ax, On Schanuel s conjecture, Annals of Math. 93 (1971), [2] J. Ax, Some topics in differential algebraic geometry I: Analytic subgroups of algebraic groups, Amer. J. Math. 94 (1972), [3] B. Bekka, M. Mayer, Ergodic Theory and Topological Dynamics of Groups Actions on Homogeneous Spaces, London Math. Soc. Lect. Note Ser [4] B. Klingler, E. Ullmo, A. Yafaev, The hyperbolic Ax-Lindemann-Weierstrass theorem, Preprint, (2013). [5] M. Orr, The Ax-Lindemann-Weierstrass theorem for abelian varieties, Preprint (2014). [6] J. Pila, U. Zannier, Rational points in periodic analytic sets and the Manin- Mumford conjecture, Rend. Math. Acc. Lincei 19 (2008), [7] J. Pila, J. Tsimermann, Ax-Lindemann for A g, Annals of Math (2), 179, (2014), n. 2, [8] E.M. Stein, Harmonic Analysis, Princeton U. Press, Princeton (1993). [9] E. Ullmo, A. Yafaev, Hyperbolic Ax-Lindemann theorem in the cocompact case, Duke Mathematical Journal, 163, (2014), n. 2,

24 24 E. ULLMO, A. YAFAEV [10] R. Walker, Algebraic curves, Princeton University Press (1950)