The Subspace Theorem and twisted heights

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1 The Subspace Theorem and twisted heights Jan-Hendrik Evertse Universiteit Leiden Heights 2011, Tossa de Mar April 29, 2011

2 Schmidt s Subspace Theorem Theorem (Schmidt, 1972) Let L 0,..., L n be linearly independent linear forms in n + 1 variables with real or complex algebraic coefficients and let a 0,..., a n be reals with a a n > 0. Then the set of solutions x = (x 0,..., x n ) Z n+1 of the system (1) L 0 (x) x a 0,..., L n (x) x an ( x = max 0 i n x i ) is contained in a finite union of proper linear subspaces of Q n+1.

3 Schmidt s Subspace Theorem Theorem (Schmidt, 1972) Let L 0,..., L n be linearly independent linear forms in n + 1 variables with real or complex algebraic coefficients and let a 0,..., a n be reals with a a n > 0. Then the set of solutions x = (x 0,..., x n ) Z n+1 of the system (1) L 0 (x) x a 0,..., L n (x) x an ( x = max 0 i n x i ) is contained in a finite union of proper linear subspaces of Q n+1. Consider the class of convex bodies parametrized by Q, B(Q) := {y R n+1 : max 0 i n L i(y) Q a i 1}. If x Z n+1 satisfies (1) then x B(Q) with Q = x. The Subspace Theorem follows from a study of the successive minima of the bodies B(Q).

4 Notation Q is an algebraic closure of Q. K Q is an algebraic number field, M K its set of places. Choose normalized absolute values v (v M K ) on K such that v extends [Kv :R]/[K:Q] if v is infinite, [Kv :Qp]/[K:Q] v extends p if v is finite, v p, p prime.

5 Notation Q is an algebraic closure of Q. K Q is an algebraic number field, M K its set of places. Choose normalized absolute values v (v M K ) on K such that v extends [Kv :R]/[K:Q] if v is infinite, [Kv :Qp]/[K:Q] v extends p if v is finite, v p, p prime. For x = (x 0,..., x n ) P n (K) define := max( x 0 v,..., x n v ) (v M K ), H(x) := v M K. H(x) is extended to P n (Q) in the usual way. The height H(F ) of F Q[X 0,..., X n ] is the height of its vector of coefficients.

6 The Subspace Theorem over number fields Let S be a finite set of places of K. For v S, let L 0v,..., L nv be linearly independent linear forms in X 0,..., X n with coefficients in K and let c 0v,..., c nv be non-negative reals. Consider the system of inequalities (2) L iv (x) v (v S, i = 0,..., n) in x P n (K).

7 The Subspace Theorem over number fields Let S be a finite set of places of K. For v S, let L 0v,..., L nv be linearly independent linear forms in X 0,..., X n with coefficients in K and let c 0v,..., c nv be non-negative reals. Consider the system of inequalities (2) L iv (x) v (v S, i = 0,..., n) in x P n (K). Theorem (Schmidt, Schlickewei, ) Assume that v S i=0 n c iv > n + 1. Then the set of solutions of (2) is contained in a finite union of proper linear subspaces of P n (K).

8 A refinement of the Subspace Theorem Theorem (Vojta, Schmidt, Faltings-Wüstholz, ) Assume again that v S i=0 n c iv > n + 1. Then there is a proper linear subspace T of P n (K) such that the system L iv (x) v (v S, i = 0,..., n) in x P n (K) has only finitely many solutions outside T.

9 A refinement of the Subspace Theorem Theorem (Vojta, Schmidt, Faltings-Wüstholz, ) Assume again that v S i=0 n c iv > n + 1. Then there is a proper linear subspace T of P n (K) such that the system L iv (x) v (v S, i = 0,..., n) in x P n (K) has only finitely many solutions outside T. Moreover, T can be determined effectively, and T belongs to a finite collection independent of the c iv.

10 Twisted heights For x P n (K), Q 1, define H Q (x) = v S ( max Liv (x) v Q c ) iv 0 i n v M K \S. Lemma Let x P n (K) and Q := H(x). If x is a solution to then H Q (x) Q. L iv (x) v (v S, i = 0,..., n)

11 The Parametric Subspace Theorem Theorem (Schlickewei, E., 2002) Suppose that v S n i=0 c iv > n + 1. Then there are Q 0 > 1, and proper linear subspaces T 1,..., T t of P n (K), such that the following holds: for every Q Q 0, there is T i {T 1,..., T t } such that {x P n (K) : H Q (x) Q} T i.

12 The Parametric Subspace Theorem Theorem (Schlickewei, E., 2002) Suppose that v S n i=0 c iv > n + 1. Then there are Q 0 > 1, and proper linear subspaces T 1,..., T t of P n (K), such that the following holds: for every Q Q 0, there is T i {T 1,..., T t } such that {x P n (K) : H Q (x) Q} T i. Proof of the Subspace Theorem. Pick a solution x P n (K) with H(x) Q 0 of L iv (x) v (v S, i = 0,..., n). Then with Q = H(x) we have H Q (x) Q, hence x i T i.

13 The successive minima of a twisted height Define λ i (Q) to be the minimum of all λ such that {x P n (K) : H Q (x) λ} contains i linearly independent points. Then 0 < λ 1 (Q) λ n+1 (Q) <. We are interested in the behaviour of the λ i (Q) as Q.

14 Weights Let c = (c iv : v S, i = 0,..., n) be a tuple of non-negative reals. For a linear subspace T of P n (K) we define w v (T ) := max A c iv (v S), w(t ) := w v (T ) v S i A where for v S the maximum is taken over all subsets A of {0,..., n} such that the restrictions L iv T (i A) are linearly independent. Let P(T ) := ( dim T + 1, w(t ) ) R 2.

15 A filtration Denote by C(c) the convex hull of the points P(T ) = ( dim T + 1, w(t ) ), for all linear subspaces T of P n (K). Lemma (Faltings-Wüstholz) There exists a unique filtration = T 0 T 1 = T t = P n (Q) of linear subspaces of P n (K), such that the lower boundary of C(c) is a concave polygon with vertices P(T 0 ),..., P(T t ).

16 An asymptotic result for the successive minima Let = T 0 T 1 T t = P n (K) be the filtration of = Faltings-Wüstholz. Put d i := dim T i + 1 (i = 0,..., t), s i := w(t i) w(t i 1 ) dim T i dim T i 1 (i = 1,..., t) (the slope of the line segment from P(T i 1 ) to P(T i )).

17 An asymptotic result for the successive minima Let = T 0 T 1 T t = P n (K) be the filtration of = Faltings-Wüstholz. Put d i := dim T i + 1 (i = 0,..., t), s i := w(t i) w(t i 1 ) dim T i dim T i 1 (i = 1,..., t) (the slope of the line segment from P(T i 1 ) to P(T i )). Theorem (Schmidt, 1993, Faltings-Wüstholz, 1994; Ferretti, E.) For every δ > 0 there is Q 1 = Q 1 (δ) > 1 such that for Q Q 1, i = 1,..., t, Q s i δ λ di 1 +1(Q) λ di (Q) Q s i +δ ; H Q (x) Q s i δ for x P n (K) \ T i 1.

18 About the spaces T i For a linear form L define H(L) to be the height of its coefficient vector. For a linear subspace T of P n (K), define H(T ) := H(a 0 a m ), where a 0,..., a m is any basis of T. Lemma Suppose that H(L iv ) H for v S, i = 0,..., n. Then H(T i ) ( nh) 4n (i = 1,..., t). Thus, the spaces T i belong to a finite, effectively computable collection independent of the c iv.

19 About the spaces T i For a linear form L define H(L) to be the height of its coefficient vector. For a linear subspace T of P n (K), define H(T ) := H(a 0 a m ), where a 0,..., a m is any basis of T. Lemma Suppose that H(L iv ) H for v S, i = 0,..., n. Then H(T i ) ( nh) 4n (i = 1,..., t). Thus, the spaces T i belong to a finite, effectively computable collection independent of the c iv. Open problem: To give more precise information about the spaces T i.

20 An example Let a 0,..., a n K and suppose that { } n {L 0v,..., L nv } X 0,..., X n, a i X i i=0 for v S. Then T i = x Pn (K) : a k x k = 0 (j = 0,..., n d i ) k A ij where d i = dim T i + 1, and A i0,..., A i,n di pairwise disjoint subsets of {0,..., n}. are non-empty,

21 Back to systems of inequalities Consider again L iv (x) v (2) (v S, i = 0,..., n) in x P n (K). Assume again c iv 0, v S n i=0 c iv > n + 1.

22 Back to systems of inequalities Consider again L iv (x) v (2) (v S, i = 0,..., n) in x P n (K). Assume again c iv 0, v S n i=0 c iv > n + 1. Corollary Let m be the smallest index such that s m > 1. Then (2) has only finitely many solutions outside T m 1.

23 Back to systems of inequalities Consider again L iv (x) v (2) (v S, i = 0,..., n) in x P n (K). Assume again c iv 0, v S n i=0 c iv > n + 1. Corollary Let m be the smallest index such that s m > 1. Then (2) has only finitely many solutions outside T m 1. Proof. Choose δ > 0 with s m δ > 1. Let x be a solution of (2) outside T m 1 with H(x) > Q 1 (δ). Put Q := H(x). Then H Q (x) Q since x satisfies (2) and H Q (x) Q sm δ since x T m 1. Contradiction. It follows that H(x) Q 1 (δ).

24 What about the solutions in T m 1? Consider L iv (x) v (2) (v S, i = 0,..., n) in x P n (K), and keep our above assumptions. Let m be as above and assume m 2, dim T m 1 > 0. Put r v := rank{l iv Tm 1 : c iv > 0}. Theorem (E.) (i) Assume there is v S with r v = dim T m Then (2) has only finitely many solutions in T m 1.

25 What about the solutions in T m 1? Consider L iv (x) v (2) (v S, i = 0,..., n) in x P n (K), and keep our above assumptions. Let m be as above and assume m 2, dim T m 1 > 0. Put r v := rank{l iv Tm 1 : c iv > 0}. Theorem (E.) (i) Assume there is v S with r v = dim T m Then (2) has only finitely many solutions in T m 1. (ii) Assume that r v dim T m 1 for all v S and s m 1 < 1. Then the solutions of (2) in T m 1 are Zariski dense in T m 1.

26 What about the solutions in T m 1? Consider L iv (x) v (2) (v S, i = 0,..., n) in x P n (K), and keep our above assumptions. Let m be as above and assume m 2, dim T m 1 > 0. Put r v := rank{l iv Tm 1 : c iv > 0}. Theorem (E.) (i) Assume there is v S with r v = dim T m Then (2) has only finitely many solutions in T m 1. (ii) Assume that r v dim T m 1 for all v S and s m 1 < 1. Then the solutions of (2) in T m 1 are Zariski dense in T m 1. Open problem: What if r v dim T m 1 for all v S and s m 1 = 1?

27 An extension of the Subspace Theorem: Assumptions Let X P R be a projective subvariety defined over K, of dimension n, degree deg X and exponential height H(X ) := H(Chow form of X ). We consider systems of inequalities f iv (x) 1/ deg f iv v (v S, i = 0,..., n) in x X (K), with homogeneous f iv K[X 0,..., X R ] and c iv 0.

28 An extension of the Subspace Theorem: Assumptions Let X P R be a projective subvariety defined over K, of dimension n, degree deg X and exponential height H(X ) := H(Chow form of X ). We consider systems of inequalities f iv (x) 1/ deg f iv v (v S, i = 0,..., n) in x X (K), with homogeneous f iv K[X 0,..., X R ] and c iv 0. We assume that {x X (Q) : f 0v (x) = = f nv (x) = 0} = for v S, n c iv = n δ with δ > 0. v S i=0

29 An extension of the Subspace Theorem: Statement Theorem (Ferretti, E., 2008; Corvaja, Zannier 2004 for X = P n ) The set of solutions of the system f iv (x) 1/ deg f iv v (v S, i = 0,..., n) in x X (K) is not Zariski dense in X.

30 An extension of the Subspace Theorem: Statement Theorem (Ferretti, E., 2008; Corvaja, Zannier 2004 for X = P n ) The set of solutions of the system f iv (x) 1/ deg f iv v (v S, i = 0,..., n) in x X (K) is not Zariski dense in X. More precisely, the set of solutions is contained in X ( ) t i=1 {G i = 0}, where G 1,..., G t K[X 0,..., X R ] are homogeneous polynomials not vanishing identically on X, of degree at most 8(n + 2) 3 n+1 (deg X )(1 + δ 1 ) where := lcm{deg f iv : v S, i = 0,..., n}.

31 A further refinement Theorem There is an effectively computable, proper projective subvariety Y X, defined over K, such that f iv (x) 1/ deg f iv v (v S, i = 0,..., n) in x X (K), has only finitely many solutions outside Y.

32 A further refinement Theorem There is an effectively computable, proper projective subvariety Y X, defined over K, such that f iv (x) 1/ deg f iv v (v S, i = 0,..., n) in x X (K), has only finitely many solutions outside Y. We may choose Y = X {G = 0}, where G K[X 0,..., X R ] is homogeneous with deg G 8(n + 2) 3 deg X n+1 (1 + δ 1 ), H(G) c 1 H c 2, where := lcm{deg f iv : v S, i = 0,..., n}, H := max{h(x ); H(f iv ) : v S, i = 0,..., n}, c 1, c 2 effectively computable in terms of R, n, deg X,, δ.

33 Idea of the proof Construct an embedding Ψ : X P M : x (g 0 (x),..., g M (x)) with homogenenous polynomials g 0,..., g M of equal degree such that if x X (K) is a solution to f iv (x) 1/ deg f iv v (v S, i = 0,..., n), then y = Ψ(x) satisfies L iv (y) v y v H(y) d iv (v S, i = 0,..., M), y P M (K) for certain linear forms L iv and reals d iv with d iv 0 and v S M i=0 d iv > M + 1.

34 Idea of the proof Construct an embedding Ψ : X P M : x (g 0 (x),..., g M (x)) with homogenenous polynomials g 0,..., g M of equal degree such that if x X (K) is a solution to f iv (x) 1/ deg f iv v (v S, i = 0,..., n), then y = Ψ(x) satisfies L iv (y) v y v H(y) d iv (v S, i = 0,..., M), y P M (K) for certain linear forms L iv and reals d iv with d iv 0 and v S M i=0 d iv > M + 1. Ferretti, E.: Such an embedding exists with deg g i = [8(n + 2) 3 deg X n+1 δ 1 ].

35 Open problems 1. The system of inequalities f iv (x) 1/ deg f iv v (v S, i = 0,..., n) has only finitely many solutions in (X \ Y )(K), where Y can be chosen from a finite collection depending on δ = v S n i=0 c iv (n + 1). Is this dependence on δ necessary? 2. More precise information on Y.

Results and open problems related to Schmidt s Subspace Theorem. Jan-Hendrik Evertse

Results and open problems related to Schmidt s Subspace Theorem Jan-Hendrik Evertse Universiteit Leiden 29 ièmes Journées Arithmétiques July 6, 2015, Debrecen Slides can be downloaded from http://pub.math.leidenuniv.nl/