MA 254 notes: Diophantine Geometry

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1 MA 254 notes: Diophantine Geometry (Distilled from [Hindry-Silverman]) Dan Abramovich Brown University January 29, 2016 Abramovich MA 254 notes: Diophantine Geometry 1 / 14

2 Height on P n (k) Let P = (x 0,..., x n ) P n (Q) with x i Z, gcd(x 0,..., x n ) = 1, Definition (Height on P n (Q)) H(P) = max{ x 0,..., x n }. This has the finiteness property: For any B the set {P P n (Q) H(P) B} is finite. Definition (Height on P n (k)) Let k be a number field, x i k, and P = (x 0,..., x n ) P n (k). Define the relative multiplicative height H k (P) = v M k max{ x 0 v,..., x n v } and the relative logarithmic height h k (P) = log H k (P) = v M k n v min{v(x 0 ),... v(x n )}. Abramovich MA 254 notes: Diophantine Geometry 2 / 14

3 Absolute height Definition The absolute multiplicative height of P P n ( Q) is H(P) = H k (P) 1/[k:Q]. The absolute logarithmic height is h(p) := log H(P) = h k (P)/[k : Q]. The absolute height of α k is H(α) := H(1, α). Proposition (Basic height properties) H k (P) is independent of choice of coordinates in k. H(P), H k (P) 1; h(p), h k (P) 0 for all k P n ( Q). H(P), h(p) are independent of choice of k or coordinates. Abramovich MA 254 notes: Diophantine Geometry 3 / 14

4 Proof of basic height properties Proof. v M k max{ cx 0 v,..., cx n v } = ( ) c v Mk v max{ x v Mk 0 v,..., x n v } = v M k max{ x 0 v,..., x n v }. If x i 0 then H k (x) = v M k max{ x 0 /x i v,..., x n /x i v } v M k 1 = 1. If k /k then H k (P) = v M k w v max{ x 0 w,..., x n w } = v M k w v max{ x 0 nw v,..., x n nw v } = v M k w v max{ x 0 nv v,..., x n nv v } [k w :kv ] = v M k w v max{ x 0 v,..., x n v } [k w :k v ] = v M k max{ x 0 v,..., x n v } [k :k] = H k (P) [k :k] as needed. Abramovich MA 254 notes: Diophantine Geometry 4 / 14

5 Invariance The Galois group G Q acts on P n ( Q). It also permutes absolute values: if σ : k k is an isomorphism then we define σ(x) σ(v) = x v, namely y σ(v) = σ 1 x v. Similarly it sends completions to completions, with n v = n σ(v), and globally [k : Q] = [σ(k) : Q]. Proposition H is invariant under G Q : we have H σ(k) (σ(x)) = H k (x) and H(σ(x) = H(x). Proof. Trace the definitions, taking the above into consideration. Abramovich MA 254 notes: Diophantine Geometry 5 / 14

6 The strong finiteness property Theorem Given B, D, the set {P P n ( Q) H(P) B, [Q(P) : Q] D} is finite. Lemma Given B, d, the set {x Q) H(x) B, [Q(x) : Q] = d} is finite. Proof of Theorem given Lemma. Given P = (x 0,..., x n ) with x j 0 we may assume by rescaling that x j = 1. Then max{ x 0 v,..., x n v } max{ x i v, 1} for each i, so H(P) H(x i ). Also Q(P) Q(x i ). There are finitely many possible choices for j, d D and, by the lemma, for x i, hence for P. Abramovich MA 254 notes: Diophantine Geometry 6 / 14

7 The strong finiteness property - continued Sublemma Suppose the minimal polynomial of x Q is X d s 1 (x)x d 1 ± + ( 1) d s d (x). Then H(1, s 1 (x),..., s d (x)) 2 d H(x) d2. Proof of Lemma given Sublemma. We have seen that {s P d (Q) : H(s) < C} is finite. Applying this to C = 2 d B d2 we find that the number of minimal polynomials of {x Q) H(x) B, [Q(x) : Q] = d} is finite. Since there are d roots per polynomial, this set itself is finite. { r v archimedean We write ε v (r) := 1 otherwise so that r i=1 x i v ε v (r) max i x i v. Note r = v ε v (r) nv /[k:q]. Abramovich MA 254 notes: Diophantine Geometry 7 / 14

8 The strong finiteness property - continued Proof of Sublemma. If x i are the d conjugates of x, and v a valuation of the splitting field k, then s r (x) v = x i1 x ir v ε v (2 d ) max x i1 x ir v ε v (2 d ) max i x i r v. Taking maximum we obtain max{1, s 1 (x) v,..., s d (x) v } ε v (2 d ) i max{ x i v, 1} d. taking products and [k : Q]-th root, we have H(1, s 1 (x),..., s d (x)) 2 d i H(x i) d = 2 d i H(x)d = 2 d H(x) d2 as needed. Abramovich MA 254 notes: Diophantine Geometry 8 / 14

9 Points of Height 1 Corollary (Kronecker s theorem) Say P = (x 0,..., x n ) P n ( Q) and x i 0. Then H(P) = 1 x j /x i µ( Q) {0} for all j. Proof. Without loss of generality i = 0 and x 0 = 1. If x j µ then x j v = 1 for all v so H(P) = 1. Assume H(P) = 1 and consider the sequence of points P r := (x r 0,... x r n). Then you check H(P r ) = H(P) r = 1, in particular bounded by 1, and by the theorem {P r } is a finite set. So there are r s such that P r = P s, so xj r = xj s, so x j µ {0} as needed. Abramovich MA 254 notes: Diophantine Geometry 9 / 14

10 Segre and Veronese embeddings Let S n,m : P n P m P nm+n+m be the Segre embedding and V n,d : P n P (n+d n ) 1 the d-th Veronese. Proposition h(s n,m (x, y)) = h(x) + h(y) and h(v n,d (x)) = dh(x). Proof. The point z = S n,m (x, y) has coordinates (..., x i y j,...). For any v we have max ij x i y j v = (max i x i v )(max j y j v ). So h(z) = log ( v max ij x i y j v = log v max i x i v max j y j nv /[k:q] v nv /[k:q] nv /[k:q] v ) = h(x) + h(y). w = V n,d (x) has coordinates the monomials M I (x) of degree d. We have M I (x) v max i x i d v so max I M I (x) v = max i x i d v, and proceed as before. Abramovich MA 254 notes: Diophantine Geometry 10 / 14

11 Functoriality of heights on projective spaces An m + 1-tuple (f 0,..., f m ) of homomgeneous forms of degree d in n + 1 variables defines a rational map φ : P n P m, which is a morphism away from the base locus Z = V (f 0,..., f m ). Theorem (Functoriality of heights on projective spaces) h(φ(p)) dh(p) + O(1) for all P P n ( Q)) Z. If X P n closed, X Z =, then h(φ(p)) = dh(p) + O(1) for all P X ( Q)). It is convenient to take absolute values of P, f j and height of φ. Represent P = (x 0,..., x n ) and in monomial notation f j = I =d a j,i x I. We write P v = max i { x i v }, f j v = max I { a j,i v } and H(φ) = H((a j,i ) j,i ) = v max j{ f j v } nv /[k:q]. Abramovich MA 254 notes: Diophantine Geometry 11 / 14

12 We write N d = ( ) n+d d. Proof of functoriality, first part. f j (P) v = I a j,i x I v ε v (N d )(max I a j,i v )(max x I v ) I We get v max j ε v (N d ) f j v (max x i d v ) i = ε v (N d ) f j v P d v. nv /[k:q] f j (P) v N d ( max f j v j v = N d H(φ)H(P) d. so h(φ(p)) dh(p) + h(φ) + log N d. nv /[k:q] )H(P) d Abramovich MA 254 notes: Diophantine Geometry 12 / 14

13 Proof of functoriality, second part, beginning Let I X = (p 1,..., p r ). By Hilbert s Nullstellensatz, after a finite extension of k we have (p1,..., p r, f 0,..., f m ) = (X 0,..., X n ). In other words, there is t d, forms g kj of degree t d and forms q lj such that for all j g 0j f g mj f m + q 1j p q rj p r = X t j. Plugging in P = (x 0,..., x n ) X we get g 0j (P)f 0 (P) + + g mj (P)f m (P) = x t j. Abramovich MA 254 notes: Diophantine Geometry 13 / 14

14 Proof of functoriality, second part, concluded. P t v = max j xj t v = max g 0j (P)f 0 (P) + + g mj (P)f m (P) v j ε v (m + 1) (max i,j g ij (x) v ) (max f i (x) v ) i (ε v (m + 1) N t d (g ij ) v ) P t d v φ(p) v. So as before H(P) t c H(P) t d H(φ(P)). Hence dh(p) h(φ(p)) + O(1) as required. Abramovich MA 254 notes: Diophantine Geometry 14 / 14