2-Selmer group of even hyperelliptic curves over function fields

Transcription

1 2-Selmer group of even hyperelliptic curves over function fields Dao Van Thinh Department of Mathematics National University of Singapore Pan Asia Number Theory Conference, 2018

2 Outline Main results The problem over Q The problem over F q (C)

3 Outline Main results The problem over Q The problem over F q (C) Proof of the main theorem Vinberg s representation of type A 2n+1 Connection to hyperelliptic curves Canonical reduction theory of G-bundles Some computations

4 Outline Main results The problem over Q The problem over F q (C) Proof of the main theorem Vinberg s representation of type A 2n+1 Connection to hyperelliptic curves Canonical reduction theory of G-bundles Some computations

5 Odd hyperelliptic curve Theorem (M. Bhargava and B. Gross (2012)) When all hyperelliptic curves of fixed genus n 1 over Q having a rational Weierstrass point are ordered by height, the average size of the 2 Selmer groups of their Jacobians is 3.

6 Odd hyperelliptic curve Theorem (M. Bhargava and B. Gross (2012)) When all hyperelliptic curves of fixed genus n 1 over Q having a rational Weierstrass point are ordered by height, the average size of the 2 Selmer groups of their Jacobians is 3. Corollary The average rank of of the Mordell-Weil groups of the Jacobians of such curves is at most 3/2.

7 Even hyperelliptic curve Theorem (A. Shankar and X. Wang (2014)) When all hyperelliptic curves of fixed genus n 2 over Q having a marked rational non-weierstrass point are ordered by height, the average size of the 2 Selmer groups of their Jacobians is 6.

8 Even hyperelliptic curve Theorem (A. Shankar and X. Wang (2014)) When all hyperelliptic curves of fixed genus n 2 over Q having a marked rational non-weierstrass point are ordered by height, the average size of the 2 Selmer groups of their Jacobians is 6. Corollary The average rank of the Mordell-Weil groups of the Jacobians of the above curves is at most 5/2.

9 Even hyperelliptic curve Theorem (A. Shankar and X. Wang (2014)) When all hyperelliptic curves of fixed genus n 2 over Q having a marked rational non-weierstrass point are ordered by height, the average size of the 2 Selmer groups of their Jacobians is 6. Corollary The average rank of the Mordell-Weil groups of the Jacobians of the above curves is at most 5/2. Theorem (A. Shankar and X. Wang (2014)) The proportion of monic even degree hyperelliptic curves having genus n 4 that have exactly two rational points is at least 1 (48n + 120)2 n.

10 Outline Main results The problem over Q The problem over F q (C) Proof of the main theorem Vinberg s representation of type A 2n+1 Connection to hyperelliptic curves Canonical reduction theory of G-bundles Some computations

11 Notation k = F q with (char(k), 2n + 2) = 1 C is a smooth, complete, geometrically connected curve over k K = k(c) the function field of C

12 Even hyperelliptic curve An even hyperelliptic curve of genus n is the smooth projective model of the affine curve defined by H : y 2 = x 2n+2 + c 2 x 2n + + c 2n+2, where c i K, and the tuple (c i ) 2 i 2n+2 is unique up to the following identification (c 2, c 3,..., c 2n+2 ) (λ 2.c 2, λ 3 c 3,..., λ 2n+2.c 2n+2 ) λ K.

13 Minimal integral model Fix the data (c 2, c 3,..., c 2n+2 ), we define the minimal integral model of H as follows: for each point v C, we can choose an integer n v which is the smallest integer satisfying that: the tuple (ϖ 2nv v c 2, ϖv 3nv c 3,, ϖ v (2n+2)nv c 2n+2 ) has coordinates in O Kv. Given (n v ) v C, we define the invertible sheaf L H K whose sections over a Zariski open U C are given by L H (U) = K ( ) ϖv nv O Kv. v U Then c i H 0 (C, L i H ) for all 2 i 2n + 2. Furthermore, the stratum (L H, c) is minimal in the sense that there is no proper subsheaf M of L H such that c i H 0 (C, M i ) for all i.

14 Height of hyperelliptic curves Definition The height of the hyperelliptic curve H is defined to be the degree of the associated line bundle L H.

15 Height of hyperelliptic curves Definition The height of the hyperelliptic curve H is defined to be the degree of the associated line bundle L H. We are going to consider the following family of hyperelliptic curves: Definition An even hyperelliptic curve H with an associated minimal data (L H, c) is called to be transversal if the discriminant (c) H 0 (C, L (2n+1)(2n+2) H ) is square-free.

16 Main theorem Denote A trans d to be the set of all transversal even hyperelliptic curves of height less than or equal to d.

17 Main theorem Denote A trans d to be the set of all transversal even hyperelliptic curves of height less than or equal to d. Theorem When all transversal even hyperelliptic curves of genus n 2 over K are ordered by height, the average size of the 2 Selmer group of their Jacobians is 6. Equivalently, lim d H A trans d H A trans d Sel 2 (H) Aut(H, ) 1 Aut(H, ) = 6.

18 Notation H: the minimal integral model of H J H and J H are Jacobian group schemes associated to H and H respectively. Observe that J H is the generic fiber of J H.

19 Restatement of the main theorem Lemma If H is transversal, then J H is the Néron model of J H. Furthermore, Sel 2 (J H ) = H 1 (C, J H [2]). The main theorem is equivalent to: lim d H A trans d H A trans d H 1 (C,J H [2]) Aut(H, ) 1 Aut(H, ) = 6.

20 Outline Main results The problem over Q The problem over F q (C) Proof of the main theorem Vinberg s representation of type A 2n+1 Connection to hyperelliptic curves Canonical reduction theory of G-bundles Some computations

21 Let (U, Q) be the split quadratic space over k of dimension 2n + 2 and discriminant 1. Then for any linear operator T : U U, we defined its adjoint T by the following equation: Tv, w Q = v, T w Q, v, w U. where v, w Q = Q(v + w) Q(v) Q(w) denotes the bilinear form associated to Q. The Vinberg s representation we are going to study is the conjugate action of on G := PSO(U) = {g GL(U) gg = I, det(g) = 1}/µ 2 V = {T : U U T = T, trace(t ) = 0} = Sym 2 0(U).

22 GIT quotient For each T V, denote f T (x) be the characteristic polynomial of T : Then f T (x) = x 2n+2 + c 2 (T )x 2n + + c 2n+1 (T )x + c 2n+2 (T ). V //G = Spec(k[c 2, c 3,..., c 2n+2 ]) = S. We denote the projection map by π : V S.

23 Regular locus Set V reg (k) = {T V (k) Stab G(k) (T ) is finite} = {T V (k) f T (x) is its minimal polynomial}

24 Regular locus Set V reg (k) = {T V (k) Stab G(k) (T ) is finite} = {T V (k) f T (x) is its minimal polynomial} = for any field extension k F and T V reg (F ), Stab G (T ) = (Res L/F µ 2 ) N=1 /µ 2, where L = F [x]/(f T (x)).

25 Stabilizer group scheme over S Theorem There exists a unique group scheme I S over S equipped with an isomorphism π I S Stab G over V reg. This isomorphism is G equivariant, thus, as a corollary, there is a G m equivariant isomorphism of stacks [BI S ] = [V reg /G], where BI S is the relative classifying stack of I S over S.

26 Outline Main results The problem over Q The problem over F q (C) Proof of the main theorem Vinberg s representation of type A 2n+1 Connection to hyperelliptic curves Canonical reduction theory of G-bundles Some computations

27 The generalized Jacobian group scheme For each c = (c 2, c 3,..., c 2n+2 ) S, the associated polynomial f c (x) = x 2n+2 + c 2 x 2n + + c 2n+1 x + c 2n+2 defines an even hyperelliptic curve y 2 = f c (x) (we allow singular hyperelliptic curves)

28 The generalized Jacobian group scheme For each c = (c 2, c 3,..., c 2n+2 ) S, the associated polynomial f c (x) = x 2n+2 + c 2 x 2n + + c 2n+1 x + c 2n+2 defines an even hyperelliptic curve y 2 = f c (x) (we allow singular hyperelliptic curves) = a group scheme J S which represents the (generalized) Jacobian functor.

29 The generalized Jacobian group scheme For each c = (c 2, c 3,..., c 2n+2 ) S, the associated polynomial f c (x) = x 2n+2 + c 2 x 2n + + c 2n+1 x + c 2n+2 defines an even hyperelliptic curve y 2 = f c (x) (we allow singular hyperelliptic curves) = a group scheme J S which represents the (generalized) Jacobian functor. Set J V reg := J S S V reg

30 Stabilizer group scheme and Jacobian Theorem There exists a canonical G equivariant isomorphism over V reg between the stabilizer scheme Stab G and J V reg [2].

31 Stabilizer group scheme and Jacobian Theorem There exists a canonical G equivariant isomorphism over V reg between the stabilizer scheme Stab G and J V reg [2]. Corollary The above isomorphism induces an isomorphism over S from I S to J S [2]. = an isomorphism between stacks BJ S [2] = [V reg /G]

32 An interpretation of H 1 (C, J H [2]) Hyperelliptic curve H (L H, c), c S(K)

33 An interpretation of H 1 (C, J H [2]) Hyperelliptic curve H (L H, c), c S(K) α H : C [S/G m ]

34 An interpretation of H 1 (C, J H [2]) Hyperelliptic curve H (L H, c), c S(K) Set A = Hom(C, [S/G m ]) α H : C [S/G m ]

35 An interpretation of H 1 (C, J H [2]) Hyperelliptic curve H (L H, c), c S(K) α H : C [S/G m ] Set A = Hom(C, [S/G m ]) Set M = Hom(C, [BJ S [2]/G m ])

36 An interpretation of H 1 (C, J H [2]) Hyperelliptic curve H (L H, c), c S(K) α H : C [S/G m ] Set A = Hom(C, [S/G m ]) Set M = Hom(C, [BJ S [2]/G m ]) = a base map b : M A

37 An interpretation of H 1 (C, J H [2]) Hyperelliptic curve H (L H, c), c S(K) α H : C [S/G m ] Set A = Hom(C, [S/G m ]) Set M = Hom(C, [BJ S [2]/G m ]) = a base map b : M A = H 1 (C, J H [2]) = b 1 (α H )

38 Counting points on stacks We also have a commutative diagram: M b π M π A Hom(C, BG m ) A

39 Counting points on stacks We also have a commutative diagram: M b π M π A Hom(C, BG m ) A = for any line bundle F over C, M F (k) = H A F (k) H 1 (C, J H [2]).

40 Now it is enough to prove that lim deg(f) M trans F A trans F (k) (k) = 6, where M trans F = b 1 (A trans F ).

41 Another interpretation of M F (k) From the isomorphism: [BJ S [2]/G m ] = [V reg /(G G m )] = M = Hom(C, [V reg /(G G m )]).

42 Another interpretation of M F (k) From the isomorphism: [BJ S [2]/G m ] = [V reg /(G G m )] = M = Hom(C, [V reg /(G G m )]). = a k point of M F is a pair (E, s), where E is a principal G bundle, and s is a section of V reg (E, F) = (V reg G E) F

43 Outline Main results The problem over Q The problem over F q (C) Proof of the main theorem Vinberg s representation of type A 2n+1 Connection to hyperelliptic curves Canonical reduction theory of G-bundles Some computations

44 As algebraic groups over k: G = PSO(U) = GSO(U)/G m, where G m denotes the center of GSO(U). = G bundles GSO(U)/G m bundles. Moreover, any GSO(U)/G m bundle can be lifted to a GSO(U) bundle uniquely up to tensor twist by a line bundle.

45 Canonical reduction of GSO(2n + 2) bundles Let E be a GSO(2n + 2) bundle. Then there exists uniquely a parabolic subgroup P GSO(U) with Levi quotient L and the associated P bundle E P such that 1. We have an isomorphism E = E P (GSO(U)), where E P (GSO(U)) is the quotient (E P GSO(U))/P with the following action of P on E P GSO(U) : for any h P, e E P, and g GSO(U) then h.(e, g) = (h.e, h 1 g).

46 Canonical reduction of GSO(2n + 2) bundles Let E be a GSO(2n + 2) bundle. Then there exists uniquely a parabolic subgroup P GSO(U) with Levi quotient L and the associated P bundle E P such that 1. We have an isomorphism E = E P (GSO(U)), where E P (GSO(U)) is the quotient (E P GSO(U))/P with the following action of P on E P GSO(U) : for any h P, e E P, and g GSO(U) then h.(e, g) = (h.e, h 1 g). 2. The Levi bundle E L associated, by extension of structure group, to E P for the projection P L is semi-stable.

47 Canonical reduction of GSO(2n + 2) bundles Let E be a GSO(2n + 2) bundle. Then there exists uniquely a parabolic subgroup P GSO(U) with Levi quotient L and the associated P bundle E P such that 1. We have an isomorphism E = E P (GSO(U)), where E P (GSO(U)) is the quotient (E P GSO(U))/P with the following action of P on E P GSO(U) : for any h P, e E P, and g GSO(U) then h.(e, g) = (h.e, h 1 g). 2. The Levi bundle E L associated, by extension of structure group, to E P for the projection P L is semi-stable. 3. For every non-trivial character χ of P which is a non-negative linear combination of simple roots with respect to some Borel subgroup contained in P, the line bundle χ E P on C has positive degree.

48 Assume that the Levi subgroup L = GL n1 GL n2 GL nt GSO(2h). = a flag of isotropic subspaces 0 = V 0 V 1 V t V t V 1 U, where dim(v i /V i 1 ) = n i for 1 i t, and dim(v t /V t ) = 2h. = a filtration of the vector bundle E GSO(U) U: 0 E P P V 1 E P P V t E P P V t E P P V 1 satisfying that the quotient bundles X i = E P P V i /(E P P V i 1 ), 1 i t and are semistable. X t+1 = (E P P V t )/(E P P V t )

49 Moreover, (E P P Vi 1)/(E P P Vi ) = Xi L and X t+1 = X t+1 L. Denote the slope of X i by µ i, then the canonical conditions imply that: µ 1 > µ 2 > > µ t > µ t+1 = d/2 if h > 0, µ 1 > µ 2 > > µ t and µ t 1 + µ t > d if h = 0.

50 Semistable filtration of (E GSO(U) V ) we obtain the following matrix filtration of Sym 2 0(E) L : Sym 2 (X 1) L X 1 X 2 L X 1 X t L X 1 X t+1 X 1 X t X 1 X 1 X 2 X 1 L Sym 2 (X 2) L X 2 X t L X 2 X t+1 X 2 X t X 2 X X t X 1 L X t X 2 L Sym 2 (X t) L X t X t+1 X t X t X t X 1 X t+1 X1 X t+1 X2 X t+1 Xt Sym2 0 (Xt+1) L X t+1 X t X t+1 X 1 X t X 1 X t X 2 X t X t X t X t+1 Sym 2 (X t ) L X t X 1 L X 2 X1 X 2 X2 X 2 Xt X 2 Xt+1 X 2 X t L X 2 X 1 L X 1 X1 X 1 X2 X 1 Xt X 1 Xt+1 X 1 X t L Sym 2 (X 1 ) L

51 Outline Main results The problem over Q The problem over F q (C) Proof of the main theorem Vinberg s representation of type A 2n+1 Connection to hyperelliptic curves Canonical reduction theory of G-bundles Some computations

52 The case P=B the Borel subgroup The main contributors to the average are M trans F,B lim (k) deg(f) A trans F E = X 1 X n+1 (X n+1 L) (X 1 L)

53 The case P=B the Borel subgroup The main contributors to the average are M trans F,B lim (k) deg(f) A trans F E = X 1 X n+1 (Xn+1 L) (X1 L) satisfying that µ i = µ i+1 + f 1 i n, where f = deg(f), µ i = deg(x i ).

54 Case 1: 2µ n+1 d = f For any (E, s) M trans F, where s is a section of (V GSO(U) E) F = Sym 2 0(E) L F,

55 Case 1: 2µ n+1 d = f For any (E, s) M trans F, where s is a section of (V GSO(U) E) F = Sym 2 0(E) L F, then s is of the following form: where x i k. x 1 0 x x x 1

56 g.s.g , for some g GSO(U)(K)

57 g.s.g , for some g GSO(U)(K) The Kostant section κ 1

58 g.s.g , for some g GSO(U)(K) The Kostant section κ 1 = This case contributes 1 to the average.

59 Any section s is of the form: where x i k. Case 2: 2µ n+1 + d = f x x n x n x n x 1

60 g.s.g 0 a b 1 a d c d e

61 g.s.g 0 a b 1 a d c d e The Kostant section κ 2

62 g.s.g 0 a b 1 a d c d e The Kostant section κ 2 = This case contributes 1 to the average.

63 The case P=B the Borel subgroup 0 a b 1 a d c 1 0 e d The Kostant section κ 1 0 a b 1 a d c d e The Kostant section κ 2

64 The whole picture B P 1... P t B G Vol(red) = 2 Vol(green) = 4 = τ(g)

65 Summary 6 = the number of Kostant sections + τ(g).

66 Summary 6 = the number of Kostant sections + τ(g). By a similar method, we also can give an upperbound for the average in general case (remove the transversal condition) if we assume char(k) is big enough.

67 Summary 6 = the number of Kostant sections + τ(g). By a similar method, we also can give an upperbound for the average in general case (remove the transversal condition) if we assume char(k) is big enough. The method that was used here, is partially similar to the method in the paper Average size of 2-Selmer groups of elliptic curves over function fields of Q.P. Ho, V.B. Le Hung, and B.C. Ngo.

68 Thank you!