Tropical Elliptic Curves and their j-invariant

Transcription

1 and their j-invariant (joint work with Eric Katz and Hannah Markwig) Elliptic Tropical Thomas Markwig Technische Universität Kaiserslautern 15th February, 2008

2 Elliptic Tropical The tropical j-invariant is the tropicalisation of the j-invariant.

3 1. Plane Cubics Main Object of Interest A plane curve of degree 3 with equation F = a 30 x 3 + a 21 x 2 y + a 12 xy 2 + a 03 y 3 +a 20 x 2 + a 11 xy + a 02 y 2 + a 10 x + a 01 y + a 00 where the coefficients a ij belong to some field K. Elliptic Tropical Notation A = {(i, j) a ij 0} = supp(f ) a = ( a ij (i, j) A ) C F = { (X, Y ) K 2 F (X, Y ) = 0 } j A 2 i

4 j-invariant When are two plane cubics C F and C F isomorphic? Elliptic Tropical Theorem There are homogeneous Polynomials A, [a] of degree 12, such that C F = CF A(a ) (a ) = A(a ) (a ). Definition j = A is the j-invariant of C F.

5 Puiseux Series Theorem (Puiseux Series) The elements of the algebraic closure of (t) have the form Elliptic Tropical a 1 t k 1 N + a2 t k 2 N + a3 t k 3 N +... with a i, k i, k 1 < k 2 < k 3 <..., N > 0.

6 Puiseux Series Theorem (Puiseux Series) The elements of the algebraic closure of (t) have the form Elliptic Tropical a 1 t k 1 N + a2 t k 2 N + a3 t k 3 N +... with a i, k i, k 1 < k 2 < k 3 <..., N > 0. General Assumption From now on K = is our base field. Advantage comes with a non-archemedian valuation ord : : a 1 t k 1 N + h.o.t k 1 N.

7 Tropicalisation Definition On ( ) 2 we have the tropicalisation Trop : ( ) 2 2 : (X, Y ) ( ord(x), ord(y ) ) Elliptic Tropical and thus for F [x, y] we have T F = Trop ( C F ( ) 2) 2.

8 Tropicalisation Definition On ( ) 2 we have the tropicalisation Trop : ( ) 2 2 : (X, Y ) ( ord(x), ord(y ) ) Elliptic Tropical and thus for F [x, y] we have T F = Trop ( C F ( ) 2) 2.

9 Tropicalisation Definition On ( ) 2 we have the tropicalisation Trop : ( ) 2 2 : (X, Y ) ( ord(x), ord(y ) ) Elliptic Tropical and thus for F [x, y] we have T F = Trop ( C F ( ) 2) 2. Problem Trop forgets an awful lot of information! The definition is not too helpful to compute T F.

10 Tropical Polynomials Definition Let F = trop(f ) : (i,j) A a ij x i y j with a ij, A 2 finite. 2 : (x, y) max (i,j) A { ord(a ij) + i x + j y} is called the tropicalisation of F. Example F = y 3 x 3 2 t xy + (1 t), A = {(0, 3), (3, 0), (1, 1), (0, 0)}. trop(f ) = max{3 y, 3 x, x + y + 1, 0}. Elliptic Tropical

11 Lifting Lemma (Newton / Kapranov) Lifting Lemma T F = {(x, y) trop(f ) non-differentiable in (x, y)} = {(x, y) maximum in trop(f ) is attained at least twice } Elliptic Tropical Where: T F = { ( ord(x), ord(y ) ) (X, Y ) CF ( ) 2}. trop(f ) : 2 : (x, y) max (i,j) A { ord(a ij) + i x + j y}

12 Lifting Lemma (Newton / Kapranov) Lifting Lemma T F = {(x, y) trop(f ) non-differentiable in (x, y)} = {(x, y) maximum in trop(f ) is attained at least twice } In particular, T F is a piece wise linear graph! Where: Elliptic Tropical T F = { ( ord(x), ord(y ) ) (X, Y ) CF ( ) 2}. trop(f ) : 2 : (x, y) max (i,j) A { ord(a ij) + i x + j y}

13 Example (continued) Find the vertices of the tropical curve T F with trop(f ) = max{3x, 3y, x + y + 1, 0}! Elliptic Tropical 3x = 3y = x + y (x, y) = (1, 1).

14 Example (continued) Find the vertices of the tropical curve T F with trop(f ) = max{3x, 3y, x + y + 1, 0}! Elliptic Tropical 3x = 3y = x + y (x, y) = (1, 1). 3x = x + y + 1 = 0 3y (x, y) = (0, 1). 3y = x + y + 1 = 0 3x (x, y) = ( 1, 0). 3x = 3y = 0 x + y + 1.

15 Example (continued) Find the vertices of the tropical curve T F with trop(f ) = max{3x, 3y, x + y + 1, 0}! Elliptic Tropical

16 Example (continued) Find the vertices of the tropical curve T F with trop(f ) = max{3x, 3y, x + y + 1, 0}! Elliptic Tropical

17 Why does this work? Theorem T F is dual to the subdivision of the Newton polygon Q F of F. Elliptic Tropical

18 Why does this work? Theorem T F is dual to the subdivision of the Newton polygon Q F of F. Example F = y 3 x 3 2 t xy + (1 t), A = {(3, 0), (0, 3), (1, 1), (0, 0)}. Elliptic Tropical Q F

19 Why does this work? Theorem T F is dual to the subdivision of the Newton polygon Q F of F. Example F = y 3 x 3 2 t xy + (1 t), A = {(3, 0), (0, 3), (1, 1), (0, 0)}. z = ord t y Elliptic Tropical x

20 Why does this work? Theorem T F is dual to the subdivision of the Newton polygon Q F of F. Example F = y 3 x 3 2 t xy + (1 t), A = {(3, 0), (0, 3), (1, 1), (0, 0)}. z y Elliptic Tropical x

21 Why does this work? Theorem T F is dual to the subdivision of the Newton polygon Q F of F. Example F = y 3 x 3 2 t xy + (1 t), A = {(3, 0), (0, 3), (1, 1), (0, 0)}. Elliptic Tropical z y x Q F

22 Why does this work? Theorem T F is dual to the subdivision of the Newton polygon Q F of F. Example F = y 3 x 3 2 t xy + (1 t), A = {(3, 0), (0, 3), (1, 1), (0, 0)}. Elliptic Tropical T F Q F

23 Example: a tropical cubic F = t 7 (x 3 +y 3 )+t 3 x 2 +t 2 (xy 2 +y 2 )+t (x 2 y +x+y +1)+xy Elliptic Tropical Q F T F

24 Example: a tropical cubic F = t 7 (x 3 +y 3 )+t 3 x 2 +t 2 (xy 2 +y 2 )+t (x 2 y +x+y +1)+xy Elliptic Tropical Q F T F g(t F )=genus of T F = # of independent circles=1

25 Aim tropical j-invariant = tropicalisation of the j-invariant Which notions do we know by now? elliptic curves over C F j-invariant of an elliptic curve j = A tropicalisation of the j-invariant ord(j) tropical elliptic curve T F It remains to define the tropical j-invariant! Elliptic Tropical

26 General Assumption From now on deg(f ) = 3 and g(t F ) = 1! Observation g(t F ) = 1 T F has a cycle (1, 1) is visible in the Newton subdivision Elliptic Tropical Q F A = Q F 2

27 Edge γ in T F dual to Γ with end points v, w The direction vector of the edge γ: v(γ) = (w 2 v 2, v 1 w 1 ) 2 Elliptic Tropical w γ Γ v Q F v(γ) = (3 1, 1 0) = (2, 1)

28 Edge γ in T F dual to Γ with end points v, w The lattice length of γ: 2 Elliptic Tropical l(γ) = euclidean length of γ euclidean length of v(γ) w γ Γ v Q F γ = ( 1, 1) (1, 0) l(γ) = ( 2, 1) (2, 1) = 1

29 Definition If g(t F ) = 1 the tropical j-invariant of T F is: j(t F ) = lattice length of the cycle of T F. Elliptic Tropical w γ Γ v Q F j(t F ) = l(γ) + l(γ ) + l(γ ) = = 3

30 Note: F = y 3 x 3 2 t xy + (1 t) Elliptic Tropical Moreover: Hence: j(t F ) = l(γ) + l(γ ) + l(γ ) = = 3 j(c F ) = 1 t t t t 12 j(t F ) = ord ( j(c F ) ).

31 Tropical j-invariant Example: F = t 7 (x 3 +y 3 )+t 3 x 2 +t 2 (xy 2 +y 2 )+t (x 2 y +x+y +1)+xy Elliptic Tropical Q F j(t F ) = = 8 = ord ( j(c F ) )

32 Tropical j-invariant??? Theorem??? If deg(f ) = 3 and g(t F ) = 1, then j(t F ) = ord ( j(c F ) ). Elliptic Tropical

33 Tropical j-invariant??? Theorem??? If deg(f ) = 3 and g(t F ) = 1, then j(t F ) = ord ( j(c F ) ). Elliptic Tropical Problem F = tx 3 + txy 2 + t 7 y 3 + xy + ty 2 + y + t Then: j(t F ) = 4 5 = ord ( j(c F ) )!!! Leading terms in (a ) cancel out!!!

34 Generic Order of the j-invariant Definition A = ω A ωa ω, = ω ωa ω [a], a = (a ij ) (i,j) A. Elliptic Tropical For u A we define (basic idea: u ij = ord(a ij )) ord u (A) = min{u ω A ω 0} und ord u ( ) = min{u ω ω 0}. The generic order of the j-invariant is then ord (j) : A : u ord u (A) ord u ( ).

35 Generic Order of the j-invariant Definition A = ω A ωa ω, = ω ωa ω [a], a = (a ij ) (i,j) A. Elliptic Tropical For u A we define (basic idea: u ij = ord(a ij )) ord u (A) = min{u ω A ω 0} und ord u ( ) = min{u ω ω 0}. The generic order of the j-invariant is then ord (j) : A : u ord u (A) ord u ( ). Note, for ord(a ij ) = u ij and F = a ij x i y j generically ord ( j(c F ) ) = ord u (j).

36 Theorem When deg(f ) = 3, g(t F ) = 1 and u ij = ord(a ij ), then Elliptic Tropical j(t F ) = ord u (j) ord ( j(c F ) ). If u lies in a full dimensional cone of the secondary fan of A, then j(t F ) = ord ( j(c F ) ). Corollary If deg(f ) = 3 and ord ( j(c F ) ) 0, then T F has no cycle.

37 Why should one expect such a result? Mikhalkin (2006) shows to tropical curves of genus 1 are equivalent (=isomorphic) if and only if their tropical j-invariant coincides. Elliptic Tropical

38 Why should one expect such a result? Mikhalkin (2006) shows to tropical curves of genus 1 are equivalent (=isomorphic) if and only if their tropical j-invariant coincides. Kerber, H. Markwig (2006) # tropical elliptic curves with j-invariant = # algebraic elliptic with fixed j-invariant (see Pandharipande) Elliptic Tropical

39 Why should one expect such a result? Mikhalkin (2006) shows to tropical curves of genus 1 are equivalent (=isomorphic) if and only if their tropical j-invariant coincides. Kerber, H. Markwig (2006) # tropical elliptic curves with j-invariant = # algebraic elliptic with fixed j-invariant (see Pandharipande) Speyer 2005: Given a tropical curve T 2 with g(t ) = 1 and j(t ) = l, and given j with ord(j) = l, then the elliptic curve with j-invariant j can be embedded into 2 such that its tropicalisation is T. Elliptic Tropical

40 Note, j(t F ) does only depend on the Elliptic trop(f ) = max (i,j) A { u ij + i x + j y} with u ij = ord(a ij ). Let T u be the tropical curve defined by the right hand side, then we can define the function Tropical cl : A : u cycle length of T u.

41 Note, j(t F ) does only depend on the Elliptic trop(f ) = max (i,j) A { u ij + i x + j y} with u ij = ord(a ij ). Let T u be the tropical curve defined by the right hand side, then we can define the function Tropical cl : A : u cycle length of T u. We then have to show that cl(u) = ord u (j), whenever u U = {v A g(t v ) = 1}.

42 We then have to show that Elliptic cl(u) = ord u (j), Tropical whenever u U = {v A g(t v ) = 1}. Idea both functions are piece-wise linear find common domains of linearity compare the rules of assignment on these domains the proof mainly relies on two fans, the secondary fan of A and the Gröbner fan of

43 The of A Definition (Regular Marked Subdivision) Each u A induces a subdivision of the convex hull Q of A with certain lattice points marked, say S u. Example 0, (i, j) = (0, 0), (3, 0), (0, 3), (1, 0), u ij = 1 (i, j) = (1, 1), 1 else z = u ij j Elliptic Tropical lower faces of E u i S u

44 of A Elliptic Definition u and u in A are called equivalent if S u = S u. Easy Facts This defines an equivalence relation. The equivalence classes are the interior of cones. Their collection is the secondary fan SF(A) of A. u is in the interior of a full-dimensional cone if and only if S u is a triangulation with only vertices marked. Tropical

45 The of A Consequence The set U = {u A g(t u ) = 1} is a union of cones of the secondary fan of A. Elliptic Tropical

46 The of A Consequence The set U = {u A g(t u ) = 1} is a union of cones of the secondary fan of A. Lemma Let T be a triangulation of Q s.t. (1, 1) is visible and let C T the cone of T in SF(A), then cl CT is linear. be Elliptic Tropical

47 The of A Consequence The set U = {u A g(t u ) = 1} is a union of cones of the secondary fan of A. Lemma Let T be a triangulation of Q s.t. (1, 1) is visible and let C T be the cone of T in SF(A), then cl CT is linear. vertex = solution of u y = u 11 + x + y = u 00 depends only on triangle and linear on u Elliptic Tropical

48 The of A Consequence The set U = {u A g(t u ) = 1} is a union of cones of the secondary fan of A. Lemma Let T be a triangulation of Q s.t. (1, 1) is visible and let C T be the cone of T in SF(A), then cl CT is linear. Elliptic Tropical length of an edge = difference of two vertices linear in u

49 s Definition Let H = ω h ω a ω [ω] and u A. Then H u = h ω a ω ω u minimal is the initial form of H w.r.t. u, and we say u u H u = H u. Elliptic Tropical

50 s Definition Let H = ω h ω a ω [ω] and u A. Then u u H u = H u. Elliptic Tropical Easy Facts This defines an equivalence relation. The equivalence classes are the interior of cones. Their collection is the Gröbner fan GF(H) of H. u is in the interior of a full dimensional cone if and only if H u is a monomial.

51 The of Theorem The secondary fan of A is a refinement of the Gröbner fan of, i.e. cones of GF( ) are unions of cones of SF(A). Elliptic Tropical some fan F a refinement of F Corollary The function u ord u ( ) = min{u ω ω 0} is linear on each cone of SF(A).

52 What about u ord u (A)? Facts SF(A) is not a refinement of GF(A)! But, U is contained in a single cone of GF(A), namely the cone with ord u (A) = 12 u 11. Elliptic Tropical

53 What about u ord u (A)? Facts SF(A) is not a refinement of GF(A)! But, U is contained in a single cone of GF(A), namely the cone with ord u (A) = 12 u 11. Elliptic Tropical Corollary The function u ord u (j) = ord u (A) ord u ( ) is linear on each cone of SF(A) contained in U.

54 What about u ord u (A)? Corollary The function u ord u (j) = ord u (A) ord u ( ) Elliptic Tropical is linear on each cone of SF(A) contained in U. Finish the Proof list all cones of SF(A) contained in U compare cl u and ord u (j) on each cone the comparison can be done using polymake and Singular (849 of 1166 cones to consider!)

55 What about u ord u (A)? Corollary The function u ord u (j) = ord u (A) ord u ( ) Elliptic Tropical is linear on each cone of SF(A) contained in U. Remark Using results from Gelfand-Kapranov-Zelevinsky one can compare the functions by hand on cones of GF( )!

56 1 Theorem If (Q, A) is any marked polygon with precisely one interior lattice point then F = (i,j) A a ij x i y j defines an elliptic curve on the toric surface X A. The analogous statement for the j-invariant then holds. Example X A = 1 Q F C F O XA (2, 2) Elliptic Tropical

57 Is j-invariant the right term? Let R N = [[ 1 ]] t N and N = (( t 1 N )). Elliptic f [x, y] = N : f N [x, y] Tropical

58 Is j-invariant the right term? Let R N = [[ 1 ]] t N and N = (( t 1 N )). Elliptic f [x, y] w.l.o.g. = f R N [x, y] Tropical

59 Is j-invariant the right term? Let R N = [[ 1 ]] t N and N = (( t 1 N )). Elliptic f [x, y] w.l.o.g. = f R N [x, y] Tropical Thus C f is defined over a discrete valuation ring, and hence: ord ( j(c f ) ) < 0 V V N (f) = N /q for some q N (f) has bad reduction Then the special fibre in the Néron model of V of N ord(q) = N ord ( j(c f ) ) projective lines. N (f) is a cycle

60 Is j-invariant the right term? Let R N = [[ 1 ]] t N and N = (( t 1 N )). Elliptic f [x, y] w.l.o.g. = f R N [x, y] Tropical Thus C f is defined over a discrete valuation ring, and hence: ord ( j(c f ) ) < 0 V V N (f) = N /q for some q N (f) has bad reduction Maybe one should consider the cycle length rather as the tropicalisation of q for elliptic curves with bad reduction.