Modular J-invariant function N. Saradha T.I.F.R. Jan 25, 2016 Student’s Seminar TIFR, Mumbai Let L = fm!1 + n!2 : m; n 2 Zg be a lattice spanned by the two linearly independent complex numbers !1 and !2: An elliptic function with respect to L is a meromorphic function f on C which satisfies f (z + !) = f (z) for all ! 2 L and z 2 C: D = fs!1 + t!2 : 0 ≤ s; t < 1g is called the fundamental parallelogram and any translate of this is referred to as a fundamental domain. L is called the period lattice. Elliptic function An elliptic function with respect to L is a meromorphic function f on C which satisfies f (z + !) = f (z) for all ! 2 L and z 2 C: D = fs!1 + t!2 : 0 ≤ s; t < 1g is called the fundamental parallelogram and any translate of this is referred to as a fundamental domain. L is called the period lattice. Elliptic function Let L = fm!1 + n!2 : m; n 2 Zg be a lattice spanned by the two linearly independent complex numbers !1 and !2: D = fs!1 + t!2 : 0 ≤ s; t < 1g is called the fundamental parallelogram and any translate of this is referred to as a fundamental domain. L is called the period lattice. Elliptic function Let L = fm!1 + n!2 : m; n 2 Zg be a lattice spanned by the two linearly independent complex numbers !1 and !2: An elliptic function with respect to L is a meromorphic function f on C which satisfies f (z + !) = f (z) for all ! 2 L and z 2 C: L is called the period lattice. Elliptic function Let L = fm!1 + n!2 : m; n 2 Zg be a lattice spanned by the two linearly independent complex numbers !1 and !2: An elliptic function with respect to L is a meromorphic function f on C which satisfies f (z + !) = f (z) for all ! 2 L and z 2 C: D = fs!1 + t!2 : 0 ≤ s; t < 1g is called the fundamental parallelogram and any translate of this is referred to as a fundamental domain. Elliptic function Let L = fm!1 + n!2 : m; n 2 Zg be a lattice spanned by the two linearly independent complex numbers !1 and !2: An elliptic function with respect to L is a meromorphic function f on C which satisfies f (z + !) = f (z) for all ! 2 L and z 2 C: D = fs!1 + t!2 : 0 ≤ s; t < 1g is called the fundamental parallelogram and any translate of this is referred to as a fundamental domain. L is called the period lattice. Here is an explicit example of an elliptic function. The Weierstrass }− function associated with L is defined by the series 1 X 1 1 }(z) = + − ; z2 (z − !)2 !2 !2L0 where L0 denotes the set of non-zero periods. It converges absolutely and uniformly on every compact subset of C=L: It is a meromorphic function having a double pole at every point of L and no other poles. The Weierstrass elliptic function }(z) The Weierstrass }− function associated with L is defined by the series 1 X 1 1 }(z) = + − ; z2 (z − !)2 !2 !2L0 where L0 denotes the set of non-zero periods. It converges absolutely and uniformly on every compact subset of C=L: It is a meromorphic function having a double pole at every point of L and no other poles. The Weierstrass elliptic function }(z) Here is an explicit example of an elliptic function. It converges absolutely and uniformly on every compact subset of C=L: It is a meromorphic function having a double pole at every point of L and no other poles. The Weierstrass elliptic function }(z) Here is an explicit example of an elliptic function. The Weierstrass }− function associated with L is defined by the series 1 X 1 1 }(z) = + − ; z2 (z − !)2 !2 !2L0 where L0 denotes the set of non-zero periods. It is a meromorphic function having a double pole at every point of L and no other poles. The Weierstrass elliptic function }(z) Here is an explicit example of an elliptic function. The Weierstrass }− function associated with L is defined by the series 1 X 1 1 }(z) = + − ; z2 (z − !)2 !2 !2L0 where L0 denotes the set of non-zero periods. It converges absolutely and uniformly on every compact subset of C=L: The Weierstrass elliptic function }(z) Here is an explicit example of an elliptic function. The Weierstrass }− function associated with L is defined by the series 1 X 1 1 }(z) = + − ; z2 (z − !)2 !2 !2L0 where L0 denotes the set of non-zero periods. It converges absolutely and uniformly on every compact subset of C=L: It is a meromorphic function having a double pole at every point of L and no other poles. Let us denote by X −2k G2k = G2k (L) = ! !2L0 the associated Eisenstein series of weight 2k. Then }(z) can be expressed as 1 1 X }(z) = + (2k + 1)G z2k : z2 2k+2 k=1 Eisenstein series Then }(z) can be expressed as 1 1 X }(z) = + (2k + 1)G z2k : z2 2k+2 k=1 Eisenstein series Let us denote by X −2k G2k = G2k (L) = ! !2L0 the associated Eisenstein series of weight 2k. Eisenstein series Let us denote by X −2k G2k = G2k (L) = ! !2L0 the associated Eisenstein series of weight 2k. Then }(z) can be expressed as 1 1 X }(z) = + (2k + 1)G z2k : z2 2k+2 k=1 It can be shown that for all z 2 C; z 62 L we have 0 2 3 } (z) = 4}(z) − 60G4}(z) − 140G6: Thus (}(z);}0(z)) lies on the curve defined by the equation 2 3 E : y = 4x − g2x − g3 where g2 = 60G4; g3 = 140G6: This cubic polynomial has the discriminant 3 2 g2 − 27g3 : It turns out that this discriminant is non-zero. Such curves are called elliptic curves (E). The differential equation for }(z) Thus (}(z);}0(z)) lies on the curve defined by the equation 2 3 E : y = 4x − g2x − g3 where g2 = 60G4; g3 = 140G6: This cubic polynomial has the discriminant 3 2 g2 − 27g3 : It turns out that this discriminant is non-zero. Such curves are called elliptic curves (E). The differential equation for }(z) It can be shown that for all z 2 C; z 62 L we have 0 2 3 } (z) = 4}(z) − 60G4}(z) − 140G6: This cubic polynomial has the discriminant 3 2 g2 − 27g3 : It turns out that this discriminant is non-zero. Such curves are called elliptic curves (E). The differential equation for }(z) It can be shown that for all z 2 C; z 62 L we have 0 2 3 } (z) = 4}(z) − 60G4}(z) − 140G6: Thus (}(z);}0(z)) lies on the curve defined by the equation 2 3 E : y = 4x − g2x − g3 where g2 = 60G4; g3 = 140G6: It turns out that this discriminant is non-zero. Such curves are called elliptic curves (E). The differential equation for }(z) It can be shown that for all z 2 C; z 62 L we have 0 2 3 } (z) = 4}(z) − 60G4}(z) − 140G6: Thus (}(z);}0(z)) lies on the curve defined by the equation 2 3 E : y = 4x − g2x − g3 where g2 = 60G4; g3 = 140G6: This cubic polynomial has the discriminant 3 2 g2 − 27g3 : Such curves are called elliptic curves (E). The differential equation for }(z) It can be shown that for all z 2 C; z 62 L we have 0 2 3 } (z) = 4}(z) − 60G4}(z) − 140G6: Thus (}(z);}0(z)) lies on the curve defined by the equation 2 3 E : y = 4x − g2x − g3 where g2 = 60G4; g3 = 140G6: This cubic polynomial has the discriminant 3 2 g2 − 27g3 : It turns out that this discriminant is non-zero. The differential equation for }(z) It can be shown that for all z 2 C; z 62 L we have 0 2 3 } (z) = 4}(z) − 60G4}(z) − 140G6: Thus (}(z);}0(z)) lies on the curve defined by the equation 2 3 E : y = 4x − g2x − g3 where g2 = 60G4; g3 = 140G6: This cubic polynomial has the discriminant 3 2 g2 − 27g3 : It turns out that this discriminant is non-zero. Such curves are called elliptic curves (E). Now we start with such an equation. Given (x; y) 2 C2; lying on the curve, can we find z such that x = }(z); y = }0(z)? If so, we can always parametrize such curves with the Weierstarss elliptic functions. Suppose }(z) = x has no solution. Then 1=(}(z) − x) is an elliptic function which is holomorphic on L: By periodicity, it is entire and bounded. Hence must be a constant- a contradiction. Hence y = ±}0(z): One can check that } is an even function and hence }0 is an odd function. So we can adjust the sign so that (x; y) = (}(z);}0(z)): We can parametrize Given (x; y) 2 C2; lying on the curve, can we find z such that x = }(z); y = }0(z)? If so, we can always parametrize such curves with the Weierstarss elliptic functions. Suppose }(z) = x has no solution. Then 1=(}(z) − x) is an elliptic function which is holomorphic on L: By periodicity, it is entire and bounded. Hence must be a constant- a contradiction. Hence y = ±}0(z): One can check that } is an even function and hence }0 is an odd function. So we can adjust the sign so that (x; y) = (}(z);}0(z)): We can parametrize Now we start with such an equation.