The Legendre family of elliptic curves

Question

The Legendre family of elliptic curves over the $\lambda$ line is given by the equation $$E_{\lambda}:y^2=x(x-1)(x-\lambda),\lambda \in P^1_{\mathbb{C}},$$ which has three singular points, $\lambda=0,1,\infty$. I remember that I read from a reference, which I could not find now, that the Legendre family is the family of elliptic curve over $\Gamma(2) \backslash \mathbb{H}$ (or $\Gamma_0(2) \backslash \mathbb{H}$, which I cannot remember clearly), where $\mathbb{H}$ is the upper half plane of $\mathbb{C}$. Could anyone explain why? How to construct an explicit "isomorphism" between $P^1_{\mathbb{C}}$ and $\Gamma(2) \backslash \mathbb{H}$ (or $\Gamma_0(2) \backslash \mathbb{H}$)?

Answer 1

Not a full answer but a suggestion, aiming at being improved

• Let $\Phi_2(z)(Y) = \prod_{\gamma \in SL_2(\mathbb{Z})\setminus A_2} (Y-j(\gamma(z))) = (Y-j(2z))(Y-j(\frac{z}2))(Y-j(\frac{z+1}2))$ where $A_2$ are the integer matrices with determinant $2$. Its coefficients are modular functions so $\Phi_2(z)(Y) = \phi_2(Y)$ where $ \phi_2 \in \mathbb{C}(j)[Y]$ is the modular polynomial, the minimal polynomial of $j_2(z) = j(2z)$.

• The splitting field of $\phi_2$ is $\mathbb{C}(X(2))$. Since $\deg(\phi_2) = 3$ and $\mathbb{C}(j_2,j)/\mathbb{C}(j)$ is not Galois then $[\mathbb{C}(X(2)):\mathbb{C}(j)] = 3!$.

• Looking at $j(E_\lambda)$ then $j = 256\frac{(1-\lambda+\lambda^2)^3}{(\lambda^2(1-\lambda)^2)}$

• So we can see $\phi_2$ as being an element $\in \mathbb{C}(\lambda)[Y]$ in which case its splits completely as shown by this magma code. Therefore $\mathbb{C}(\lambda) \supset \mathbb{C}(X(2))$

    phi2 := ClassicalModularPolynomial(2); 
    F<lambda> := FunctionField(Rationals()); 
    j := 256*(1-lambda+lambda^2)^3/(lambda^2*(1-lambda)^2); 
    P<j2> := PolynomialRing(F); h := Evaluate(phi2,[j,j2]); 
    Factorization(h); 
  

• As $ [\mathbb{C}(\lambda):\mathbb{C}(j)]=6$ then $\mathbb{C}(\lambda) = \mathbb{C}(X(2))$ and $z \mapsto \lambda(z)$ is an isomorphism of Riemann surface $X(2) \to P^1(\mathbb{C})$

What is $\lambda(z)$ here ? You can define it as a function of $j(z),j(2z),j(\gamma(z))=j(z/2)$

phi2 := ClassicalModularPolynomial(2);
u<j> := FunctionField(Rationals());
P<T> := PolynomialRing(u); 
v<j2> := ext<u|Evaluate(phi2,[j,T])>;
P<T> := PolynomialRing(v);
fac := Factorization(Evaluate(phi2,[j,T]));
w<j2g> := ext<v|fac[2][1]>;
P<lambda> := PolynomialRing(w);
fac := Factorization((lambda^2*(1-lambda)^2)*j-256*(1-lambda+lambda^2)^3);
fac[1][1];

Otherwise you can say locally around some $z_0$ it is any continuous function such that $j(z) = 256\frac{(1-\lambda(z)+\lambda(z)^2)^3}{(\lambda(z)^2(1-\lambda(z))^2)}$ extended by analytic continuation to the upper half-plane and to the modular curve, the obtained function will be related to the usual $\lambda$-function by $z\mapsto \gamma(z)$ for some $\gamma \in SL_2(\mathbb{Z}/2\mathbb{Z})$ ie. an automorphism of $X(2)/X(1)$. That's why wiki's article starts with the first few coefficients at $i\infty$ to distinguish between $\lambda(z)$ and $\lambda(\gamma(z))$.

Answer 2

The Legendre family is isomorphic to $X(2) = H/\Gamma(2)$. The latter space is easily seem to be the same as the space of complex tori E with the choice of an isomorphism between $Z/2 \times Z/2$ and the 2-torsion $E[2]$. It is a covering space of $M_1 = H/SL_2(Z)$ with deck group $SL_2(Z/2)$. In the Legendre family, the origin of $E_\lambda$ is at infinity, and the points of order exactly 2 are the points with $x=0,1$ and $\lambda$ and $y=0$. (Note that the tangent line to $E$ at a point $P$ passes through $E$ at the point $Q = -2P$ in the group law on $E$; and the points of order 2 just listed are where the cubic curve has a vertical tangent, passing through the flex at infinity.)

In concrete terms, if $E$ in $X(2)$ has marked 2-torsion $0,a,b,c$, then there is a unique isomorphism between $E/(z \rightarrow -z)$ and the projective line that sends $0$ to $\infty$ and $(a, b, c)$ to $(0,1,\lambda)$ for some $\lambda$. In other words $\lambda$ is the cross-ratio of the branch points of the Wierstrass $p$--function, ordered by the marking. This allows one to write down an explicit formula for $\lambda$ as a function of $\tau$ in $ H$. See Ahlfors, Complex Analysis, Chapter 7: Elliptic functions, section 3.4: The modular function $\lambda(\tau)$.

More geometrically, a fundamental domain for $\Gamma(2)$ is given by the ideal hyperbolic quadrilateral with vertices $(0,1,-1,\infty)$. The side identifications yield as quotient a sphere with 3 points removed. This sphere is naturally identified with the $\lambda$-line with $0,1,\infty$ removed.