Error report: the modular curve X_0(11).

[Yang, Yuan]

Hi LMFDB team, I'd like to report an error about the defining equation of the modular curve X_0(11).

In your page LMFDB - Elliptic curve with LMFDB label 11.a2 (Cremona label 11a1), you said this is a model of the modular curve X_0(11), but I'd like to point out that this is only a model of X_0(11) over C, the complex number, not over Q.

The reason is as follows.

The modular curve X_0(11) over Q, should be defined as a smooth projective curve over Q with function field Q(j(z),j(11z)), where j(z) is the modular j-function. (See Diamond-Shurman GTM 228) However, assuming Gross-Zagier formula is right, the function field of curve 11.a2 is actually not isomorphic to Q(j(z),j(11z)).

We consider the Heegner point (denoted by P) of discriminant (-8) on X_0(11). There is only one such point(up to automorphisms of X_0(11)), corresponding to z=(3+\sqrt{-2})/11, represented by a degree 11 isogeny of CM elliptic curves C/aO_K-->C/O_K, here O_K is the ring Z[\sqrt{-2}], the ring of integers of K=Q(\sqrt{-2}), and a=3+\sqrt{-2}, a principle ideal. P is defined over Q because j(z)=j(11z)=8000 are both rational. If 11.a2 is the modular curve X_0(11), then since rank(11.a2)=0, P is torsion. Then the Gross-Zagier formula tell us that L'(E/K,s) is non-zero. But the twist of E by -8 is 704.c2 has rank 1, which implies L'(E/K,s) is non-zero, contradiction! 

The reference you might refer to is Yifan Yang's Defining Equation of Modular Curves, in this paper the author defined a Weierstrass equation exactly the same equation with 11.a2. However, we have to notice that in this paper the author only required C(X,Y)=C(j(z), j(11z)), which means that the change of variables (X,Y)-->(j(z),j(11z)) might not be defined over Q.

As above, with respect,

Yuan

[John Voight]

Dear Yuan,

Thank you for writing!  We hope the LMFDB has been useful to you, it's great to be able to discuss such concrete questions. 

The equation for X_0(11) is correct, and it's about as classical as it gets.  Tom Weston (https://www.math.arizona.edu/~swc/aws/2001/01Weston1.pdf) has a nice, self-contained, from scratch derivation of this equation.

We consider the Heegner point (denoted by P) of discriminant (-8) on X_0(11). There is only one such point(up to automorphisms of X_0(11)), corresponding to z=(3+\sqrt{-2})/11, represented by a degree 11 isogeny of CM elliptic curves C/aO_K-->C/O_K, here O_K is the ring Z[\sqrt{-2}], the ring of integers of K=Q(\sqrt{-2}), and a=3+\sqrt{-2}, a principle ideal. P is defined over Q because j(z)=j(11z)=8000 are both rational. If 11.a2 is the modular curve X_0(11), then since rank(11.a2)=0, P is torsion. Then the Gross-Zagier formula tell us that L'(E/K,s) is non-zero. But the twist of E by -8 is 704.c2 has rank 1, which implies L'(E/K,s) is non-zero, contradiction! 

The apparent contradiction is due to a lack of specifying what "the Heegner point" means.  Over what field? 

• If you meant that P = P_K is in X_0(11)(K), then what you say is correct: it has infinite order in the Mordell-Weil group over K, and this is consistent with Gross-Zagier. 
  
• If you meant that P = P_QQ is in X_0(11)(QQ), then you have to take a trace of P_K from K to QQ, i.e. P_QQ = P_K + sigma(P_K) where sigma is the nontrivial Galois automorphism of K, and the result is indeed torsion.  This is consistent with it giving a point of infinite order on the quadratic twist (up to torsion). 
  

But you can't use P to mean both simultaneously, or you indeed get a contradiction!

JV