Heegner Points on Cartan Non-split Curves

Abstract

Abstract. Let E Q be an elliptic curve of conductor N, and let K be an imaginary quadratic eld such that the root number of E K is − . Let O be an order in K and assume that there exists an odd prime p, such that p N, and p is inert in O. Although there are no Heegner points on X (N) attached to O, in this article we construct such points on Cartan non-split curves. In order to do that we give a method to compute Fourier expansions for forms on Cartan non-split curves, and prove that the constructed points form a Heegner system as in the classical case.

A Gross-Kohnen-Zagier theorem for non-split Cartan curves

“…We compute for negative discriminants D prime to 17 such that 17 is inert in Q( √ D) the special points Q + D,s (which, according to Remark 5.2, do not depend on s) and we give the integer m(D) such that, up to torsion, Q + D,s = m(D) Q. These points can be computed using the non-split Cartan curve as explained in [Koh17], [KP16]. They are constructed by giving an explicit modular parametrization π + : X + ns (17) → E, which amounts to finding an explicit cuspform for Γ + ns (17) with the same eigenvalues as f E for the Hecke operators, where f E is the cuspform of weight 2 and level 289 corresponding to E. In Table 1 below we record the computations for all valid discriminants of absolute value less than 200.…”

On Heegner points for primes of additive reduction ramifying in the base field

“…Our goal is to give an explicit construction in all cases where the local sign of Table 1 equals +1. The cells colored in light grey correspond to the classical construction, and the ones colored with dark grey are considered in the article [KP15]. In the present article we will consider the following cases:…”

A Moduli Interpretation for the Non-Split Cartan Modular Curve

“…Now, from the fact that Γ ∩ α −1 Γα = Γ ∩ Γ 0 ( ) or Γ ∩ αΓα −1 = Γ ∩ Γ 0 ( ), we deduce in a classical manner, as explained for instance in Section 6.3 in [DS05] taking Γ 1 = Γ 2 = Γ, that the double coset description of T coincide with the moduli-theoretic description we gave above. Compare with Theorem 1.11 in [KP14] for a different proof.…”

“…See Proposition 1.1 and Remark 1.3 in [KP14]. The following defines a PGL E[p] -equivariant bijection between the set of such endomorphisms φ and the set of necklaces on E. The endomorphism φ defines an element of order two in PGL E[p] without fixed point; so it belongs to a unique non-split Cartan subgroup H whose normaliser is the stabiliser of a necklace v. Conversely, every stabiliser of a necklace contains a unique element in PGL E[p] that lifts to an element φ ∈ GL E[p] with φ 2 = ε.…”

“…While finalising this article, we learnt of yet another moduli interpretation given by Kohen and Pacetti in [KP14]: Fix a choice of a quadratic non-residue ε modulo p. They represent each point in Y + nsp (k) by ak-isomorphism class of (E, φ) where E/k is an elliptic curve and φ ∈ GL E[p] is an element such that φ 2 is the multiplication by ε. See Proposition 1.1 and Remark 1.3 in [KP14].…”