Supersingular module, Gross-Kudla formula and rational points of modular curves
In this well-written article, the author proves that for all prime number \(p\) outside a set \(\mathcal{C}\) of analytic density \(9/2^{10}\), the quotient of the modular curve \(X_0(p^r)\), \(r>1\), by the Atkin-Lehner involution has no rational points except cusps and CM points (theorem 0.6). The set \(\mathcal{C}\) is determined by explicit quadratic congruences. The method consists in applying a criterion due to \textit{P. J. R. Parent} [Compos. Math. 141, No. 3, 561--572 (2005; Zbl 1167.11310)] to well-chosen degree zero elements in the supersingular module annihilated by the winding ideal. More precisely, for \(p>3\) any prime number let \(\mathcal{P}\) denote the free abelian group generated by the isomorphism classes of supersingular elliptic curves over \(\overline{\mathbb{F}}_p\), let \(\mathcal{P}^0\) denote the subgroup of degree zero elements in \(\mathcal{P}\), let \(I_e\) denote the annihilator in the Hecke algebra of the set of primitive cusps forms \(f\) of level \(\Gamma_0(p)\) such that \(L(f,1)\neq 0\), and let \(\mathcal{P}^0[I_e]\) denote the subgroup of \(\mathcal{P}^0\) on which \(I_e\) acts trivially. Thanks to the famous formula involving triple products of \(L\)-functions due to \textit{B. H. Gross} and \textit{S. S. Kudla} [Compos. Math. 81, No. 2, 143--209 (1992; Zbl 0807.11027)], the author constructs an explicit family \((y^0_m)\), \(m>0\), of elements in \(\mathcal{P}^0[I_e]\otimes \mathbb{Q}\). She then applies Parent's criterion to this family. She proves in fact that the sequence \((y^0_m)\), \(m>0\), generates \(\mathcal{P}^0[I_e]\otimes \mathbb{Q}\) as a module over the Hecke algebra. Previous results were due to Parent [loc. cit.] replacing the set \(\mathcal{C}\) by one of analytic density \(7/2^9\). Parent used another family of \(\mathcal{P}^0[I_e]\otimes \mathbb{Q}\) which also generates this vector space. The author provides the link between these two families: up to some Eisenstein element, \(y_m^0\) is a linear combination of Parent's elements (proposition 0.2).
scientific article; zbMATH DE number 5366303