Rank-one twists of a certain elliptic curve
Let \(E\) denote a modular elliptic curve with corresponding newform \(F(z) = \sum_{n \geq 1} a(n) q^n\), where \(q = e^{2\pi i z}\). We define the \(D\)-th quadratic twist of \(F\), as \(F_D(z) = \sum_{n \geq 1} (\frac{D}{N})a(n) q^n\) and we define the \(D\)-th quadratic twist of \(E\) as \(E_D:Dy^2=P(x)\) where \(E:y^2=P(x)\). We note that these two notions of twisting are compatible in the sense that, up to a finite number of Euler factors, the \(L\)-functions \(L(E_D,s)\) and \(L(F_D,s)\) are the same. The Birch and Swinnerton-Dyer conjecture predicts that the rank of the elliptic curve \(E_D\) will be the same as the order of vanishing of either of these \(L\)-functions at \(s=1\). Thus it is natural to consider \[ M^r_F(X) = \#\{D: |D|< X; \text{ Ord}_{s=1}L(F_D,s) = r \}. \] It was conjectured by \textit{D. Goldfeld} [Number Theory, Carbondale, Springer Lect. Notes Math. 751, 108-118 (1979; Zbl 0417.14031)] that if \(r=0\) or 1, then \(M^r_F(X) \sim X/2\) and for \(r>1\), \(M^r_F(X) = o(1)\). For some special elliptic curves it is known that \(M^0_E(X) \gg X\). This is due to work of the reviewer [J. Am. Math. Soc. 11, 635-641 (1998; Zbl 0904.11015)] and the author [Duke Math. J. 98, No. 2, 397-419 (1999; Zbl 0979.11027) (see also C. R. Math. Acad. Sci., Soc. R. Can. 20, No. 3, 77-82 (1998; Zbl 0943.11028))]. The best general result for \(r=0\) is presently due to \textit{K. Ono} and \textit{C. Skinner} [Invent. Math. 134, 651-660 (1998)]. They prove that \(M^0_F(X) \gg X/\log(X)\). In the case that \(r\neq 0\), even less seems to be known. For instance, the best general result known to the reviewer when \(r=1\), due to \textit{A. Perelli} and \textit{J. Pomykała} [Acta Arith. 80, 149-163 (1997; Zbl 0878.11022)], states that \(M^1_F(X)\gg X^{1-\varepsilon}\). In this interesting paper, the author proves that if \(E\) is the modular elliptic curve \(X_0(19)\) and if \(F\) is its corresponding newform, then \(M^0_F(X)\gg X\) and \(M^1_F(X)\gg X\). The proof proceeds as follows. The author makes use of a theorem of \textit{H. Davenport} and \textit{H. Heilbronn} [Proc. R. Soc. Lond., Ser. A 322, 405-420 (1971; Zbl 0212.08101)] as refined by \textit{J. Nakagawa} and \textit{K. Horie} [Proc. Am. Math. Soc. 104, 20-24 (1988; Zbl 0663.14023)] which along with a classical reflection theorem of \textit{A. Scholz} [J. Reine Angew. Math. 166, 201-203 (1932; Zbl 0004.05104)] guarantees that for any suitable arithmetic progression \(A\), \[ \#\{0<c<X:c\in A; 3\nmid h(-3c); 3\nmid h(c)\} \gg X. \] The author then makes use of a theorem of \textit{B. H. Gross} [Modular forms, Symp. Durham/Engl. 1983, 87-105 (1984; Zbl 0559.14011)] which guarantees the non-triviality of the Heegner point on \(X_0(19)\) associated to the quadratic fields \(\mathbb Q(\sqrt{-3c})\) and \(\mathbb Q(\sqrt{c})\) for \(c\) as in the previous equation. He then uses the Gross-Zagier formula [\textit{B. H. Gross} and \textit{D. B. Zagier}, Invent. Math. 84, 225-320 (1986; Zbl 0608.14019)] to determine that the non-triviality of this Heegner point simultaneously forces a simple zero at \(s=1\) for \(L(F_c,s)\) and forces \(L(F_{-3c},1) \neq 0\).
scientific article; zbMATH DE number 1221873