Canonical periods and congruence formulae
The results in this paper state roughly that congruences between the Fourier coefficients of modular forms give rise to congruences between the critical values for their associated \(L\)-functions. The author shows that periods may be selected for the associated \(L\)-functions so that one may precisely formulate the notion of a congruence between critical values. More precisely, the author proves the following. Let \(p\) be any odd prime and let \(f=\sum a_n q^n\) and \(g=\sum b_nq^n\) be cuspidal newforms of weight 2 on \(\Gamma_1(M)\), such that \(a_n\equiv b_n\pmod{\mathfrak p^r}\), for some prime \(\mathfrak p\) above \(p\) in \(\overline{\mathbb Q}\). Assume that \((M,p)=1\) and that the residual representation attached to \(f\) is irreducible. Fix an isomorphism \(\mathbb C_p \cong \mathbb C\), such that the prime \(p\) of \(\overline{\mathbb Q} \subset \mathbb C\) induces the usual absolute value on \(\mathbb C_p\). Then there exist canonical periods \(\Omega^{\pm}_f\) and \(\Omega^{\pm}_g\) such that the congruence \[ \tau(\overline{\chi}) \frac{L(1,f,\chi)}{-2\pi i \Omega^{\pm}_f} \equiv \tau(\overline{\chi}) \frac{L(1,g,\chi)}{-2\pi i \Omega^{\pm}_g} \pmod{\mathfrak p^r} \] holds for every character \(\chi\). Furthermore, there exists a \(\chi\) such that both sides of the congruence are nonzero modulo \(\mathfrak p\). The author also has some results in the residually reducible case as well. As a corollary, the author shows that most elliptic curves \(E\) with a rational point of order 3 have the property that a positive proportion of its quadratic twists \(E_D\) have rank zero. He does this by employing a theorem of Davenport and Heilbronn on the distribution of 3-torsion in the class groups of quadratic number fields. These results generalize previous examples by \textit{W. Kohnen} [J. Reine Angew. Math. 508, 179-187 (1999; Zbl 0968.11023)] and the reviewer [J. Am. Math. Soc. 11, 635-641 (1998; Zbl 0904.11015)].
scientific article; zbMATH DE number 1425220
scientific article