This article relates the Gross-Zagier formula with a simpler formula of Gross for special values of L-series, via the theory of congruences between modular forms. Given two modular forms f and g (of different levels) which are congruent but whose functional equations have sign - 1 and 1 respectively, and an imaginary quadratic field K satisfying certain auxiliary conditions, the main result gives a congruence between the algebraic part of L′(f/K, 1) (expressed in terms of Heegner points) and the algebraic part of the special value L(g/K, 1). Congruences of this type were anticipated by Jochnowitz, and for this reason are referred to as "Jochnowitz congruences.".
Euler systems and Jochnowitz congruences / M. Bertolini, H. Darmon. - In: AMERICAN JOURNAL OF MATHEMATICS. - ISSN 0002-9327. - 121:2(1999), pp. 259-281. [10.1353/ajm.1999.0010]
Euler systems and Jochnowitz congruences
M. Bertolini;
1999
Abstract
This article relates the Gross-Zagier formula with a simpler formula of Gross for special values of L-series, via the theory of congruences between modular forms. Given two modular forms f and g (of different levels) which are congruent but whose functional equations have sign - 1 and 1 respectively, and an imaginary quadratic field K satisfying certain auxiliary conditions, the main result gives a congruence between the algebraic part of L′(f/K, 1) (expressed in terms of Heegner points) and the algebraic part of the special value L(g/K, 1). Congruences of this type were anticipated by Jochnowitz, and for this reason are referred to as "Jochnowitz congruences.".
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