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Non-vanishing of the Central Derivative of Canonical Hecke L-functions

Stephen D. Miller·Tonghai Yang
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Abstract

In the early 1980s, Rohrlich began a study of canonical Hecke characters, which are closely related to the simplest examples of CM elliptic curves. He and Montgomery showed the non-vanishing of the central value when the L-function has an even functional equation, and we now show the non-vanishing of the central derivative when the functional equation is odd. Using the results of Gross-Zagier and Kolyvagin-Logachev, we can apply the non-vanishing to the ranks and Shafarevitch-Tate groups of the Q-curves "A(p)" studied by Gross in his thesis. In particular, their rank is determined by a congruence condition. http://www.math.yale.edu/users/steve/milleryang

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Stephen D. Miller, Tonghai Yang. 2000-03-20. Non-vanishing of the Central Derivative of Canonical Hecke L-functions. https://arxiv.org/abs/math/0003114

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