Non-vanishing of class group $L$-functions at the central point

Heegner points and Eisenstein series

“…Previous aspect is of practical interest: when mollifying a family, an explicit expression for the main term of the asymptotic is required (as clearly explained in the discussion preceding Theorem 1.2 of [11]). In the present case, Blomer [1,Lemma 3.1] observed that a cancellation occurred in the explicit expression of c 00 given in [4]. At page 13 we check that the resulting expression is consistent with Theorem 1.…”

“…1 It is interesting to compare the content of our work with [24]. We carry out the discussion with a lot of details because the comparison applies to other contexts.…”

“…Moments of special values of L-functions associated to class group characters can often be interpreted as periods of particular automorphic form against Heegner points. In the present paper we focus on the special values |L(s, χ)| 2 for ℜe s = 1 2 , so that the period formula is well-known and due to Hecke. Our first result is the following: Theorem 1.…”

On the central value of symmetric square L-functions

“…1 The set B (η) is probably empty (under GRH), but in any case small. General zero-density estimates [11,Theorem 2] (see also [19, (29) …”

“…The calculation of the second moment is still not completely straightforward; we need a delicate cancellation of a certain portion of the diagonal term by some part of the off-diagonal term. This resembles a similar analysis for the second moment of class group L-functions [1]. The similarity is no coincidence: Let K := Q( √ −D) be an imaginary quadratic field with discriminant −D, u an even positive integer, and ξ a primitive Hecke Größencharacter on K satisfying ξ((α)) = (α/|α|) u for α ∈ K .…”