Addendum: Exponential decay in the frequency of analytic ranks of automorphic L-functions
نویسندگان
چکیده
Since the paper [KM1] was released much progress has been made on the problem of bounding the analytic rank of automorphic forms on average. For example, in [KMV], a uniform bound for the square of the analytic rank of automorphic L-functions was obtained. This was used in getting a sharp numerical upper bound for average of the analytic rank. However, this improvement used only a slight variant of the methods of [KM1]. In fact, it is possible to pursue this idea further and it turns out that much more is true. Recall the notations from [KM1]: for q a prime number, let S2(q)∗ be the set of primitive forms of weight 2 and level q, normalized so that their first fourier coefficient is 1. For f ∈ S2(q)∗, let L(f, s) be the associated (normalized) L-function, and rf := ords=1/2L(f, s) be the analytic rank of f . We prove the following:
منابع مشابه
Exponential Decay in the Frequency of Analytic Ranks of Automorphic L-functions
This note should be seen as an appendum to the work of E. Kowalski and the second author [KM1] and deals with the problem of bounding, unconditionally, the order of vanishing at the critical point in a family of L-functions. This problem is illustrated in the particular case of L-functions of weight-2 primitive modular forms of prime level. Recall the notation from [KM1]: For q a prime number, ...
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