Optimal windowed test functions for bounding analytic ranks of elliptic curves
Assuming the Generalized Riemann Hypothesis, the explicit formula for the L-function of an elliptic curve E/Q bounds the analytic rank of E in terms of the traces a_p for p < e^(2L), for any admissible test function supported on a window [-L,L]. In rank computations this test function is usually fixed in advance, typically to a Fejer-type kernel; optimising it for an individual L-function goes back to Booker, and an earlier test on quadratic twists of rank about 7 found the gain to be small. We choose, for each individual curve, the test function that minimises the resulting bound within a…
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