Twists of elliptic curve L-functions by cubic and higher order characters

David, Fearnley, and Kisilevsky have made conjectures about the frequency of vanishing of cubic and higher twists of a fixed elliptic curve. For cubic, the conjecture is that about $D_E X^{1/2}\log^A X$ cubic twists with conductor up to $X$ will vanish for a certain $A$.

• Can we conjecture the constant $D_E$ here?

• Is there any variation for moduli in arithmetic progressions (as there is in the quadratic case)?

• Is there any sensible way to interpret the collection of special values of cubic twists of a fixed L-function (as in the quadratic case where coefficients of half-integral weight forms give the special values)?

See the website by Fearnley and Kisilevsky for more information and lots of data on these questions.

Back to the main index for Random Matrix Theory and central vanishing of L-functions.