By a result of Serre, it’s known that a cusp form of weight $k\geq2$ and level $\Gamma_0(N)$ with some $\chi$ is lacunary if and only if it is in the space of CM newforms. Is there a similar result for the case of weight 1? Say, can we also characterize lacunary modular forms of weight 1 in terms of some special modular forms such as theta series? Any help is appreciated! Thank you!
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
3
Serre proved in 1985 that weight 1 cusp forms are always lacunary; see Theorem 19 of Serre's article
Serre, Jean-Pierre, Quelques applications du théorème de densité de Chebotarev, Publ. Math., Inst. Hautes Étud. Sci. 54, 123-202 (1981). ZBL0496.12011.
in which he proves that for any weight 1 cusp form $f$, we have $$\#\{ n \le X : a_n(f) \ne 0\} = O(X / (\log x)^{1/4}).$$
Certain special weight 1 forms satisfy estimates of the above form with powers of log larger than $1/4$; the largest possible exponent is $3/4$. In contrast, weight $\ge 2$ CM forms satisfy an analogous estimate with exponent $1/2$, so these special weight 1 forms are in a sense strictly more lacunary than any form of weight $\ge 2$.
answered Aug 31, 2023 at 7:39
-
$\begingroup$ Thanks so very much, David, for your response and explanation! I have a subsequent question that what about holomorphic modular forms of weight 1, not necessarily cuspidal. Is it still true that any holomorphic modular form of weight 1 is lacunary? $\endgroup$ Commented Sep 2, 2023 at 3:12
-
$\begingroup$ Yes, I think it is still true. Try to prove it for an Eisenstein eigenfrom (where the Galois rep is isomorphic to the direct sum of two characters). $\endgroup$ Commented Sep 3, 2023 at 18:07
-
$\begingroup$ Thank you, David, for your help! $\endgroup$ Commented Sep 4, 2023 at 12:44