On the slopes of the {U_5} operator acting on overconvergent modular forms

Abstract

We show that the slopes of the U5 operator acting on slopes of 5adic overconvergent modular forms of weight k with primitive Dirichlet character χ of conductor 25 are given by eitherdepending on k and χ.

On the Up operator acting on p-adic overconvergent modular forms when X0(p) has genus 1

“…. , E 4 · z 6 , E 4 · (17/z) 6 } then we find that the characteristic power series modulo 17 13 these are the same slopes as those predicted by Buzzard's conjecture. We also note that the space S 4+16·17 (SL 2 (Z)) of classical modular forms is effectively computable by a computer algebra system such as Magma or Sage, and we find that the first few slopes of the Hecke operator U 17 acting on these modular forms are {1, 1, 1, 1, 3, 4}; it can be seen that the largest of these classical slopes is exactly k − 1.…”

“…. , z 6 , (11/z) 6 }, we find that the characteristic power series modulo 11 13 is given by There are conjectures of Buzzard (see [4,Section 2]) which predict what the slopes of the Hecke operators acting on classical modular forms of weight greater than 1 will be; we can use the work of Wan [17] on families of modular forms, for instance, which proves a weak version of the Gouvêa-Mazur conjecture (see [9]) which allows us to relate these slopes of weight k classical forms to the slopes of overconvergent forms of weight 0 that we compute. One can therefore check explicitly what the first few slopes should be in weight 0, and we find that the first few slopes are those we have computed above.…”

“…Let p = 11 and k = 0. We can compute the characteristic power series of U p acting on overconvergent modular forms of weight 0 using the procedure we have outlined above; for instance, if we compute the action of the U 11 operator modulo 11 13 acting on the basis {1, z, 11/z, . .…”

Slopes of the U7 operator acting on a space of overconvergent modular forms

“…. , 7, these can be extended to all i 1 by means of a 7th-order linear recurrence relation with coefficients inL(t), as in [11,Section 4]. The reason for this is essentially the same key fact which was used in the previous section, namely that inside the function field of X 0 (49), s is algebraic of degree 7 overL(y).…”

Spectral Expansions of Overconvergent Modular Functions

“…The results obtained will be independent of r, since it is known that any overconvergent U -eigenform of finite slope must extend to a function on X 0 (1) ≥p −r for all r < p p+1 (see [Buz03]). The slopes of U are somewhat mysterious; the complete list of slopes is known only for p = 2, tame level 1 and weight 0 by [BC05], and for 2-adic, 3-adic and 5-adic weights near the boundary of weight space by [BK05], [Jac03] and [Kil06] respectively. There are conjectures ( [Buz05], [Cla05]) for a general weight, prime and level, but these appear to be rather inaccessible at present.…”