Computations with classical and p-adic modular forms

Abstract

We present p-adic algorithms for computing Hecke polynomials and Hecke eigenforms associated to spaces of classical modular forms, using the theory of overconvergent modular forms. The algorithms have a running time which grows linearly with the logarithm of the weight and are well suited to investigating the dimension variation of certain p-adically defined spaces of classical modular forms.

Stark Points and -Adic Iterated Integrals Attached to Modular Forms of weight one

“…Whenγ and/orh arise from weight-two Eisenstein series, these expressions are also related to p-adic regulators of Beilinson elements in the K -groups K 2 (X 1 (N )) or in K 1 (X 1 (N ) 2 ) (see [BD2] and [BDR1], [BDR2], respectively). On the computational side, using a method for computing with overconvergent modular forms via Katz expansions developed in [La1], the article [La2] describes an algorithm for the efficient numerical evaluation of these p-adic iterated integrals, and uses it to calculate certain Chow-Heegner points on the elliptic curve E which were first defined and studied by Shouwu Zhang. These global points, which arise when k 2 andγ andh are eigenvectors for T N with the same eigenvalues, have a well-understood geometric provenance, and can also be calculated by complex analytic means following the strategy described in [DRS, DDLR].…”

Computing overconvergent forms for small primes

“…We now sketch how Theorems A and B enable us to generalise previous work in the literature due to Wan [17], Lauder [14] and Darmon et al [8]. We work with Γ = Γ 0 (N ) for computational simplicity when appropriate; see Remark 1.…”

“…The previous subsection removed the restriction p 5 from the algorithm in [14] to compute this action, and it is now straightforward to compute the desired special value of the Rankin triple product p-adic L-function, as described in detail in [15]. We have implemented a version in Magma that works for all p, which is available on the author's web page.…”

“…There is an explicitly computable quadratic polynomial P ∈ Q[x], depending only on p and the level, such that N α P (α). [14]. By an application of Coleman's trick [5, equation (3.3)], it is particularly useful when k becomes very large.…”

“…As a result, we obtain a Z p -lattice which we compare to the one considered in [13]. In § 2, we present a number of applications of these results by removing the restriction p 5 from both Wan's quadratic bound for the Gouvêa-Mazur conjecture [17] and Lauder's algorithms for computing characteristic series for U p and constructing Chow-Heegner points on elliptic curves [14,15].…”