Differential eigenforms

Abstract

The aim of this paper is to show how differential characters of Abelian varieties (in the sense of [A. Buium, Differential characters of Abelian varieties over p-adic fields, Invent. Math. 122 (1995) 309-340]) can be used to construct differential modular forms of weight 0 and order 2 (in the sense of [A. Buium, Differential modular forms, Crelle J. 520 (2000) 95-167]) which are eigenvectors of Hecke operators. These differential modular forms will have "essentially the same" eigenvalues as certain classical complex eigenforms of weight 2 (and order 0).

Differential modular forms attached to newforms mod p

“…So ρ(O f [1/N, ζ N , ζ p ]) ⊂ R π . Exactly as in [7], Proposition 4.5, and using the same argument as on the top of page 992, [7], we get:…”

“…A basic construction in [7] attaches to classical newforms f = a n q n over C, of level N (i.e. on Γ 1 (N )) and weight 2, some δ p -modular forms f of level N , order 2, and weight 0, with δ p -Fourier expansion nicely expressible in terms of f (−1) (q) := a n n q n .…”

“…on Γ 1 (N )) and weight 2, some δ p -modular forms f of level N , order 2, and weight 0, with δ p -Fourier expansion nicely expressible in terms of f (−1) (q) := a n n q n . The forms f in [7] are arithmetic-differential objects that have no classical analogue but, rather, can be viewed as "dual" to the classical objects f ; by the way, the forms f introduced in [7] played a key role in [8] where the theory of arithmetic differential equations was used to prove finiteness results for Heegner-like points.…”

“…The Serre lift f being of weight 2 and level Np one can use the method in [7] to attach to it an element f in the ring of δ π -modular forms of level Np and weight 0. Using a construction from [9] one can interpret this f as a sum of δ π -modular forms of level N and weights 0, −1, −2, .…”

“…Note that each given κ has a priori several conjugates and the theorem does not tell which of these conjugates gives the weight of f ; be that as it may the weight −κ of f is always non-zero (in contrast to the weight of f in [7,9] which is equal to 0). Finally note that, as in [7], f in our Theorem 1.1 is not unique.…”