Hecke operators on differential modular forms mod p

Abstract

A description is given of all primitive δ-series mod p of order 1 which are eigenvectors of all the Hecke operators nTκ(n), "pTκ(p)", (n, p) = 1, and which are δ-Fourier expansions of δ-modular forms of arbitrary order and weight w with deg(w) = κ ≥ 0; this set of δ-series is shown to be in a natural one-to-one correspondence with the set of series mod p (of order 0) which are eigenvectors of all the Hecke operators T κ+2 (n), T κ+2 (p), (n, p) = 1 and which are Fourier expansions of (classical) modular forms of weight ≡ κ+2 mod p−1.

Foundations of Arithmetic Differential Geometry

“…A theory of arithmetic differential equations was developed in a series of papers [23]- [42], [7] and was partly summarized in our monograph [35] (cf. also the survey papers [43,102]); this theory led to a series of applications to invariant v vi PREFACE theory [27,7,28,35], congruences between modular forms [27,37,38], and Diophantine geometry of Abelian and Shimura varieties [24,36]. On the other hand an arithmetic differential geometry was developed in a series of papers [42]- [47], [8]; the present book follows, and further develops, the theory in this latter series of papers.…”

Differential calculus with integers

“…There are two paths towards such a theory so far. One path is via δ-symmetry [24,26]; this is a characteristic 0 analogue of the concept of δ-p-symmetry mod p discussed above. Another path is via δ-overconvergence [27].…”

“…Cf. [15,2,16,26,17,25]. Let X 1 (N ) be the complete modular curve over R of level N > 4 and let L 1 (N ) → X 1 (N ) be the line bundle with the property that the sections of its various powers are the classical modular forms on Γ 1 (N ) of various weights.…”

“…In contrast to this T (p) and hence U are still mysterious in the theory of δ-modular forms. An step in understanding U was taken in [26] where the theory mod p for series of order 1 was given a rather definitive treatment. However the theory in characteristic zero seems elusive at this point.…”

Arithmetic jet spaces

“…It is noteworthy to mention that this theory was developed in the framework of p-adic formal geometry. Since then it has been extensively studied and applied to diophantine geometry [Bui96], [BP], differential modular forms [Bui00], [BuSa1], [BuSa2] and p-adic Hodge theory [BoSa1], [BoSa2]. Very recently the theory of δ-rings (rings endowed with a p-derivation δ) have led to the development of the prismatic cohomology by Bhatt and Scholze [BhSc].…”