Arithmetic Jet Spaces and Modular Forms

In this thesis, we would look into the theory of arithmetic jet spaces and its application in modular forms. The arithmetic jet spaces can be thought of as an analogue of jet spaces in differential algebra. In the case of arithmetic jet spaces, a derivation is replaced by p-derivation delta. This theory was initiated by A. Buium in [7]. The results in the first chapter are concerning the connection between arithmetic jet spaces and Witt vectors. Let R = Zurp &d14; be the p-adic completion of the maximal unramified extension of Zp . If A is an R-algebra and we denote JnA its n-th jet ring. Firstly, we show the adjunction property which says that the arithmetic jet functor from rings to rings is the left adjoint of the Witt vector functor. This property was also shown by Borger in [3]. However, we give an explicit proof of this fact and the highlight of this proof is the construction of a ring homomorphism P:A→Wn JnA which is the analogue to the exponential map exp : A → A[t]/(tn +1) given by exp(a) = i=0n6 iai!ti . If we denote by Dn (B) := B[t]/( tn+1) then we show that there is a family of ring homomorphisms indexed by alpha ∈ Bn+1, Psialpha : D1&j0;Wn B→W n&j0;D1 (B) for any ring B and n . This gives yields the relation between a usual derivation ∂ and a p-derivation delta given by ∂delta x = pdelta∂x + (∂ x)p - xp -1∂x. This interaction is used to analyse the ring homomorphisms eta : TJnA → JnTA where T associates the tangent ring to the ring A..;In the second chapter of the thesis, we apply the theory of arithmetic jet spaces to modular forms. Let M denote the ring of modular forms over an affine open embedding X ⊂ X 1(N) where X1( N) is the modular curve that parametrises elliptic curves and level N structures on it. Let Minfinity be the direct limit of the jet rings of M which we call the ring of delta-modular forms. Then from the universality property of jet spaces, there are ring homomorphism En : Mn → R((q)).;[q', ..., q(n)].; which are prolongationof the given Fourier expansion map E : M → R((q)). Hence En is the delta-Fourier expansion of Minfinity. Denote by Sinfinity = limn Im(En). If Sinfinity denote the reduction mod p of Sinfinity then, one of our main results says that Sinfinity can be realised as an Artin-Schrier extension over Sinfinity where S is the coordinate ring of X. If we set all the indeterminates q' = .. = q (n) = 0 then we obtain a ring homomorphism Minfinity → W where W is the ring of generalised p-adic modular forms. Our next result shows that the image of the above homomorphism is p-adically dense in W . We also classify the kernel of this homomorphism which is the p-adic closure of the delta-ideal (f∂ ) - 1, f1, delta( f∂ - 1), deltaf 1, ..., ) where f∂ and f1 are delta-modular forms with weights. This should be viewed as delta-analogue of the Theorem of Swinnerton-Dyer and Serre where the Fourier expansion over Fp of the modular forms has the kernel (E p-1 - 1), Ep -1 is the Hasse invariant.;In the third chapter, we take the step to understand the 'delta-Fourier expansion principle' and the action of the Hecke operators on the Fourier expansion of differential modular forms. We work on k[[ q]][q'] which is the reduction mod p of R[[q]].;[q'].;, the "holomorphic subspace" of R((q)).;[q'].;. The definitionof the Hecke operators away from the prime p extends naturally from the classical definition of Hecke operators. At the prime p, we define Tkappa(p) on a "delta-symmetric subspace" of delta-modular forms using the definition of A. Buium introduced in [11]. Our main result states that there is a one-to-one correspondence between the classical cusp forms which are eigenvectors of all Hecke operators with "primitive" delta-modular forms whose delta-Fourier series lies in k[[q]][q'] and are eigenvectors of all Hecke operators. This chapter should be viewed as the first attempt to understand the structure of eigenforms on the Fourier side of delta-modular forms.