# Arithmetic jet spaces and modular forms

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# Arithmetic jet spaces and modular forms

 Title: Arithmetic jet spaces and modular forms Author: Saha, Arnab Advisor(s): Buium, Alexandru Committee Member(s): Boyer, CharlesNakamaye, MichaelBorger, James Department: University of New Mexico. Dept. of Mathematics and Statistics Subject: Mathematics, Algebraic Geometry, Number Theory LC Subject(s): Jets (Topology)Forms, Modular.Ring extensions (Algebra)Homeomorphisms.Geometry, Algebraic.Arithmetical algebraic geometry.Number theory.Fourier integral operators.Hecke operators. Degree Level: Doctoral Abstract: 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 $\d$. This theory was initiated by A. Buium in \cite{char}. The results in the first chapter are concerning the connection between arithmetic jet spaces and Witt vectors. Let $R = \widehat{\Z_p^{ur}}$ be the $p$-adic completion of the maximal unramified extension of $\Z_p$. If $A$ is an $R$-algebra and we denote $J^nA$ 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 \cite{jim2}. However, we give an explicit proof of this fact and the highlight of this proof is the construction of a ring homomorphism $\mcal{P}:A \map \bb{W}_n(J^nA)$ which is the analogue to the exponential map $\exp:A \map A[t]/(t^{n+1})$ given by $\exp(a) = \sum_{i=0}^n \frac{\partial^i a}{i!} t^i$. If we denote by $\bb{D}_n(B):= B[t]/(t^{n+1})$ then we show that there is a family of ring homomorphisms indexed by $\alpha \in B^{n+1},~ \Psi_\alpha: \bb{D}_1 \circ \bb{W}_n(B) \map \bb{W}_n \circ \bb{D}_1(B)$ for any ring $B$ and $n$. This gives yields the relation between a usual derivation $\partial$ and a $p$-derivation $\d$ given by $\partial \d x = p \d \partial x + (\partial x)^p - x^{p-1}\partial x$. This interaction is used to analyse the ring homomorphisms $\eta : TJ^nA \map J^nTA$ 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 $X \subset X_1(N)$ where $X_1(N)$ is the modular curve that parametrises elliptic curves and level $N$ structures on it. Let $M^\infty$ be the direct limit of the jet rings of $M$ which we call the ring of $\d$-modular forms. Then from the universality property of jet spaces, there are ring homomorphism $E^n: M^n \map R((q))^{\hat{}}[q',..., q^{(n)}]^{\widehat{}}$ which are prolongation of the given Fourier expansion map $E:M \map R((q))$. Hence $E^n$ is the $\d$-Fourier expansion of $M^\infty$. Denote by $\bb{S}^\infty = \lim_n Im(E^n)$. If $\overline{\bb{S}^\infty}$ denote the reduction mod $p$ of $\bb{S}^\infty$ then, one of our main results says that $\overline{\bb{S}^\infty}$ can be realised as an Artin-Schrier extension over $\overline {S^\infty}$ where $S$ is the coordinate ring of $X$. If we set all the indeterminates $q'=..=q^{(n)}=0$ then we obtain a ring homomorphism $M^\infty \map \mcal{W}$ where $\mcal{W}$ 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 $\mcal{W}$. We also classify the kernel of this homomorphism which is the $p$-adic closure of the $\d$-ideal $(f^\partial -1, f^1, \d(f^\partial -1), \d f^1,..., )$ where $f^\partial$ and $f^1$ are $\d$-modular forms with weights. This should be viewed as $\d$-analogue of the Theorem of Swinnerton-Dyer and Serre where the Fourier expansion over $\bb{F}_p$ of the modular forms has the kernel $(E_{p-1}-1)$, $E_{p-1}$ is the Hasse invariant. In the third chapter, we take the step to understand the $\d$-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 $R[[q]]^{\hat{}}[q']^{\widehat{}}$, the holomorphic subspace'' of $R((q))^{\hat{}}[q']^{\widehat{}}$. The definition of the Hecke operators away from the prime $p$ extends naturally from the classical definition of Hecke operators. At the prime $p$, we define $T_\kappa(p)$ on a $\d$-symmetric subspace'' of $\d$-modular forms using the definition of A. Buium introduced in \cite{eigen}. 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'' $\d$-modular forms whose $\d$-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 $\d$-modular forms. Graduation Date: May 2012 URI: http://hdl.handle.net/1928/20871
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