Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Rank 2 Drinfeld modular forms have been studied for more than 30 years, and while it is known that a higher rank theory could be possible, higher rank Drinfeld modular forms have only recently been de ned. In 1988 Gekeler published [Ge2] in which he studies the coe cients of rank 2 Drinfeld modular forms. The goal of this thesis is to perform a similar study of the coe cients of higher rank Drinfeld modular forms. The main results are that the coe cients themselves are (weak) Drinfeld modular forms, a product formula for the discriminant function, the rationality of certain naturally de ned modular forms, and the computation of some Hecke eigenforms and their eigenvalues. / AFRIKAANSE OPSOMMING: Drinfeld modulêre vorme van rang 2 word al vir meer as 30 jaar bestudeer en alhoewel dit lankal bekend is dat daar Drinfeld modulêre vorme van hoër rang moet bestaan, is die de nisie eers onlangs vasgepen. In 1988 het Gekeler die artikel [Ge2] gepubliseer waarin hy die koeffisiënte van Fourier reekse van rang 2 Drinfeld modulêre vorme bestudeer. Die doel van hierdie proefskrif is om dieselfde studie vir Drinfeld modulêre vorme van hoër rang uit te voer. Die hoofresultate is dat die koeffi siënte self (swak) Drinfeld modulêre vorme is, `n produk formule vir die diskriminant funksie, die feit dat sekere natuurlik gede finiëerde modulêre vorme rasionaal is, en die vasstelling van Hecke eievorme en hul eiewaardes.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第22231号 / 理博第4545号 / 新制||理||1653(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 池田 保, 教授 雪江 明彦, 教授 並河 良典 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

In this dissertation, we give a complete classification and list all identities of the form h = fg, where f , g and h are Hecke eigenforms of any weight with respect to Γ₁(N). This result extends the work of Ghate [Gha02] who considered this question for eigenforms with respect to Γ₁(N), with N square-free and f and g of weight 3 or greater. We remove all restrictions on the level N and the weights of f and g. For N = 1 there are only 16 eigenform identities, which are classically known. We first give a new proof of the level N = 1 case. We then give a proof which classifies all such eigenform identities for all levels N > 1. The identities fall into two categories. There are two infinite families of identities, given in Table 7.2. There are 209 other identities, listed (up to conjugacy) in Table 7.1. Thus any eigenform identity h = f g with respect to Γ₁(N) is either conjugate to an identity in Table 7.1 or takes the form of an identity described in Table 7.2.

In this thesis we present the connection between vertex operator algebras and modular forms which lies at the heart of Borcherds’ proof of the Monstrous Moonshine conjecture. In order to do so we introduce modular forms, vertex algebras, vertex operator algebras and their partition functions. Each notion is illustrated with examples.

We give congruences modulo powers of 2 for the Fourier coefficients of certain level 2 modular functions with poles only at 0, answering a question posed by Andersen and Jenkins. The congruences involve a modulus that depends on the binary expansion of the modular form's order of vanishing at infinity. We also demonstrate congruences for Fourier coefficients of some level 4 modular functions.

The main results of this thesis describe a relationship between two families of arithmetic divisors on an integral model of a Shimura curve. The first family, studied by Kudla, Rapoport and Yang, parametrizes abelian surfaces with specified endomorphism structure. The second family is comprised of pullbacks of arithmetic cycles on integral models of Shimura varieties associated to unitary groups of signature (1,1). In the thesis, we construct these families of cycles, and describe their relationship, which is expressed in terms of the ``Shimura lift", a classical tool in the theory of modular forms of half-integral weight. This relations can be viewed as further evidence for the modularity of generating series of arithmetic "special cycles" for U(1,1), and fits broadly into Kudla's programme for unitary groups.

Let V( ) denote a local system of weight on X = A2;n(C), where X is the moduli space of principle polarized abelian varieties of genus 2 over C with xed n-level structure. The inner cohomology of X with coe cients in V( ), H3 ! (X;V( )), has a Hodge ltration of weight 3. Each term of this Hodge ltration can be presented as space of cuspidal automorphic representations of genus 2. We consider the purely non-holomorphic part of H3 ! (X;V( )) denoted by H3 Ends(X;V( )). First of all we show that there is a non-zero subspace of H3 Ends(X;V( )) denoted by V (K), where K is an open compact subgroup of GSp(4;A), such that elements of V (K) are obtained by the global theta lifting of cuspidal automorphic representations of GL(2) GL(2)=Gm. This means that there is a non-zero part of H3 Ends(X;V( )) which is endoscopic. Secondly, we consider the local theta correspondence and nd an explicit answer for the level of lifted cuspidal automorphic representations to GSp(4; F) over a non-archimedean local eld F. Therefore, we can present an explicit way for nding a basis for V (K) for a xed level structure K. ii There is a part of the Hodge structure that only contributes in H(3;0) ! (X;V( )) H(0;3) ! (X;V( )). This part is endoscopic and coming from the Yoshida lift from O(4). Finally, in the case X = A2, if eendo(A2;V( )) denotes the motive corresponded to the strict endoscopic part (the part that contributes only in non-holomorphic terms of the Hodge ltration), then we have eendo(A2;V( )) = s 1+ 2+4S[ 1 2 + 2]L 2+1; (1) where = ( 1; 2) and is far from walls. Here S[k] denotes the motive corresponded to Sk, the space of cuspidal automorphic forms of weight k and trivial level, and sk = dim(Sk). ii

The main results of this thesis describe a relationship between two families of arithmetic divisors on an integral model of a Shimura curve. The first family, studied by Kudla, Rapoport and Yang, parametrizes abelian surfaces with specified endomorphism structure. The second family is comprised of pullbacks of arithmetic cycles on integral models of Shimura varieties associated to unitary groups of signature (1,1). In the thesis, we construct these families of cycles, and describe their relationship, which is expressed in terms of the ``Shimura lift", a classical tool in the theory of modular forms of half-integral weight. This relations can be viewed as further evidence for the modularity of generating series of arithmetic "special cycles" for U(1,1), and fits broadly into Kudla's programme for unitary groups.

Let V( ) denote a local system of weight on X = A2;n(C), where X is the moduli space of principle polarized abelian varieties of genus 2 over C with xed n-level structure. The inner cohomology of X with coe cients in V( ), H3 ! (X;V( )), has a Hodge ltration of weight 3. Each term of this Hodge ltration can be presented as space of cuspidal automorphic representations of genus 2. We consider the purely non-holomorphic part of H3 ! (X;V( )) denoted by H3 Ends(X;V( )). First of all we show that there is a non-zero subspace of H3 Ends(X;V( )) denoted by V (K), where K is an open compact subgroup of GSp(4;A), such that elements of V (K) are obtained by the global theta lifting of cuspidal automorphic representations of GL(2) GL(2)=Gm. This means that there is a non-zero part of H3 Ends(X;V( )) which is endoscopic. Secondly, we consider the local theta correspondence and nd an explicit answer for the level of lifted cuspidal automorphic representations to GSp(4; F) over a non-archimedean local eld F. Therefore, we can present an explicit way for nding a basis for V (K) for a xed level structure K. ii There is a part of the Hodge structure that only contributes in H(3;0) ! (X;V( )) H(0;3) ! (X;V( )). This part is endoscopic and coming from the Yoshida lift from O(4). Finally, in the case X = A2, if eendo(A2;V( )) denotes the motive corresponded to the strict endoscopic part (the part that contributes only in non-holomorphic terms of the Hodge ltration), then we have eendo(A2;V( )) = s 1+ 2+4S[ 1 2 + 2]L 2+1; (1) where = ( 1; 2) and is far from walls. Here S[k] denotes the motive corresponded to Sk, the space of cuspidal automorphic forms of weight k and trivial level, and sk = dim(Sk). ii

The modularity theorem implies that for every elliptic curve E /Q there exist rational maps from the modular curve X_0(N) to E, where N is the conductor of E. These maps may be expressed in terms of pairs of modular functions X(z) and Y(z) that satisfy the Weierstrass equation for E as well as a certain differential equation. Using these two relations, a recursive algorithm can be constructed to calculate the q - expansions of these parameterizations at any cusp. These functions are algebraic over Q(j(z)) and satisfy modular polynomials where each of the coefficient functions are rational functions in j(z). Using these functions, we determine the divisor of the parameterization and the preimage of rational points on E. We give a sufficient condition for when these preimages correspond to CM points on X_0(N). We also examine a connection between the algebras generated by these functions for related elliptic curves, and describe sufficient conditions to determine congruences in the q-expansions of these objects.