We are interested in investigating the number of integral points on quadrics. First, we consider non-degenerate plane conic curves defined over Z. In particular we look at two types of conic sections: hyperbolas with two rational points at infinity, and ellipses. We give upper bounds for the number of integral points on such curves which depends on the number of divisors of the determinant of a given conic. Next we consider quadratic surfaces of the form q(x, y, z) = k, where k is an integer and q is a non-degenerate homogeneous quadratic form defined over Z. We give an upper bound for the number of integral points (x, y, z) with bounded height.

We are interested in investigating the number of integral points on quadrics. First, we consider non-degenerate plane conic curves defined over Z. In particular we look at two types of conic sections: hyperbolas with two rational points at infinity, and ellipses. We give upper bounds for the number of integral points on such curves which depends on the number of divisors of the determinant of a given conic. Next we consider quadratic surfaces of the form q(x, y, z) = k, where k is an integer and q is a non-degenerate homogeneous quadratic form defined over Z. We give an upper bound for the number of integral points (x, y, z) with bounded height.

In this thesis we study moduli stacks \calM_\Phi^n, indexed by an integer n>0 and a CM-type (K,\Phi), which parametrize abelian schemes equipped with action by \OK and an \OK-linear principal polarization, such that the representation of \OK on the relative Lie algebra of the abelian scheme consists of n copies of each character in \Phi. We do this by systematically applying Serre's tensor construction, and for that we first establish a general correspondence between polarizations on abelian schemes M\otimes_R A arising from this construction and polarizations on the abelian scheme A, along with positive definite hermitian forms on the module M. Next we describe a tensor product of categories and apply it to the category \Herm_n(\OK) of finite non-degenerate positive-definite \OK-hermitian modules of rank n and the category fibred in groupoids \calM_\Phi^1 of principally polarized CM abelian schemes. Assuming n is prime to the class number of K, we show that Serre's tensor construction provides an identification of this tensor product with a substack of the moduli space \calM_\Phi^n, and that in some cases, such as when the base is finite type over \CC or an algebraically closed field of characteristic zero, this substack is the entire space. We then use this characterization to describe the Galois action on \calM_\Phi^n(\overline{\QQ}), by using the description of the action on \calM_\Phi^1(\overline{\QQ}) supplied by the main theorem of complex multiplication.

In this thesis we study moduli stacks \calM_\Phi^n, indexed by an integer n>0 and a CM-type (K,\Phi), which parametrize abelian schemes equipped with action by \OK and an \OK-linear principal polarization, such that the representation of \OK on the relative Lie algebra of the abelian scheme consists of n copies of each character in \Phi. We do this by systematically applying Serre's tensor construction, and for that we first establish a general correspondence between polarizations on abelian schemes M\otimes_R A arising from this construction and polarizations on the abelian scheme A, along with positive definite hermitian forms on the module M. Next we describe a tensor product of categories and apply it to the category \Herm_n(\OK) of finite non-degenerate positive-definite \OK-hermitian modules of rank n and the category fibred in groupoids \calM_\Phi^1 of principally polarized CM abelian schemes. Assuming n is prime to the class number of K, we show that Serre's tensor construction provides an identification of this tensor product with a substack of the moduli space \calM_\Phi^n, and that in some cases, such as when the base is finite type over \CC or an algebraically closed field of characteristic zero, this substack is the entire space. We then use this characterization to describe the Galois action on \calM_\Phi^n(\overline{\QQ}), by using the description of the action on \calM_\Phi^1(\overline{\QQ}) supplied by the main theorem of complex multiplication.

For a fixed rational prime p and primitive p-th root of unity ζ, we consider the Jacobian, J, of the complete non-singular curve give by equation yᵖ = xᵃ(1 − x)ᵇ. These curves are quotients of the p-th Fermat curve, given by equation xᵖ+yᵖ = 1, by a cyclic group of automorphisms. Let k = Q(ζ) and k(S) be the maximal extension of k unramified away from p inside a fixed algebraic closure of k. We produce a formula for the image of certain coboundary maps in group cohomology given in terms of Massey products, applicable in a general setting. Under specific circumstance, stated precisely below, we can use this formula and a pairing in the Galois cohomology of k(S) over k studied by W. McCallum and R. Sharifi in [MS02] to produce non-trivial elements in the Tate-Shafarevich group of J. In particular, we prove a theorem for predicting when the image of certain cyclotomic p-units in the Selmer group map non-trivially into X(k, J). / Q(zeta) and k_S be the maximal extension of k unramified away from p inside a fixed algebraic closure of k. We produce a formula for the image of certain coboundary maps in group cohomology given in terms of Massey products, applicable in a general setting. Under specific circumstance, stated precisely below, we can use this formula and a pairing in the Galois cohomology of k_S over k studied by W. McCallum and R. Sharifi to produce non-trivial elements in the Tate-Shafarevich group of J. In particular, we prove a theorem for predicting when the image of certain cyclotomic p-units in the Selmer group map non-trivially into Shah(k,J).

We give algorithms to compute Coleman integrals on superelliptic curves over unramified extensions of the p-adics, and apply these via Chabauty methods to determine the set of rational points on such curves. We also determine the solution to an explicit instance of the Shafarevich conjecture by finding all elliptic curves with good reduction outside of the first 6 primes, subject to a heuristic. We use a combination of non-abelian Chabauty and the Mordell--Weil sieve to determine the rational points on several quotient modular curves, and therefore classify pairs of elliptic curves over the rationals with 67-, 73-, and 107-isogenies. We give methods to explicitly compute Coleman integrals on modular curves using a canonical lift of Frobenius and canonical local coordinates in each residue disk, and discuss the problem of computing the Weil pairing in finite rings.

In this thesis we investigate practical aspects related to point counting problems on algebraic varieties over finite fields. In particular, we present significant improvements to Lauder’s deformation method for smooth projective hypersurfaces, which allow this method to be successfully applied to previously intractable instances. Part I is dedicated to the deformation method, including a complete description of the algorithm but focussing on aspects for which we contribute original improvements. In Chapter 3 we describe the computation of the action of Frobenius on the rigid cohomology space associated to a diagonal hypersurface; in Chapter 4 we develop a method for fast computations in the de Rham cohomology spaces associated to the family, which allows us to compute the Gauss–Manin connection matrix. We conclude this part with a small selection of examples in Chapter 6. In Part II we present an improvement to Lauder’s fibration method. We manage to resolve the bottleneck in previous computation, which is formed by so-called polynomial radix conversions, employing power series inverses and a more efficient implementation. Finally, Part III is dedicated to a comprehensive treatment of the arithmetic in unramified extensions of Qp , which is connected to the previous parts where our computations rely on efficient implementations of p-adic arithmetic. We have made these routines available for others in FLINT as individual modules for p-adic arithmetic.

Doctor of Philosophy (PhD) / In this work, we investigate methods for computing explicitly with homomorphisms (and particularly endomorphisms) of Jacobian varieties of algebraic curves. Our principal tool is the theory of correspondences, in which homomorphisms of Jacobians are represented by divisors on products of curves. We give families of hyperelliptic curves of genus three, five, six, seven, ten and fifteen whose Jacobians have explicit isogenies (given in terms of correspondences) to other hyperelliptic Jacobians. We describe several families of hyperelliptic curves whose Jacobians have complex or real multiplication; we use correspondences to make the complex and real multiplication explicit, in the form of efficiently computable maps on ideal class representatives. These explicit endomorphisms may be used for efficient integer multiplication on hyperelliptic Jacobians, extending Gallant--Lambert--Vanstone fast multiplication techniques from elliptic curves to higher dimensional Jacobians. We then describe Richelot isogenies for curves of genus two; in contrast to classical treatments of these isogenies, we consider all the Richelot isogenies from a given Jacobian simultaneously. The inter-relationship of Richelot isogenies may be used to deduce information about the endomorphism ring structure of Jacobian surfaces; we conclude with a brief exploration of these techniques.

In the first three chapters of this thesis we study two conjectures relating arithmetic with geometry, namely Tate and Lang&rsquo / s conjectures, for a certain class of algebraic surfaces. The surfaces we are interested in are assumed to be defined over a number field, have irregularity two and admit a genus two fibration over an elliptic curve. In the final chapter of the thesis we prove the isomorphism of the Picard motives of an arbitrary variety and its Albanese variety.

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.