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In modern number theory L-series play a prominent role. They encode many deep properties of number fields and primes and are objects of intense interest. The analogous L-functions over global function fields play an equally prominent role. Here we will prove estimates for mean values of such L-functions, where the averaging is done over quadratic extensions of a fixed global function field. Our estimates cover a much wider range of cases than the similar estimates of Hoffstein and Rosen [1992] and those of Andrade and Keating (for values on the critical line) [2012]. Our methods are akin to those used by Siegel [1944], where he estimates the average number of quadratic forms with given discriminant and signature. For a prime p, let Fp denote the finite field with p elements and let X be transcendental over Fp, so that Fp.X/ is a field of rational functions. Fix algebraic closures Fp of Fp and Fp.X/ Fp of Fp.X/. In what follows, by global function field (or simply function field) we mean a finite algebraic extension K Fp.X/ contained in Fp.X/. For such a field K we have K\FpD FqK for some finite field FqK with qK elements; this field is called the field of constants of K . We write gK for the genus of K and JK for the number of divisor classes of degree 0. We denote the set of places of K by M.K/ and the divisor group (i.e., the free abelian group generated by the places) by Div K . The reader can refer to Chapters I and V of [Stichtenoth 1993] for a thorough background on these notions. We will use capital script German letters to denote divisors A, B, etc., with the sole exception of the zero divisor 0. For any divisor A2 Div K we write AD P ordv.A/ v, where the

Let K be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let [Formula: see text] denote an algebraic closure of K. We count points in projective space [Formula: see text] with given height and generating a quadratic extension of K. If n > 2, we derive an asymptotic estimate for the number of such points as the height tends to infinity. Such estimates are analogous to previous results of Schmidt where the field K is replaced by the field of rational numbers ℚ.

We formulate an analog of Hermite's constant for function fields over a finite field and state a conjectural value for this analog. We prove our conjecture in many cases, and prove slightly weaker results in all other cases.

We consider Diophantine equations of the kind $|F(x,y)|=m$, where $F(X,Y)\in\mathbb{Z}[X,Y]$ is a homogeneous polynomial of degree at least 3 that has non-zero discriminant, $m$ is a fixed positive integer and $x,y$ are relatively prime integer solutions. Our results improve upon previous theorems due to Bombieri and Schmidt and also Stewart. We further provide reasonable heuristics for conjectures of Schmidt and Stewart regarding such equations.

We consider the number of integral solutions to the inequality | F ( x ) | ≤ m |F(\mathbf {x}) |\le m , where F ( X ) ∈ Z [ X ] F(\mathbf {X} )\in \mathbb {Z} [\mathbf {X} ] is a decomposable form of degree n + 1 n+1 in n n variables. We show that the number of such solutions is finite for all m m only if the discriminant of F F is not zero. We get estimates for the number of such solutions that display appropriate behavior in terms of the discriminant. These estimates sharpen recent results of the author for the general case of arbitrary degree.

We show that some power of the discriminant must appear in the upper bound of the formulation of Siegel's Lemma over a number field.

We count the number of subspaces of affine space with a given dimension defined over an algebraic number field with height less than or equal to B B . We give an explicit asymptotic estimate for the number of such subspaces as B B goes to infinity, where the constants involved depend on the classical invariants of the number field (degree, discriminant, class number, etc.). The problem is reformulated as an estimate for the number of lattice points in a certain bounded domain.

We estimate the number of integer solutions to decomposable form inequalities (both asymptotic estimates and upper bounds are provided) when the degree of the form and the number of variables are relatively prime. These estimates display good behavior in terms of the coefficients of the form, and greatly improve upon previous estimates in this regard., version 2, 30 pages. Corrects some typos and statements of propositions 5 and 6 (all theorems and other propositions remain as before). The last reference is updated, as well

We consider Diophantine inequalities of the kind |f(x)| \le m, where F(X) \in Z[X] is a homogeneous polynomial which can be expressed as a product of d homogeneous linear forms in n variables with complex coefficients and m\ge 1. We say such a form is of finite type if the total volume of all real solutions to this inequality is finite and if, for every n'-dimensional subspace S\subseteq R^n defined over Q, the corresponding n'-dimensional volume for F restricted to S is also finite. We show that the number of integral solutions x \in Z^n to our inequality above is finite for all m if and only if the form F is of finite type. When F is of finite type, we show that the number of integral solutions is estimated asymptotically as m\to \infty by the total number of integral solutions is estimated asymptotically as m\to \infty by the total volume of all real solutions. This generalizes a previous result due to Mahler for the case n=2. Further, we prove a conjecture of W. M. Schmidt, showing that for F of finite type the number of integral solutions is bounded above by c(n,d)m^(n/d), where c(n,d) is an effectively computable constant depending only on n and d., 38 pages, published version

We give asymptotic estimates for the number of subspaces of height m in affine n -space defined over a function field. These are analogous to similar estimates for number fields.

For homogeneous decomposable forms F( X ) in n variables with real coefficients, we consider the associated volume of all real solutions x ∈ R n to the inequality |F( x )|⩽1 . We relate this to the number of integral solutions z ∈ Z n to the Diophantine inequality |F( z )|⩽m in the case where F has rational coefficients. We find quantities which bound the volume and which yield good upper bounds on the number of solutions to the Diophantine inequality in the rational case.

Let F ∈ Z[X1, . . . ,Xn] be a decomposable form, that is, a homogeneous polynomial of degree d which can be factored into linear forms over C. Denote by NF (m) the number of integer solutions to the inequality |F(x)| ≤ m and by VF (m) the volume of the set{x ∈ Rn : |F(x)| ≤ m}. In 2001, Thunder [19] proved a conjecture of W.M. Schmidt, stating that, under suitable finiteness conditions, one has NF (m) << mn/d where the implicit constant depends only on n and d. Further, he showed an asymptotic formula NF (m) = mn/dV (F) + OF (mn/(d+n−2)) where, however, the implicit constant depends on F. In subsequent papers, Thunder’s concern was to obtain a similar asymptotic formula, but with the upper bound of the error term |NF (m)−mn/dV (F)| depending only on n and d. In [20] and [22], hemanaged to prove that if gcd(n, d) = 1, the implicit constant in the error term can indeed be made depending only on n and d.The main objective of this thesis is to extend Thunder’s results to the p-adic setting. Namely, we are interested in solutions to the inequality |F(x)| · |F(x)|p1 . . . |F(x)|pr ≤ m in x = (x1, x2, . . . ,xn) ∈ Zn with gcd(x1, x2, . . . ,xn, p1 · · · pr) = 1. (5.4.3)where p1, . . . , pr are distinct primes and | · |p denotes the usual p-adic absolute value.Chapter 1 is devoted to the p-adic set-up of this problem and to the proofs of the auxiliary lemmas. Chapter 2 is devoted to extending Thunder’s results from [19]. In chapter 3, we show the effectivity of the condition under which the number of solutions of (5.4.3) is finite. Chapter 4 and chapter 5 generalize Thunder’s results from [20], [21] and [22].; Soit F ∈ Z[X1, . . . ,Xn] une forme décomposable, c’est-à-dire un polynôme homogène de degré d qui peut être factorisé en formes linéaires sur C. Notons NF (m) le nombre de solutions entières à l’inégalité |F(x)| ≤ m et VF (m) le volume de l’ensemble {x ∈ Rn :|F(x)| ≤ m}. En 2001, Thunder [19] a prouvé une conjecture de W.M. Schmidt, énonçant que, sous des conditions de finitude appropriées, on a NF (m) << m n/d où la constante implicite ne dépend que de n et d. En outre, il a montré une formule asymptotique NF (m) = m n/d V (F) + OF (m n/(d+n−2)) où, cependant, la constante implicite dépend de F. Dans des articles ultérieurs, la préoccupation de Thunder était d’obtenir une formule asymptotique similaire, mais avec la borne supérieure du terme d’erreur |NF (m) −m n/dV (F)| ne dépendant que de n et d. Dans [20] et [22], il a réussi à prouver que si gcd(n, d) = 1, la constante implicite dans le terme d’erreur peut en effet être fonction uniquement de n et d. L’objectif principal de cette thèse est d’étendre les résultats de Thunder au cadre p-adique. `A savoir, nous sommes intéressés par les solutions à l’inégalité |F(x)| · |F(x)|p1 . . . |F(x)|pr ≤ m en x = (x1, x2, . . . ,xn) ∈ Zn avec gcd(x1, x2, . . . ,xn, p1 · · · pr) = 1. (5.4.9) où p1, . . . , pr sont des nombres premiers distincts et |·|p désigne la valeur absolue p-adique habituelle. Le chapitre 1 est consacré au cadre p-adique de ce problème et aux preuves des lemmes auxiliaires. Le chapitre 2 est consacré à l’extension des résultats de Thunder de [19]. Dans le chapitre 3, nous montrons l’effectivité de la condition sous laquelle le nombre de solutions de (5.4.9) est fini. Le chapitre 4 et le chapitre 5 généralisent les résultats de Thunder dans [20], [21] et [22].