Counting integral points on indefinite ternary quadratic equations over number fields

Let N ( n , A n , X ) N(n, A_n, X) be the number of number fields of degree n n whose Galois closure has Galois group A n A_n and whose discriminant is bounded by X X . By a conjecture of Malle, we expect that N ( n , A n , X ) ∼ C n ⋅ X 1 2 ⋅ ( log ⁡ X ) b n N(n, A_n, X)\sim C_n\cdot X^{\frac {1}{2}} \cdot (\log X)^{b_n} for constants b n b_n and C n C_n . For 6 ≤ n ≤ 84393 6 \leq n \leq 84393 , the best known upper bound is N ( n , A n , X ) ≪ X n + 2 4 N(n, A_n, X) \ll X^{\frac {n + 2}{4}} , by Schmidt’s theorem, which implies there are ≪ X n + 2 4 \ll X^{\frac {n + 2}{4}} number fields of degree n n . (For n > 84393 n > 84393 , there are better bounds due to Ellenberg and Venkatesh.) We show, using the important work of Pila on counting integral points on curves, that N ( n , A n , X ) ≪ X n 2 − 2 4 ( n − 1 ) + ϵ N(n, A_n, X) \ll X^{\frac {n^2 - 2}{4(n - 1)}+\epsilon } , thereby improving the best previous exponent by approximately 1 4 \frac {1}{4} for 6 ≤ n ≤ 84393 6 \leq n \leq 84393 .

Let N ( d , G , X ) N(d, G, X) be the number of degree d d number fields K K with Galois group G G and whose discriminant D K D_K satisfies | D K | ≤ X |D_K| \le X . Under standard conjectures in diophantine geometry, we show that N ( 4 , A 4 , X ) ≪ ϵ X 2 / 3 + ϵ N(4, A_4, X) \ll _\epsilon X^{2/3+\epsilon } , and that there are ≪ ϵ N 3 + ϵ \ll _\epsilon N^{3+\epsilon } monic, quartic polynomials with integral coefficients of height ≤ N \le N whose Galois groups are smaller than S 4 S_4 , confirming a question of Gallagher. Unconditionally we have N ( 4 , A 4 , X ) ≪ ϵ X 5 / 6 + ϵ N(4, A_4, X) \ll _\epsilon X^{5/6 + \epsilon } , and that the 2 2 -class groups of almost all Abelian cubic fields k k have size ≪ ϵ D k 1 / 3 + ϵ \ll _\epsilon D_k^{1/3+\epsilon } . The proofs depend on counting integral points on elliptic fibrations.

Ternary quadratic forms over number fields with small class number

On almost strong approximation for algebraic groups

The following is an expanded version of the notes for a survey lecture on the theta correspondence given at Trichy in January, 1996. The aim of this lecture was to give an idea of this theory and some of the directions in which the theory has been developing. Perhaps some of the material contained in sections 2,3 and 9 is new. The theory of theta correspondence gives one of the few general methods of constructing automorphic forms of groups over number fields, or admissible representations of groups over local fields. The method has its origin in the classical construction of theta functions which are modular forms–perhaps of half integral weight on the upper half plane. Here is a special case of this: Let q be a positive definite quadratic form on a lattice L ⊆ IR, q : L→ Z . Let p be a homogeneous polynomial on IR which is harmonic with respect to q, i.e. ∆qp = 0 where ∆q denotes the Laplacian with respect to q (∆q is the unique homogeneous differential operator of order 2 invariant under the orthogonal group of q). Then,

For a positive definite integral ternary quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. The famous Minkowski–Siegel formula implies that if the class number of [Formula: see text] is one, then [Formula: see text] can be written as a constant multiple of a product of local densities which are easily computable. In this paper, we consider the case when the spinor genus of [Formula: see text] contains only one class. In this case the above also holds if [Formula: see text] is not contained in a set of finite number of square classes which are easily computable. By using this fact, we prove some extension of the recent results on both the representations of generalized Bell ternary forms and the representations of ternary quadratic forms with some congruence conditions.

Refining the notion of regularity introduced by Dickson, an integral quadratic form is said to be spinor regular if it represents all integers represented by its spinor genus. Examples of positive definite primitive integral ternary quadratic forms which have this property are presented, and it is proved that there exist only finitely many equivalence classes containing such forms.

In this note we prove an elementary result concerning the distribution of the positive integers which are represented by a positive-definite integral binary quadratic form Theorem 1. Let f(X, Y) = aX ~ + b X Y + c y2 be a positive-definite integral binary quadratic form of discriminant - A (= b 2 - 4ac = mi , there exist integers x and y such that n < ax 2 q- bxy + cy 2 < n + 2ml/~A1/4n ~m + m i . P r 0 0 f. Replacing the form f by an equivalent form we may suppose that m~ = c. We define integers x and y by =F(4cn~l/2q I!4cn-Ax2)l/2-bx l x L\~) /' Y= 2c -+1.

Small representations of integers by integral quadratic forms

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer N N , first we give the classification of positive definite ternary quadratic forms of level 4 N 4N explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level 4 N 4N , which are involved with modified Hurwitz class number. In the proof of the main results, we use the relations among ternary quadratic forms, quaternion algebras, and Jacobi forms. As a corollary, we get the formula for the class number of positive ternary quadratic forms of level 4 N 4N . As applications, we derive an explicit base of Eisenstein series space of modular forms of weight 3 / 2 3/2 and level 4 N 4N , and give new proofs of some interesting identities involving representation number of ternary quadratic forms.

We investigate the existence of a weak solution u ∈ W 1,p 0 (Ω) to the degenerate quasi-linear Dirichlet boundary value problem (P) -Δ p u = λ 1 |u| p-2 u + f(x) in Ω; u = 0 on ∂Ω. It is assumed that 1 < p < ∞, p ¬= 2, Ω is a bounded domain in R N , f ∈ L∞ (Q) is a given function, and the number λ 1 stands for the first (smallest) eigenvalue of the positive p-Laplacian -Ap, where Δ p u ≡ div(|⊇u| p-2 ⊇u). The eigenvalue λ 1 being simple, let φ 1 denote the eigenfunction associated with λ 1 . We show the existence of a solution for problem (P) when f satisfies the orthogonality condition ⊃ Ω fφ 1 dx = 0, in which case the set of solutions is bounded in C 1 (Ω) provided p ¬= 2 and f ≢ 0 in Ω. A key role in our proofs is played by a second-order Taylor formula in its integral form (near φ 1 ) which contains certain Gâteaux derivatives and a positive semidefinite quadratic form. This quadratic form compensates for lack of coercivity in the energy functional corresponding to problem (P). When combined with well-known regularity results, its positive semidefiniteness renders a priori estimates from which existence and boundedness (for p ¬= 2) follow.

Mennicke (Proc R Soc Edinb Sect A 67:309–352, 1963; Proc R Soc Edinb Sect A 88:151–157, 1981) investigated the number of conic points with isotropies of orders 3, 4 and 6 that can appear in the orientable twofold orbifold covering of the orbifold $$Q_{f}$$ associated with an integral ternary quadratic form. Mennicke made essential use of a theorem of Jones (The Arithmetic Theory of Quadratic Forms. The Carus Mathematical Monographs, vol. 10. MAA, Baltimore, 1950, Theorem 86). In this paper we revisit Mennicke’s results and we extend them to $$Q_{f},$$ that is, even when $$Q_{f}$$ is non-orientable. Our method is new and independent of Jones’ Theorem. We also study the possible cusp points.

The basic objects in supergeometry are the superspaces, “spaces on which some of the coordinates commute and the rest anticommute.” A precise definition of superspaces and their mappings is given in terms of sheaves. This chapter is devoted to the exposition of the basic facts about superspaces. In § 1 the fundamental definitions are introduced, especially the concept of a supermanifold, a space with local systems of independent coordinate functions. In § 2 it is explained how supermanifolds and superfibrations are classified up to isomorphism in the smooth category. The obstructions to the validity of the analogous theorems for complex manifolds are cohomological invariants. In § 3 flag superspaces are introduced; these are important in their own right and also provide good illustrations of the general theory. In § 4 are gathered the obvious supergeometric definitions in the theory of integrability, including the Frobenius theorem and connections. In § 5 objects specific to supergeometry are introduced—integral forms. The complex of integral forms is constructed using a canonical right connection on the sheaf of Berezin volume forms of the supermanifold; the idea of a right connection on an arbitrary vector superbundle is also defined and studied here. Integration on supermanifolds is the subject of §§ 6–9. The fundamental idea here is the Berezin integral of a volume form with compact support on a supermanifold; this is defined in § 6. Various derived constructions which permit one to define integrals over immersed supermanifolds of any dimension are described in §§ 7 and 9. The Stokes formula for integral forms is proved in § 8. Finally, in § 10 there is a list of finite-dimensional simple Lie superalgebras over the field of complex numbers.

For each positive integer n n , let g Z ( n ) g_\mathbb {Z}(n) be the smallest integer such that if an integral quadratic form in n n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g Z ( n ) g_\mathbb {Z}(n) squares of integral linear forms. We show that every positive definite integral quadratic form is equivalent to what we call a balanced Hermite–Korkin–Zolotarev-reduced form and use it to show that the growth of g Z ( n ) g_\mathbb {Z}(n) is at most an exponential of n \sqrt {n} . Our result improves the best known upper bound on g Z ( n ) g_\mathbb {Z}(n) which is on the order of an exponential of n n . We also define an analogous number g O ∗ ( n ) g_{\mathcal O}^*(n) for writing Hermitian forms over the ring of integers O \mathcal O of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1 1 , we show that the growth of g O ∗ ( n ) g_{\mathcal O}^*(n) is at most an exponential of n \sqrt {n} . We also improve on results of both Conway and Sloane and Kim and Oh on s s -integrable lattices.

In this paper we prove that every positive definite n-ary integral quadratic form with 12 &lt; n &lt; 13 (respectively 14 ≦ n ≤ 20) that can be represented by a sum of squares of integral linear forms is represented by a sum of 2 · 3n + n + 6 (respectively 3 · 4n + n + 3) squares. We also prove that every positive definite 7-ary integral quadratic form that can be represented by a sum of squares is represented by a sum of 25 squares.