Counting Integral Points Research Articles

Counting integral points on indefinite ternary quadratic equations over number fields

Abstract We study an asymptotic formula for counting integral points over an equation defined by a non-degenerate indefinite integral ternary quadratic form f representing a non-zero integer a such that $-a\cdot det(\,f)$ is square over a number field. In particular, we prove that the leading coefficient of this asymptotic formula is given by the product of local densities normalized by $1-p^{-1}$ over all finite primes p.

Geometric quantization of almost toric manifolds

Kostant gave a model for the real geometric quantization associated to polarizations via the cohomology associated to the sheaf of flat sections of a pre-quantum line bundle. This model is well-adapted for real polarizations given by integrable systems and toric manifolds. In the latter case, the cohomology can be computed counting integral points inside the associated Delzant polytope. In this article we extend Kostant's geometric quantization to semitoric integrable systems and almost toric manifolds. In these cases the dimension of the acting torus is smaller than half of the dimension of the manifold. In particular, we compute the cohomology groups associated to the geometric quantization if the real polarization is the one associated to an integrable system with focus-focus type singularities in dimension four. As application we determine models for the geometric quantization of K3 surfaces, a spin-spin system, the spherical pendulum, and a spin-oscillator system under this scheme.

Vinberg’s representations and arithmetic invariant theory

Recently, Bhargava and others have proved very striking results about the average size of Selmer groups of Jacobians of algebraic curves over ℚ as these curves are varied through certain natural families. Their methods center around the idea of counting integral points in coregular representations, whose rational orbits can be shown to be related to Galois cohomology classes for the Jacobians of these algebraic curves. In this paper we construct for each simply laced Dynkin diagram a coregular representation (G,V) and a family of algebraic curves over the geometric quotient V∕∕G. We show that the arithmetic of the Jacobians of these curves is related to the arithmetic of the rational orbits of G. In the case of type A2, we recover the correspondence between orbits and Galois cohomology classes used by Birch and Swinnerton-Dyer and later by Bhargava and Shankar in their works concerning the 2-Selmer groups of elliptic curves over ℚ.

Upper bounds for the number of number fields with alternating Galois group

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 .

Exploiting polyhedral symmetries in social choice

A large amount of literature in social choice theory deals with quantifying the probability of certain election outcomes. One way of computing the probability of a specific voting situation under the Impartial Anonymous Culture assumption is via counting integral points in polyhedra. Here, Ehrhart theory can help, but unfortunately the dimension and complexity of the involved polyhedra grows rapidly with the number of candidates. However, if we exploit available polyhedral symmetries, some computations become possible that previously were infeasible. We show this in three well known examples: Condorcet’s paradox, Condorcet efficiency of plurality voting and in Plurality voting vs Plurality Runoff.

A generalization of the Bombieri–Pila determinant method

The so-called determinant method was developed by Bombieri and Pila in 1989 for counting integral points of bounded height on affine plane curves. In this paper we give a generalization of that method to varieties of higher dimension, yielding a proof of Heath-Brown's 'Theorem 14' by real-analytic considerations alone.

Imaginary quadratic fields with class group exponent 5

We show that there are finitely many imaginary quadratic number fields for which the class group has exponent 5. Indeed there are finitely many with exponent at most 6. The proof is based on a method of Pierce [13]. The problem is reduced to one of counting integral points on a certain affine surface. This is tackled using the author’s “square-sieve”, in conjunction with estimates for exponential sums. The latter are derived using the q-analogue of van der Corput’s method. Mathematics Subject Classification (2000): 11R29 (11D45, 11L07, 11N36, 11LR29) Let −d be a negative fundamental discriminant and let E(−d) denote the exponent of the ideal class group of the imaginary quadratic field Q( √ −d). Under the Generalized Riemann Hypothesis one has E(−d) (log d)/(log log d), see Boyd and Kisilevsky [3], and Weinberger [15]. However without any unproved hypothesis it is not even known that E(−d) → ∞. The problem of Euler’s “Numeri Idonei” is essentially equivalent to the case E(−d) = 2, and here it has long been known that there are finitely many admissible values of d. Both Boyd and Kisilevsky loc. cit., and Weinberger loc. cit., showed that there are finitely many fundamental discriminants for which E(−d) = 3. It should be remarked that none of the proofs, either for E(−d) = 2 or for E(−d) = 3, are effective. In this paper we shall handle the case in which E(−d) = 5. Theorem 1 There is an (ineffective) constant d5 such that E(−d) 6= 5 for every fundamental discriminant −d with d > d5. While this is our principal result we also record the following easy observation. Theorem 2 Let r be a non-negative integer, and let E = 2 or E = 3.2. Then there is an (ineffective) constant dE such that E(−d) 6= E for every fundamental discriminant −d with d > dE. It follows from these two results that E(−d) ≥ 7 for all large enough d. We begin by proving Theorem 2. If E(−d) = E with E as above, then the class group takes the form C × C2r × C2a × . . .× C2b

Efficient verification of Tunnell’s criterion

An integern is congruent if there is a triangle with rational sides whose area isn. In the 1980s Tunnell gave an algorithm to test congruence which relied on counting integral points on the ellipsoids 2x 2 +y 2 + 8z 2 =n and 2x 2 +y 2 + 32z 2 =n. The correctness of this algorithm is conditional on the conjecture of Birch and Swinnerton-Dyer. The known methods for testing Tunnell’s criterion useO(n) operations. In this paper we give several methods with smaller exponents, including a randomized algorithm using timen 11/2+o(1) and spacen o(1) and a deterministic algorithm using space and timen 1/2+o(1).

Unitary periods, Hermitian forms and points on flag varieties

Let E be an imaginary quadratic extension of \({\mathbb{Q}}\) of class number one. We examine certain representation numbers associated to Hermitian forms over E , which involve counting integral points on flag varieties.

Densities of quartic fields with even Galois groups

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.

A Polynomial Time Algorithm for Counting Integral Points in Polyhedra When the Dimension is Fixed

We prove that for any dimension d there exists a polynomial time algorithm for counting integral points in polyhedra in the d-dimensional Euclidean space. Previously such algorithms were known for dimensions d = 1, 2, 3, and 4 only.

Computing the volume, counting integral points, and exponential sums

We design polynomial-time algorithms for some particular cases of the volume computation problem and the integral points counting problem for convex polytopes. The basic idea is a reduction to the computation of certain exponential sums and integrals. We give elementary proofs of some known identities between these sums and integrals and prove some new identities.

On the Complexity of Computing the Volume of a Polyhedron

We show that computing the volume of a polyhedron given either as a list of facets or as a list of vertices is as hard as computing the permanent of a matrix.