8,666 research outputs found
The triangular theorem of eight and representation by quadratic polynomials
We investigate here the representability of integers as sums of triangular
numbers, where the -th triangular number is given by . In
particular, we show that ,
for fixed positive integers , represents every nonnegative
integer if and only if it represents 1, 2, 4, 5, and 8. Moreover, if
`cross-terms' are allowed in , we show that no finite set of positive
integers can play an analogous role, in turn showing that there is no
overarching finiteness theorem which generalizes the statement from positive
definite quadratic forms to totally positive quadratic polynomials
Two-dimensional lattices with few distances
We prove that of all two-dimensional lattices of covolume 1 the hexagonal
lattice has asymptotically the fewest distances. An analogous result for
dimensions 3 to 8 was proved in 1991 by Conway and Sloane. Moreover, we give a
survey of some related literature, in particular progress on a conjecture from
1995 due to Schmutz Schaller.Comment: 21 pages, final version, accepted for publication in L'Enseignement
Math\'ematiqu
Supersingular abelian surfaces and Eichler class number formula
Let be a totally real field with ring of integers , and be a
totally definite quaternion algebra over . A well-known formula established
by Eichler and then extended by K\"orner computes the class number of any
-order in . In this paper we generalize the Eichler class number
formula so that it works for arbitrary -orders in . The
motivation is to count the isomorphism classes of supersingular abelian
surfaces in a simple isogeny class over a prime finite field . We
give explicit formulas for the number of these isomorphism classes for all
primes .Comment: 29 pages, 3 numerical tables, shortened revised version with same
results, Sections 7-9 of v2 are remove
Principal forms X^2 + nY^2 representing many integers
In 1966, Shanks and Schmid investigated the asymptotic behavior of the number
of positive integers less than or equal to x which are represented by the
quadratic form X^2+nY^2. Based on some numerical computations, they observed
that the constant occurring in the main term appears to be the largest for n=2.
In this paper, we prove that in fact this constant is unbounded as n runs
through positive integers with a fixed number of prime divisors.Comment: 10 pages, title has been changed, Sections 2 and 3 are new, to appear
in Abh. Math. Sem. Univ. Hambur
On the cohomology of linear groups over imaginary quadratic fields
Let Gamma be the group GL_N (OO_D), where OO_D is the ring of integers in the
imaginary quadratic field with discriminant D<0. In this paper we investigate
the cohomology of Gamma for N=3,4 and for a selection of discriminants: D >=
-24 when N=3, and D=-3,-4 when N=4. In particular we compute the integral
cohomology of Gamma up to p-power torsion for small primes p. Our main tool is
the polyhedral reduction theory for Gamma developed by Ash and Koecher. Our
results extend work of Staffeldt, who treated the case n=3, D=-4. In a sequel
to this paper, we will apply some of these results to the computations with the
K-groups K_4 (OO_{D}), when D=-3,-4
Linear correlations amongst numbers represented by positive definite binary quadratic forms
Given a positive definite binary quadratic form f, let r(n) = |{(x,y):
f(x,y)=n}| denote its representation function. In this paper we study linear
correlations of these functions. For example, if r_1, ..., r_k are
representation functions, we obtain an asymptotic for sum_{n,d} r_1(n) r_2(n+d)
... r_k(n+ (k-1)d).Comment: 60 pages. Small correction
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