Theta series of inhomogeneous quadratic forms

Ohne Zusammenfassung     

On ternary quadratic forms

A result from 1988 on the square-free integers represented by a positive definite ternary quadratic form with integral coefficients is made uniform in the determinant of the form.     

The correspondence between theta series of ternary and quaternary quadratic forms

This article uses orthogonal groups and quaternion algebras to describe the Shimura lifting for theta series of general three-dimensional lattices with spherical coefficients.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniye Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 196, pp. 61–82, 1991.     

Theta series of imaginary quadratic function fields

LetL be an imaginary quadratic extension of the rational function field . We prove transformation rules for the theta series corresponding to partial zeta functions of the extension .     

Overpartitions and ternary quadratic forms

Let \(\overline{p}(n)\) denote the number of overpartitions of n. Recently, Mahlburg showed that \(\overline{p}(n) \equiv 0 \pmod {64}\) and Kim showed that \(\overline{p}(n) \equiv 0 \pmod {128}\) for almost all integers n. In this paper, with the help of some ternary quadratic forms, we prove that \(\overline{p}(n) \equiv 0 \pmod {256}\) for almost all integers n, which was conjectured by Mahlburg.     

Eisenstein series of weight-3/2 and singular Hardy-Littlewood series for ternary quadratic forms

By means of analytic continuation Eisenstein series of weight-3/2 are constructed with respect to a main congruence subgroup (N). This makes it possible to define the singular series of an arbitrary integral ternary quadratic form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 50, pp. 156–168, 1975.     

Theta-series associated with indefinite ternary quadratic forms and the related Dirichlet series

One obtains theoretic-numerical results connected with the investigation of the Fourier coefficients of theta-series, associated with indefinite ternary quadratic forms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 41–50, 1981.     

Line-bundle-valued ternary quadratic forms over schemes

We study degenerations of rank 3 quadratic forms and of rank 4 Azumaya algebras, and extend what is known for good forms and Azumaya algebras. By considering line-bundle-valued forms, we extend the theorem of Max-Albert Knus that the Witt-invariant—the even Clifford algebra of a form—suffices for classification. An algebra Zariski-locally the even Clifford algebra of a ternary form is so globally up to twisting by square roots of line bundles. The general, usual and special orthogonal groups of a form are determined in terms of automorphism groups of its Witt-invariant. Martin Kneser’s characteristic-free notion of semiregular form is used.     

Theta functions of indefinite quadratic forms over real number fields

We define theta functions attached to indefinite quadratic forms over real number fields and prove that these theta functions are Hilbert modular forms by regarding them as specializations of symplectic theta functions. The eighth root of unity which arises under modular transformations is determined explicitly.

     

Representation of algebraic integers by ternary quadratic forms

The discrete ergodic method is generalized to totally positive-ternary quadratic forms over totally real algebraic number fields. We obtain estimates for the number of representations of elements in maximal orders of such number fields which are precise in the sense of the order of growth. We prove that the representations are asymptotically uniformly distributed with respect to a given module.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 157–168, 1983.     

Equidistribution of Heegner points and ternary quadratic forms

We prove new equidistribution results for Galois orbits of Heegner points with respect to single reduction maps at inert primes. The arguments are based on two different techniques: primitive representations of integers by quadratic forms and distribution relations for Heegner points. Our results generalize an equidistribution result with respect to a single reduction map established by Cornut and Vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow. Moreover, for fixed fundamental discriminant and variable conductor, we deduce an effective surjectivity theorem for the reduction map from Heegner points to supersingular points at a fixed inert prime. Our results are applicable to the setting considered by Kolyvagin in the construction of the Heegner points Euler system.     

Integral quadratic forms and Dirichlet series

A Dirichlet series with multiplicative coefficients has an Euler product representation. In this paper we consider the special case where these coefficients are derived from the numbers of representations of an integer by an integral quadratic form. At first we suppose this quadratic form to be positive definite. In general the representation numbers are not multiplicative. Instead we consider the average number of representations over all classes in the genus of the quadratic form. And we consider only representations of integers of the form tk ² with t square-free. If we divide the average representation number for these integers by a suitable factor, we do get a multiplicative function. Using results from Siegel (Ann. Math. 36:527–606, 1935), we derive a uniform expression for the Euler product expansion of the corresponding Dirichlet series. As a special case, we consider the standard quadratic form in n variables corresponding to the identity matrix. Here we use results from Shimura (Am. J. Math. 124:1059–1081, 2002). For 2≤n≤8, the genus of this particular quadratic form contains only one class, and this leads to a rather simple expression for the Dirichlet series, where the coefficients are just the number of representations of a square as the sum of n squares. Finally we consider the indefinite case, where we can get results similar to the definite case.     

Positive ternary quadratic forms with finitely many exceptions

An integral quadratic form is said to be almost regular if globally represents all but finitely many integers that are represented by the genus of . In this paper, we study and characterize all almost regular positive definite ternary quadratic forms.

     

An asymmetric inequality for non-homogeneous ternary quadratic forms

LetQ(x,y,z) be an indefinite ternary quadratic form of type (2,1) and determinantD(<0). Let 0≤t≤1/3 and \(f(t) = \frac{4}{{(1 + t)^2 (1 + 5t)}}\) . Then given any real numbersx ₀,y ₀,z ₀ there exist integersx,y,z satisfying $$ - t(f(t)|D|)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}< Q (x + x_0 ,y + y_0 ,z + z_0 ) \leqslant (f(t)|D|)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} $$ All the cases when equality holds are also obtained.