The trace-form of an algebraic number field

Let F be the rational field or a p-adic field, and let K an algebraic number field over F. If ω₁,…, ω_n is an integral basis for the ring $\text{D}$_L of integers in K, then the quadratic form Q whose matrix is (trace_K∥F(ω_iω_j)) has integral coefficients, and is called an integral trace-form. Q is determined by K up to integral equivalence. The purpose of this paper is to show that the genus of Q determines the ramification of primes in K.     

On the average difference between two conjugates of an algebraic number

We give a lower bound for |α−1|, where α is an algebraic number, and also an upper bound for the number of real zeros of a polynomial. A lower bound for the maximal modulus of conjugates of a totally real algebraic integer is also obtained. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 35, No. 4, pp. 421–431, October–December, 1995.     

Computation of Weng's rank 2 zeta function over an algebraic number field

In this paper, we study the zeta function, named non-abelian zeta function, defined by Lin Weng. We can represent Weng's rank r zeta function of an algebraic number field F as the integration of the Eisenstein series over the moduli space of the semi-stable OF-lattices with rank r. For r=2, in the case of F=Q, Weng proved that it can be written by the Riemann zeta function, and Lagarias and Suzuki proved that it satisfies the Riemann hypothesis. These results were generalized by the author to imaginary quadratic fields and by Lin Weng to general number fields. This paper presents proofs of both these results. It derives a formula (first found by Weng) for Weng's rank 2 zeta functions for general number fields, and then proves the Riemann hypothesis holds for such zeta functions.     

On the representations of projective geometries in algebraic combinatorial geometries

In this paper, we show that the full algebraic combinatorial geometry is not a projective geometry, it is only semimodular, but the p-polynomial points give a projective subgeometry. Also, we show that the subgeometry can be coordinatized by a skew field, which is quotient ring of an Ore domain. As a corollary, we prove the existence of algebraic representations over fields of prime characteristic of the non-Pappus matroid and its dual matroid. Regarding the existence of algebraic representations of the non-Pappus matroid, this result was earlier proved by Lindström [7] for finite fields.     

On the sign of regular algebraic polarizable automorphic representations

We remove a parity condition from the construction of automorphic Galois representations carried out in the Paris Book Project. We subsequently generalize this construction to the case of ‘mixed-parity’ (but still regular essentially self-dual) automorphic representations over totally real fields, finding associated geometric projective representations. Finally, we optimize some of our previous results on finding geometric lifts, through central torus quotients, of geometric Galois representations, and apply them to the previous mixed-parity setting.     

Theta series of a real algebraic number field

We investigate transformation formulas for theta series with spherical functions on a Hilbert-Siegel space. As an application we show that some of Hilbert-Siegel modular varieties are of general type.     

On the large sieve in algebraic number fields

In the present note Bombieri's central theorem concerning the average distribution of the prime numbers in arithmetic progressions is generalized to arbitrary algebraic number fields.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 673–680, December, 1967.Finally, I express my profound gratitude to B. V. Levin for setting the problem and the help he rendered and to A. I. Vinogradov for valuable suggestions.