Unitriangular actions on quadratic forms and character degrees

On Character Sums of Binary Quadratic Forms

We establish character sum bounds of the form ∣∣∣∣ ∑ a≤x≤a+H b≤y≤b+H χ(x + ky) ∣∣∣∣ < p−τH2, where χ is a nontrivial character (mod p), p 1 4 +ε < H < p, and |a|, |b| < p H. As an application, we obtain that given k ∈ Z\{0}, x + k is a quadratic non-residue (mod p) for some 1 ≤ x < p 1 2e. Introduction. Let k be a nonzero integer. Let p be a large prime and let H ≤ p. We are interested in the c...

Applications of quadratic D-forms to generalized quadratic forms

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.

On Finite Groups With Two Irreducible Character Degrees

AFFINE SUBGROUPS OF THE CLASSICAL GROUPS AND THEIR CHARACTER DEGREES

In this paper we describe how the degrees of the irreducible characters of the affine subgroups of the classical groups under consideration can be found inductively. In [4] Gow obtained certain character degrees for all of the affine subgroups of the classical groups. We apply the method of Fischer to the above groups and, in addition to the character degrees given in [4], we obtain some ne...

On the Quadratic Character of Quadratic Units

Let p ≡ 1 (mod 4) be a prime. Let a, b ∈ Z with p a(a2 + b2). In the paper we mainly determine ( b+ √ a2+b2 2 ) p−1 2 (mod p) by assuming p = c2 + d2 or p = Ax2 + 2Bxy + Cy2 with AC − B2 = a2 + b2. As an application we obtain simple criteria for εD to be a quadratic residue (mod p), where D > 1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the qu...