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Congruence classes in M3(Fq)(q even)

B. Corbas, G.D. Williams
2002 Discrete Mathematics
In dimensions 2 and 3, the sizes of the congruence classes of matrices over a ÿnite ÿeld of characteristic 2 are determined.  ...  This complements the authors' earlier solution of the corresponding problem in odd characteristic.  ...  The sizes of the congruence classes in M 2 (F q ) (q odd) are given in Table 1 . Theorem 2. The sizes of the congruence classes in M 2 (F q ) (q even) are given in Table 2 .  ...

Congruence classes in M3(Fq) (q odd)

B. Corbas, G.D. Williams
2000 Discrete Mathematics
Over a ÿnite ÿeld, these representatives then enable us very easily to determine the sizes of the various congruence classes.  ...  In dimensions 2 and 3, matrices over a ÿeld of characteristic = 2 are classiÿed up to congruence by means of certain representatives which are sums of diagonal and antisymmetric matrices.  ...  The sizes of the congruence classes in M 2 (F q ) ( q odd) are given by: Representative Stabilizer |Class| 1 O 2; 1 2 q(q − 1)(q + (− )) 1 ÿ −ÿ SO 2; q(q − 1)(q + (− )) 0 ±1 c d 1 2 (q 2 − 1) 1 −1 0 ±  ...

Triple product p-adic L-function attached to p-adic families of modular forms [article]

Kengo Fukunaga
2019 arXiv   pre-print
We generalize his result in the unbalanced case and construct a three-variable triple product p-adic L-function attached to a primitive Hida family and two more general p-adic families of modular forms  ...  Let Ω FQ 1 be the canonical period defined in Definition 3.3.4 and E FQ 1 (Π Q,p ) be the modified p-Euler factor defined in (3.4.1). Our main theorem is as follows. Main Theorem.  ...  We put (π ′ 1 , π ′ 2 , π ′ 3 ) = (π FQ 1 ⊗(χ Q ) A , π G (2) (m2) , π G (3) (m3) ). The following proposition is proved in [Hsi17, Proposition 6.12]. Proposition 5.1.4.  ...

A Fractal-Like Algebraic Splitting of the Classifying Space for Vector Bundles

V. Giambalvo, David J. Pengelley, Douglas C. Ravenel
1988 Transactions of the American Mathematical Society
The pieces in the splittings are finite, and the grading extends that of H*n2S3 which splits it into Brown-Gitler modules.  ...  There are fractal A-algebra maps fq: Bo -* B0 for q > 1 satisfying: (1) Each fq(ui) is an indecomposable in dimension i + 2m^+q, and thus fq is a monomorphism. (2) fq(Bn)EBn+1. PROOF.  ...  In this case, clearly fq(pdl) = gq(pdi), so we only need show that p(di+2m+q) -gqdz = 0.  ...

A fractal-like algebraic splitting of the classifying space for vector bundles

V. Giambalvo, David J. Pengelley, Douglas C. Ravenel
1988 Transactions of the American Mathematical Society
The pieces in the splittings are finite, and the grading extends that of H*n2S3 which splits it into Brown-Gitler modules.  ...  There are fractal A-algebra maps fq: Bo -* B0 for q > 1 satisfying: (1) Each fq(ui) is an indecomposable in dimension i + 2m^+q, and thus fq is a monomorphism. (2) fq(Bn)EBn+1. PROOF.  ...  In this case, clearly fq(pdl) = gq(pdi), so we only need show that p(di+2m+q) -gqdz = 0.  ...

Involutory elliptic curves over \$\mathbb{F}_q(T)\$

Andreas Schweizer
1998 Journal de Théorie des Nombres de Bordeaux
The author, being supported by CICMA in form of a post-doc position at Concordia University and McGill University, wishes to express his gratitude to all three institutions.  ...  Suppose that q is even and n = [m with (1, m) = 1 and deg(m) &#x3E; 1. Write m = m2m1 with m1, m2 E Fq [T], where ml is square- free.  ...  It is given on (resp. rm for some q E F:. From this it is obvious that Wml Wm2 = with m3 = (ml-.  ...

Lectures on Applied ℓ-adic Cohomology [article]

Etienne Fouvry, Emmanuel Kowalski, Philippe Michel, Will Sawin
2019 arXiv   pre-print
For any squarefree integer q, let a q (mod q) be the unique congruence class modulo q such that ∀p|q, a q ≡ a p (mod p); in particular a q ∈ (Z/qZ) × .  ...  a set of integers has finite or infinite intersection with some congruence class.  ...  In [FM98] , this was shown to hold more generally for the trace functions K(x) = e q (x −k + ax), a ∈ F q , k 1. (2) For more general trace functions, the first condition in (16.10) and (16.13) can be  ...

Quaternions and Projective Geometry

C. J. Joly
1903 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Ml5 M3 and M3.  ...  The type of a function of this class I4 is fq (a d d 'd ") = a (a q d ' a ") + a ") + a" (ada'q) in which a, a , a"and d "are arbitrary quaternions.  ...  Dividing each member of the identities by (abed), we obtain the biquadratics and J (q),J' (q), J " (q),J' n (g), of the fourth, third, second and first ord in q, are the invariants of f( p q ) considered  ...

On Ree's Series of Simple Groups

Harold N. Ward
1966 Transactions of the American Mathematical Society
Condition I implies that q = 4 + e (mod 8) where e = + 1. IV. If denotes a cyclic subgroup of order (q + e)/2 in L, then the normalizer NC((R0}) of any subgroup of is contained in  ...  For some element J of order 2 (an "involution") in G, the centralizer CG(J) of J in G is the direct product of and L where L is isomorphic to the linear fractional group LF(2,q).  ...  If then £¡ has multiplicity m¡ in 0, 2 :£ 1 + m2 + m3 + m4 and 2 ;£ 1 -m2 + m3 -m4. Thus 1 ^ m3; as £3 has degree <j3, 9 = Çx + ¿3. Thus [13] this action is doubly transitive.  ...

On Ree's series of simple groups

Harold N. Ward
1966 Transactions of the American Mathematical Society
Condition I implies that q = 4 + e (mod 8) where e = + 1. IV. If denotes a cyclic subgroup of order (q + e)/2 in L, then the normalizer NC((R0}) of any subgroup of is contained in  ...  For some element J of order 2 (an "involution") in G, the centralizer CG(J) of J in G is the direct product of and L where L is isomorphic to the linear fractional group LF(2,q).  ...  If then £¡ has multiplicity m¡ in 0, 2 :£ 1 + m2 + m3 + m4 and 2 ;£ 1 -m2 + m3 -m4. Thus 1 ^ m3; as £3 has degree <j3, 9 = Çx + ¿3. Thus [13] this action is doubly transitive.  ...

On Inverses of Permutation Polynomials of Small Degree over Finite Fields [article]

Yanbin Zheng, Qiang Wang, Wenhong Wei
2018 arXiv   pre-print
In this list, the explicit inverse of a class of fifth degree PPs is our main result,which is obtained by using some congruences of binomial coefficients, the Lucas' theorem, and a known formula for the  ...  Permutation polynomials (PPs) and their inverses have applications in cryptography, coding theory and combinatorial design.  ...  If i is even, then r i + 2j is even, and so the coefficients of odd powers of x in (6) are all 0. Also note k(q − 1) + (q − 2) is odd. We have b i,q−2 = 0.  ...

Equations and monoid varieties of dot-depth one and two

1994 Theoretical Computer Science
In this paper, we first simplify the infinite defining sequence of equations for VI ,m given in [S].  ...  For m > 1, a sequence of equations satisfied in (but not necessarily complete for) V2,m is given. Parts of the present paper are also to be published in [S].  ...  Let m3 1.  ...

Matrix representatives for three-dimensional bilinear forms over finite fields

B. Corbas, G.D. Williams
1998 Discrete Mathematics
If ~ is symmetric or alternating, then explicit normal forms for the congruence classes over various fields are well known, but this is not the case for general asymmetric forms.  ...  Therefore, = {(aij) E M3(K) l a12 = a13 = a21 = a31 = 0}. Lemma 6.  ...

The cusp amplitudes and quasi-level of a congruence subgroup of SL2 over any Dedekind domain

A. W. Mason, Andreas Schweizer
2012 Proceedings of the London Mathematical Society
We study cusp amplitudes and the level of a (congruence) subgroup of SL_2(D) for any Dedekind domain D, as ideals of D. In particular, we extend a remarkable result of Larcher.  ...  We extend some algebraic properties of the classical modular group SL_2(Z) to equivalent groups in the theory of Drinfeld modules, in particular properties which are important in the theory of modular  ...  In addition ql(∆(Q)) = Q, by [M3, Theorem 3.8] . Part (iii) follows. Suppose that l(∆(Q)) = (f ).  ...

The Eisenstein ideal and Jacquet-Langlands isogeny over function fields [article]

Mihran Papikian, Fu-Tsun Wei
2015 arXiv   pre-print
Let p and q be two distinct prime ideals of F_q[T].  ...  Our results are stronger than what is currently known about the analogues of these problems over Q.  ...  Part of this work was carried out while the first author was visiting Taida Institute for Mathematical Sciences in Taipei and National Center for Theoretical Sciences in Hsinchu.  ...
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