Proceedings of the Steklov Institute of Mathematics, Vol. 218, 1997, pp. 146–159. Translated from Trudy Matematicheskogo Instituta imeni V. A. Steklova, Vol. 218, 1997, pp. 151–164. Original Russian Text Copyright \begin{matrix}c \\ ◯ \end{matrix} 1997 by Zhuravlev. English Translation Copyright \begin{matrix}c \\ ◯ \end{matrix} by ���� �����/Interperiodica Publishing (Russia).

Orbits of Representation of Numbers by Local Quadratic Forms

V. G. Zhuravlev

Received February, 1997

INTRODUCTION

In this article an attempt is made to investigate the structure of orbits {b} of primitive representations Q[b] = A of numbers by integer, over the odd p-adic ring ℤ_{p}, quadratic forms Q. Theorem 2.1 from [1], to be more precise, its p-adic analog, serves as a basis. It is proved that all orbits are divided into sets pb_{Q,A}({G}) numbered by a finite number of classes {G} of quadratic forms from the orthogonal complement of A. This division gives a “rough” classification of the orbits. Within the indicated sets there exists a one-to-one correspondence between the orbits {b} and O(G)-orbits {C} which encircle the forms A and G of the matrices C, this is a “fine” classification of the orbits {b}. In the one-dimensional case, the matrices form a set of vectors C mod A with a natural action of the orthogonal group O(G) over ℤ_{p}. If we restrict our consideration to the numbers A = εp^{ν} with ν = 1, 2, 3, then we shall have the cases enveloped by the Witt theorem within whose framework we remain to avoid an increase in the volume of the article.

The article consists of three sections. Following [1], in Section 1 we introduce the principal concepts and notation and define the one-to-one correspondence between the orbits {b} and {C}. In Section 2 all vectors C that encircle A and G are found and the local invariants of the resulting forms Q_{G}^{A}(C) are calculated. The main results concerning the orbits {b} can be found in Section 3. Here the orbits {C} are classified according to the vectors C of minimal length contained in them and the corresponding minimal forms X = Q(C) and their weights w are written out. Using these forms, we can find the number of orbits {b} for solving the quadratic equation Q[b] = A: the number of orbits is equal to the weighted number of representations of the form Q as the direct sum X ⊕ Y with the arbitrary second form Y.

Example 1. Suppose that the quadratic form Q of dimension ≥ 3 has a determinant |Q| which is not divisible by p. Then the equation Q[b] = A has an orbit {b}, i.e., its solutions exist and can be obtained from any fixed b_{1} by means of the transformations σ from the orthogonal group O(Q) over ℤ_{p}.

It is impossible to briefly formulate the conditions for the arbitrary forms Q under which one orbit {b} remains since upon the decomposition of Q in the sum X ⊕ Y a strong ramification is observed, and its translation into the language of relationships between the invariants of the form Q only makes the situation still more intricate. Instead, we shall give an example of the forms Q with a large number of orbits {b}.

Example 2. The set of all solutions (x_{1}, …, x_{4}) from the ring ℤ_{p} of the quadratic equation x_{1}^{2} + px_{2}^{2} + p^{2}x_{3}^{2} + p^{3}x_{4}^{2} = p^{3} disintegrates into (7p − 3)/4 or (5p + 1)/4 orbits according as we have the condition p ≡ 1 or −1 (mod 4).

1. CLASSIFICATION OF PRIMITIVE REPRESENTATIONS

All quadratic forms, lattices, and representations are defined over the local ring of integer p-adic numbers ℤ_{p} for an odd p.

1.1. Suppose that Q = ^{t}Q is a nondegenerate symmetric matrix of order n from the set of matrices M_{n}(ℤ_{p}) with coefficients from ℤ_{p} and A = ^{t}A is a similar matrix from M_{m}(ℤ_{p}) of dimension 1 ≤ m < n. We shall also call them quadratic forms, or simply forms, and will investigate the primitive representations of quadratic forms by forms of larger dimensions:

Q[b] = ^{t}bQb = A  for b ∈ M_{n,m}(ℤ_{p})  (1.1)

under the condition that d(b) = d_{p}(b) = 1, where we denote by d(b) the greatest common divisor (over ℤ_{p}) of minors of order m of the matrix b. Then we follow [1]. Primitive matrices b are pulled back to the matrices B = (b | e) from SL_{n}(ℤ_{P}), and the transformated form

Q_{B} = Q[B] = ( \begin{matrix}A & C \\ ^{t}C & Q′ \end{matrix} )

again belongs to the class {Q} of forms which are equivalent to Q relative to the group SL_{n}(ℤ_{p}). The matrix

M_{C} = ( \begin{matrix}1 & −A^{−1}C \\ 0 & 1 \end{matrix} )

which is, in general, not p-integer, reduces the form Q_{B} to the form

Q_{B}[M_{C}] = ( \begin{matrix}A & 0 \\ 0 & G′ \end{matrix} ) = A ⊕ G′,

equal to the direct sum of the forms A and G′ = Q′ − A^{−1}[C]. We fix the nonzero p-integer a = p^{ν} · {a} (p^{−ν} = |a|_{p} means, as usual, the p-norm of a), such that the matrix a · A^{−1} has integer coefficients and ν = 0, 1, 2, … is minimal under this condition, i.e., a is a p-adic analog of the level of the quadratic form over the ring of integer rational numbers ℤ. Then the form

a · Q_{B}[M_{C}] = ( \begin{matrix}aA & 0 \\ 0 & G \end{matrix} )

with the block G = aQ′ − aA^{−1}[C] is p-integer, and

|G| = a^{k} \frac{|Q|}{|A|},  (1.2)

where k = n − m and | · | denotes the determinant of the matrix. Excluding from the sequence b → B → C → G the dependence on the choice of the matrix B, we arrive at the representation [1, (1.16)]

[Q, A, a]: b ↦ {G} = {G(b)},

where b defines the class {G(b)} but not the form G(b) itself. We denote by pb_{Q,A} the set of all primitive representations b (1.1) and by pb_{Q,A}({G}) the set of b corresponding to the class {G}.

Since |G| is defined by the forms A and Q, it follows that pb_{Q,A} is divided into a finite number of sets pb_{Q,A}({G}) which are easier to investigate:

pb_{Q,A} = ∑_{{G}} pb_{Q,A}({G}).  (1.3)

1.2. Returning to the beginning of the section, for each solution C mod A of the congruence

E[C] + G ≡ 0 (mod a),  (1.4)

where E = a·A^{−1}, mod a is an abbreviation for mod aℤ_{p}, and C ≡ C′ (mod A) denotes the integrity of the matrix A^{−1}(C − C′), we define the form

Q_{G}^{A}(C) = \frac{1}{a} ( \begin{matrix}aA & 0 \\ 0 & G \end{matrix} ) [M_{C}^{−1}] = ( \begin{matrix}A & C \\ ^{t}C & Q′ \end{matrix} )  (1.5)

with the block Q′ = \frac{1}{a}(E[C] + G). Let C_{G}^{A}(Q) be the set of C for which form (1.5) belongs to the class {Q}. If we now fix the class {G} and exclude from b → B → C the dependence on B, then we arrive at the orbits {C}^{+} of the solutions C of congruence (1.4): C · S/ mod A for all S from the proper orthogonal group O^{+}(G) over ℤ_{p}. In turn, the mapping b → {C}^{+} divides the solutions b (1.1) into its own orbits {b}^{+} relative to the group O^{+}(Q). The quotient mapping

pb_{Q,A}(\{G\}) ⊃ \{b\}^{+} \overrightarrow{∼} \{C\}^{+} ⊂ C_{G}^{A}(Q)  (1.6)

becomes a bijection (cf. [1, p. 38]). Both sets from (1.6) are empty only if the determinant |G| does not satisfy condition (1.2).

If in the preceding definitions we replace the groups O^{+} by complete orthogonal groups O (note that the index [O : O^{+}] = 2 over ℤ_{p}), then the orbits { · }^{+} are replaced by { · } with the preservation of the bijection (1.6):

pb_{Q,A}(\{G\}) ⊃ \{b\} \overrightarrow{∼} \{C\} ⊂ C_{G}^{A}(Q).  (1.7)

The forms A and G can be defined (cf. with glueing [2, p. 133]) if the set Ext(G, A) of the encircling matrices C (1.4) is nonempty. The division (1.3) has a dual and, according to (1.2), finite partitioning

Ext(G, A) = ∑_{{Q}} C_{G}^{A}(Q).

2. ENCIRCLING VECTORS

Let us investigate the sets Ext(G, A) in the one-dimensional case m = 1 of the forms A = p^{ν}·{A}. Suppose that

G = ⨁_{α} G_{α}

with blocks G_{α} = p^{α} · G_{p^{α}} of dimensions k_{α} = dim G_{α}, α = 0, 1, 2, …, and with the determinants |G_{p^{α}}| from the group of the units ℤ_{p}^{×}. We divide the encircling vectors (in this case one-dimensional matrices) C = (C_{0} | … | C_{α} | …) into blocks C_{α} of length k_{α} respectively.

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

2.1. For ν = 1 we have E[C_{0}] ≡ −G_{0} (mod p) according to (1.4). Therefore k_{0} = 0 or k_{0} = 1 and E = {a}/{A} ∼ −G_{0}, where ∼ denotes the equivalence modulo the squares of the units (ℤ_{p}^{×})^{2}. In the first case we can take C = (0 | … | 0), and then (1.5) implies

Q_{G}^{A}(C) = A ⊕ \frac{1}{a}G.

Now if k_{0} = 1, then C = (±C_{0,fix} | … | 0), and again, according to (1.5), we have

Q_{G}^{A}(C) = ( \begin{matrix}A & C_{0} \\ C_{0} & 0 \end{matrix} ) ⊕ \frac{1}{a}G_{≠0} ∼ 1_{2}^{−} ⊕ \frac{1}{a}G_{≠0},  (2.1)

where in G_{≠0} the block G_{0} is excluded, 1_{2}^{−} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), and ∼ denotes the equivalence over the

ring ℤ_{p}.

Proposition 2.1. If A = p · {A}, a = p · {a}, then every encircling vector C from Ext(G, A) produces the form

Q_{G}^{A}(C) ∼ \{ \begin{matrix}1_{2}^{−} ⊕ \frac{1}{a}G_{≠0} & if k_{0} = 1, E ∼ −G_{0} \\ A ⊕ \frac{1}{a}G & if k_{0} = 0. \end{matrix}

In this case, Ext(G, A) consists of ext(G, A) = 2 or 1 vectors respectively.

2.2. For ν = 2, we have, by analogy, E[C_{0}] ≡ −G_{0} (mod p^{2}), and again k_{0} = 0 or k_{0} = 1 and E ∼ −G_{0}. In addition, by virtue of (1.4), we have k_{1} = 0. In the case of k_{0} = 1, we arrive at form (2.1).

For k_{0} = 0 we have

G = ⨁_{α≥2} G_{α}.

Therefore C = pC′ according to (1.4) and C′ = (0 | … | 0), and then Q_{G}^{A}(C) = p^{2}\{a\}E ⊕ \frac{1}{a}G or C′ = (C′_{2} | … | C′_{α} | 0 …) with the largest α, for which the block C′_{α} has a nonzero mod p rank r_{p}(C′_{α}) ≠ 0. The definition of matrix (1.5) implies the equivalence

Q_{G}^{A}(C) ∼ Q_{G}^{A}(p · (C′σ/ mod p)) for every σ ∈ O(G).

Therefore, using Witt’s theorem (see, e.g., [3, 4]), we can assume that C′_{α} = (1 0 … 0) and that the α-block of the matrix G is equal to

G_{p^{α}} = \{ \begin{matrix}( \begin{matrix}c & 0 \\ 0 & c^{⊥}(G_{p^{α}}) \end{matrix} ) & with c = c_{α}^{−1} if G_{p^{α}}^{−1}[^{t}C′_{α}] = c_{α} ≢ 0 (mod p) \\ ( \begin{matrix}J_{2} & 0 \\ 0 & J_{2}^{⊥}(G_{p^{α}}) \end{matrix} ) & with J_{2} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) if c_{α} ≡ 0 (mod p). \end{matrix}  (2.2)

Next we shall need the matrix realization of the orthogonal group O(Q) of the form that can

be written as Q = Q_{1} ⊕ qQ_{q}, q = p^{α}. The relation Q[M] = Q for the matrix M = ( \begin{matrix}A & B \\ C & D \end{matrix} ) implied

B = q · B_{1} with the p-integer B_{1} and

Q_{1}[A] + qQ_{q}[C] = Q_{1},  ^{t}AQ_{1}B_{1} +^{t}CQ_{q}D = 0,  qQ_{1}[B_{1}] + Q_{q}[D] = Q_{q}.  (2.3)

We infer that the block A belongs to the group O(Q_{1}/ mod q) and the block D belongs to the group O(Q_{q}/ mod q) which are both orthogonal groups over the ring of residues ℤ_{p}/qℤ_{p}. Moreover, for arbitrary A, D of this kind and for arbitrary B_{1} or C, we can use (2.3) to reconstruct the whole matrix M from the group O(Q).

Applying successively this property of orthogonal transformations to the blocks of the matrix C′, we find for it a convenient representation

C′ = C′_{fix} · σ  for C′_{fix} = ( 0 | C′_{α} | 0 ) and some σ ∈ O(G),  (2.4)

where the α-block is equal to C′_{α} = (1 0 … 0) as before. This representation and Witt’s theorem lead to the following parametrization (see (3.3)) of the orbits {C} of solutions C of congruence (1.4) with the one-dimensional form A = p^{2} ·{A}: {C} = {α, c_{α}}, where α ≥ 2 and c_{α} mod p is arbitrary, nontrivially representable by the form G_{p^{α}}^{−1} which is equivalent to the form G_{p^{α}} itself.

Represenation (2.4) makes it possible to easily calculate the blocks Q′ of the forms Q_{G}^{A}(C_{fix}) for C_{fix} = p · C′_{fix}, namely,

Q′ = \frac{p^{2}}{a} c′ ⊕ \frac{p^{2}}{a} c^{⊥} ⊕ \frac{1}{a}G_{≠α}  for c_{α} ≢ 0 (mod p),  (2.5)

where c′ = E + p^{α−2}c, c^{⊥} = c^{⊥}(G_{p^{α}}), and

Q′ = \frac{p^{2}}{a} ( \begin{matrix}E & p^{α−2} \\ p^{α−2} & 0 \end{matrix} ) ⊕ \frac{p^{α}}{a} J_{2}^{⊥} ⊕ \frac{1}{a}G_{≠α} for c_{α} ≡ 0 (mod p),

where we have also used the abbreviation J_{2}^{⊥} = J_{2}^{⊥}(G_{p^{α}}).

For what follows, we shall need the matrix analog of the well-known algorithm of diagonalization of quadrtic forms over the odd local ring ℤ_{p}. The transformation

M = ( \begin{matrix}1 & 0 & 0 \\ −Q_{1}^{−1} ·^{t}C & 1 & −Q_{1}^{−1} · C_{2} \\ 0 & 0 & 1 \end{matrix} )

of the form

Q = ( \begin{matrix}A & C & C_{1} \\ ^{t}C & Q_{1} & C_{2} \\ ^{t}C_{1} & ^{t}C_{2} & Q_{2} \end{matrix} )

which is proper and integer under the condition |Q_{1}| ≢ 0 (mod p), splits off the block Q_{1}. By a direct calculation we get the decomposition of the form Q in the direct sum

Q ∼ Q_{1} ⊕ ( \begin{matrix}Ã & C̃_{1} \\ ^{t}C̃_{1} & Q̃ \end{matrix} ) ,  (2.6)

where Ã = A − Q_{1}^{−1}[^{t}C], C̃_{1} = C_{1} − CQ_{1}^{−1}C_{2}, and Q̃ = Q_{2} − Q_{1}^{−1}[C_{2}].

For c_{α} ≢ 0 (mod p) we use the relations (1.5) and (2.5) to find

Q_{G}^{A}(C_{fix}) = E_{c} ⊕ \frac{p^{α}}{a} c^{⊥} ⊕ \frac{1}{a}G_{≠α}, E_{c} = ( \begin{matrix}aE^{−1} & p \\ p & \frac{p^{2}}{a} c′ \end{matrix} ) .

Setting

Q_{1} = \frac{p^{2}}{a} c′ = \frac{1}{\{a\}}c′,

for α > 2, we obtain, from relation (2.6), the decomposition of the form E_{c} ∼ {a}E ⊕ p^{α}{a}c. For α = 2, c′ ≢ 0 (mod p) and c′ ≡ 0 (mod p) we obtain, by analogy, E_{c} ∼ {a}(E+c)⊕p^{2}{a}(E^{−1}+c^{−1}) and E_{c} ∼ p · 1_{2}^{−}, respectively. If c_{α} ≡ 0 (mod p), then

Q_{G}^{A}(C_{fix}) = E_{0} ⊕ \frac{p^{α}}{a} J_{2}^{⊥} ⊕ \frac{1}{a}G_{≠α}

with the block

E_{0} = ( \begin{matrix}aE^{−1} & p & 0 \\ p & \{a\}^{−1}E & \{a\}^{−1}p^{α−2} \\ 0 & \{a\}^{−1}p^{α−2} & 0 \end{matrix} ) ∼ \{a\}E ⊕ ( \begin{matrix}0 & −E^{−1}p^{α−1} \\ −E^{−1}p^{α−1} & −\{a\}^{−1}E^{−1}p^{α−2} \end{matrix} ) .

The last block

E′_{0} ∼ \{ \begin{matrix}p^{α−1} · 1_{2}^{−} & if α > 2 \\ − \{a\}E ⊕ p^{2}\{a\}E & if α = 2. \end{matrix}

Finally, it remains to calculate the number of rows of the orbits {α, c_{α}} which have the form

C = (pC′_{2} | … | pC′_{α−1} | pC′_{α} | 0 …)

with arbitrary blocks C′_{2}, …, C′_{α−1} mod p, and, consequently, the number of combinations is equal to p^{k_{<α}} with k_{<α} = k_{2} +⋯+k_{α−1} and with the block C′_{α} which satisfies the congruence G_{p^{α}}^{−1}[^{t}C′_{α}] ≡ 0 (mod p). According to the Witt’s theorem, all solutions belong to the same orbit, and, hence, their number is equal to o(G_{p^{α}}/ mod p), i.e., to the order of the orthogonal group mod p for the form G_{p^{α}} ∼ G_{p^{α}}^{−1} divided by stab(C′_{α}), which is the order of the stabilizer of the fixed C′_{α}. According to the conditions c_{α} ≢ 0 (mod p) and c_{α} ≡ 0 (mod p), the stabilizers contain elements o(c^{⊥}(G_{p^{α}})/ mod p) and p^{k_{α}−1}o(J_{2}^{⊥}(G_{p^{α}})/ mod p), and the relations for the orders of orthogonal groups are known (see, e.g., [1, (10.2)]). We have thus proved

Proposition 2.2. Suppose that A = p^{2}{A}, E = aA^{−1}, a = p^{2}{a} is fixed, and the form G = ⨁_{α} G_{α} with blocks G_{α} = p^{α} · G_{p^{α}} of dimension k_{α}.

I. If k_{0} = 1, k_{1} = 0, and E ∼ −G_{0}, then every encircling vector C from Ext(G, A) gives the form

Q_{G}^{A}(C) ∼ 1_{2}^{−} ⊕ \frac{1}{a}G_{≠0}  and  ext(G, A) = 2.

II. If k_{0} = k_{1} = 0, then there exists a partitioning of the set of encircling vectors from Ext(G, A) into the orbits

Ext(G, A) = {0} + ∑_{α≥2,c_{α}}{α, c_{α}}  (2.7)

with the summation over all c_{α} mod p which are nontrivially representable by the form G_{p^{α}}.

1. The zero vector C = (0 | … | 0) gives the form

Q_{G}^{A}(C) ∼ p^{2}\{a\}E ⊕ \frac{1}{a}G.

2. If C is from the orbit {α, c_{α}} for c_{α} ≢ 0 (mod p) and c = c_{α}^{−1}, then

Q_{G}^{A}(C) ∼ E_{c} ⊕ \frac{p^{α}}{a} c^{⊥}(G_{p^{α}}) ⊕ \frac{1}{a}G_{≠α},  (2.8)

every vector C performs the 2-encircling E_{c}, i.e., minimal form of dimension 2 that can be written as

E_{c} ∼ \{ \begin{matrix}\{a\}E ⊕ p^{α}\{a\}c & if α > 2 \\ \{a\}(E + c) ⊕ p^{2}\{a\}(E^{−1} + c^{−1}) & if α = 2, c ≢ −E (mod p) \\ p · 1_{2}^{−}, & if α = 2, c ≡ −E (mod p), \end{matrix}

and the orbit {α, c_{α}} has the order

(α, c_{α}) = p^{k_{<α}} \frac{o(G_{p^{α}}/ mod p)}{o(c^{⊥}(G_{p^{α}})/ mod p)},  (2.9)

which depends only on the sign of (\frac{c}{p}) = ±1.

3. Now if the vector C belongs to the zero orbit {α, 0}, then

Q_{G}^{A}(C) ∼ E_{0} ⊕ \frac{p^{α}}{a} J_{2}^{⊥}(G_{p^{α}}) ⊕ \frac{1}{a}G_{≠α},

where the form E_{0} is now a 3-encircling that can be written as

E_{0} ∼ \{ \begin{matrix}\{a\}E ⊕ p^{α−1} · 1_{2}^{−} & if α > 2 \\ 1_{2}^{−} ⊕ p^{2}\{a\}E & if α = 2, \end{matrix}

and

(α, 0) = p^{k_{<α}} \frac{o(G_{p^{α}}/ mod p)}{p^{k_{α}−1}o(J_{2}^{⊥}(G_{p^{α}})/ mod p)}.  (2.10)

In all others cases ext(G, A) = 0.

Remark. For applications, we must know the distribution by classes 1^{η_{1}} ⊕ p^{α} · 1^{η_{2}}, where η = (η_{1}, η_{2}) = (±, ±) and (\frac{1^{−}}{p} ) = −1, of the forms of the second kind from (2.8). We denote by (η) the number of residues c ≢ 0 (mod p) under the conditions

(\frac{\{a\}(E + c)}{p}) = η_{1}, (\frac{\{a\}(E^{−1} + c^{−1})}{p}) = η_{2}.

By analogy [5, p. 86], we calculate

(η) = \{ \begin{matrix}\frac{1}{4}(p − 2 − 3(\frac{−1}{p})) & if η_{1} = η_{2} = (\frac{−\{a\}E}{p} ) \\ \frac{1}{4}(p − 2 + (\frac{−1}{p})) & in the other cases. \end{matrix}

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

2.3. As we shall see later, the primitive representations of cubes A = p^{3}{A} is the last case, where we can directly use Witt’s theorem. If k_{0} = 1, then the two dimensions k_{1} = k_{2} = 0, and we again arrive at relation (2.1).

Let k_{0} = k_{1} = k_{2} = 0 and, hence, G = G_{3} ⊕ G_{4} ⊕ · · ·. By virtue of (1.4), C = p^{2}C′ with any C′ mod p, i.e., C′ = (0 | … | 0). Then

Q_{G}^{A}(C) = p^{3}\{a\}E ⊕ \frac{1}{a}G,

or C′ = (C′_{3} | … | C′_{α} | 0 …) with the largest α ≥ 3, when the rank r_{p}(C′_{α}) ≠ 0.

In the case of C′ ≢ 0 (mod p), we can assume that the α-block has form (2.2) and the vector C′ = C′_{fix}, where C′_{fix} = (0 | C′_{α} | 0) and C′_{α} = (1 0 … 0).

For c_{α} ≢ 0 (mod p), by virtue of (1.5), we have

Q′ = \frac{1}{\{a\}}c′ ⊕ \frac{p^{α}}{a} c^{⊥} ⊕ \frac{1}{a}G_{≠α},

where c′ = pE + p^{α−3}c, and therefore, for C_{fix} = p^{2}C′_{fix}, we have

Q_{G}^{A}(C_{fix}) ∼ E_{c} ⊕ \frac{p^{α}}{a} c^{⊥} ⊕ \frac{1}{a}G_{≠α} with E_{c} = ( \begin{matrix}aE^{−1} & p^{2} \\ p^{2} & \{a\}^{−1}c′ \end{matrix} )

as 2-encircling whose decomposition can be found by means of algorithm (2.6). Thus c′ = pc′′ and c′′ ≡ E ≢ 0 (mod p) for α > 4, whence it follows that

E_{c} ∼ p\frac{c′′}{\{a\}} ⊕ pÃ,  Ã = \frac{a}{p}E^{−1} − \frac{\{a\}}{c′′}p^{2} ∼ p^{α−2}\{a\}c,

and, hence, E_{c} ∼ p{a}E ⊕ p^{α−1}{a}c. The other cases can be considered by analogy. For c_{α} ≡ 0 (mod p) we obtain

Q_{G}^{A}(C_{fix}) ∼ E_{0} ⊕ \frac{p^{α}}{a} J_{2}^{⊥} ⊕ \frac{1}{a}G_{≠α}

with the block

E_{0} = ( \begin{matrix}aE^{−1} & p^{2} & 0 \\ p^{2} & \{a\}^{−1}Ep & \{a\}^{−1}p^{α−3} \\ 0 & \{a\}^{−1}p^{α−3} & 0 \end{matrix} )

as a 3-encircling now, just as in item 2.2.

The case k_{0} = k_{1} = 0 and k_{2} = 1, is a new one, where, in accordance with (1.4), C_{2} = pC′_{2} and C′_{2} mod p^{2} satisfies the congruence E[C′_{2}] ≡ −G_{p^{2}} (mod p). We can assume here that E = −G_{p^{2}} and C′_{2} has the form C′_{2,fix} = 1 + pC′′_{2} with any C′′_{2} mod p. Therefore C = pC′, where C′ = (C′_{2} | 0 …) or C′ = (C′_{2} | pC′_{3} | … | pC′_{α} | 0 …) with the α-block C′_{α} = (1 0 … 0). In this case, the block G_{p^{α}} has form (2.2). The vectors C′ = (C′_{2} | 0 …) give the form

Q_{G}^{A}(C) ∼ E_{C} ⊕ \frac{1}{a}G_{>2}, where E_{C} = ( \begin{matrix}aE^{−1} & pC′_{2} \\ pC′_{2} & \{a\}^{−1}c′ \end{matrix} )

and c′ = 2EC′′_{2} + pEC′′_{2}^{2}. If c′ ≢ 0 (mod p), i.e., C′′_{2} ≢ 0 (mod p), then

E_{C} ∼ 1 ⊕ p^{2} · (−1)  or  E_{C} ∼ −1 ⊕ p^{2} · 1,

and every form appears 2 · (p − 1)/2 = p − 1 times. Now if c′ ≡ 0 (mod p), then E_{C} ∼ p · 1_{2}^{−} and the form on the right appears twice.

The maximal 4-encirclings give the vectors C′ = (C′_{2} | … | pC′_{α} | 0 …) from the O(G)-orbit of the vector

C′_{fix} = (1 | 0 … | pC′_{α} | 0 …)  with  C′_{α} = (1 0 … 0)  (2.11)

and the form G_{p^{α}} of the kind (2.2) in the case of c_{α} ≡ 0 (mod p):

E_{0} = ( \begin{matrix}aE−1 & p & p^{2} & 0 \\ p \\ p^{2} & & \{a\}^{−1}E′_{0} \\ 0 \end{matrix} )

with the block

E′_{0} = ( \begin{matrix}0 & E & 0 \\ E & pE & p^{α−3} \\ 0 & p^{α−3} & 0 \end{matrix} ) .

Setting

Q_{1} = \frac{1}{\{a\}} ( \begin{matrix}0 & E \\ E & pE \end{matrix} ) ,

in the relation (2.6), we find the decomposition for E_{0}.

Proposition 2.3. Let A = p^{3}{A} and E = aA^{−1} for a = p^{3}{a}.

I. If k_{0} = 1, k_{1} = k_{2} = 0, and E ∼ −G_{0}, then the encircling vectors C from ext(G, A) give the form

Q_{G}^{A}(C) ∼ 1_{2}^{−} ⊕ \frac{1}{a}G_{≠0}  and  ext(G, A) = 2.

II. If k_{0} = k_{1} = k_{2} = 0, then the set of encircling vectors from Ext(G, A) decompose, just as in (2.7), into orbits {0} and {α, c_{α}}, where α ≥ 3.

1. The vector C = (0 | … | 0) gives the form

Q_{G}^{A}(C) ∼ p^{3}\{a\}E ⊕ \frac{1}{a}G.

2. If C belongs to the orbit {α, c_{α}} with c_{α} ≢ 0 (mod p) and c = c_{α}^{−1}, then the form

Q_{G}^{A}(C) ∼ E_{c} ⊕ \frac{p^{α}}{a} c^{⊥}(G_{p^{α}}) ⊕ \frac{1}{a}G_{≠α}

with the 2-encirclings

E_{c} ∼ \{ \begin{matrix}p\{a\}E ⊕ p^{α−1}\{a\}c & if α > 4 \\ p\{a\}(E + c) ⊕ p^{α−1}\{a\}(E^{−1} + c^{−1}) & if α = 4, c ≢ −E (mod p) \\ p^{2} · 1_{2}^{−} & if α = 4, c ≡ −E (mod p) \\ \{a\}c ⊕ p^{3}\{a\}E & if α = 3, \end{matrix}

and the orbit {α, c_{α}} is of order (α, c_{α}) (2.9).

3. In the case of a vector from the orbit {α, 0}, the form

Q_{G}^{A}(C) ∼ E_{0} ⊕ \frac{p^{α}}{a} J_{2}^{⊥}(G_{p^{α}}) ⊕ \frac{1}{a}G_{≠α}

with 3-encirclings

E_{0} ∼ \{ \begin{matrix}p\{a\}E ⊕ p^{α−2} · 1_{2}^{−} & if α > 4 \\ p · 1_{2}^{−} ⊕ p^{3}\{a\}E & if α = 4 \\ 1_{2}^{−} ⊕ p^{3}\{a\}E & if α = 3, \end{matrix}

and the order of orbit (α, 0) is equal to (2.10).

III. Finally, let k_{0} = k_{1} = 0, k_{2} = 1, and E ∼ −G_{p^{2}}. Then the set of encircling vectors from Ext(G, A) has partitioning (2.7), where the zero orbit {0} consists of C = (pC′_{1} | 0 …) with the block C′_{2} mod p^{2} and E[C′_{2}] ≡ −G_{p^{2}} (mod p), and {α, c_{α}} with α ≥ 3 is an O(G)-orbit of the vector C_{fix} = p · C′_{fix} (2.11).

1. The vectors C from the orbit {0} lead to the form

Q_{G}^{A}(C) ∼ E_{C} ⊕ \frac{1}{a}G_{>2},  (2.12)

where the blocks E_{C} ∼ 1 ⊕ p^{2} · (−1), E_{c} ∼ −1 ⊕ p^{2} · 1, and E_{C} ∼ p · 1_{2}^{−} appear p − 1, p − 1, and 2 times, respectively.

2. For C, from the orbit {α, c_{α}} with c_{α} ≢ 0 (mod p) we have

Q_{G}^{A}(C) ∼ E_{c} ⊕ \frac{p^{α}}{a} c^{⊥}(G_{p^{α}}) ⊕ \frac{1}{a}G_{≠2,α}  (2.13)

with 3-encircling

E_{c} ∼ 1_{2}^{−} ⊕ p^{α−1}{a}c, α ≥ 3,

and with the order of the orbit (α, c_{α}) twice exceeding the order (2.9).

3. Now if C belongs to the orbit {α, 0}, then

Q_{G}^{A}(C) ∼ E_{0} ⊕ \frac{p^{α}}{a} J_{2}^{⊥}(G_{p^{α}}) ⊕ \frac{1}{a}G_{≠2,α}  (2.14)

with a 4-encircling

E_{0} ∼ 1_{2}^{−} ⊕ p^{α−1} · 1_{2}^{−}, α ≥ 3,

and now the order of the orbit (α, 0) twice exceeds the order (2.10).

In all the other cases ext(G, A) = 0.

3. ORBITS OF ONE-DIMENSIONAL REPRESENTATIONS

The existence of the one-to-one correspondence (1.7) between the orbits {b} of the primitive solutions of (1.1) and the orbits {C} of the solutions C mod A of congruence (1.4) imply the finiteness of the number of orbits {b}. In Section 2 we classified the orbits {C} in the case of one-dimensional matrices A = p^{ν}{A} with ν ≤ 3. Preserving the possibility of classifying the orbits {b}, we shall restrict ourselves to the calculation of the number of these orbits.

3.1. Suppose that in the notation of Section 2 C = (C_{0} | … | C_{α} | …) and G = G(C) ⊕ G_{res}, where G(C) is the direct sum of the blocks G_{α}, for which G_{α} ≢ 0 (mod p^{ν}). Then, similarly, Q′ ∼ Q′(C) ⊕ \frac{1}{a}G_{res}, and therefore, the form (1.5) decomposes in the direct sum

Q_{G}^{A}(C) ∼ Q(C) ⊕ \frac{1}{a}G_{res},  Q(C) = ( \begin{matrix}A & C_{≢0} \\ ^{t}C_{≢0} & Q′(C) \end{matrix} ) .  (3.1)

Thus, choosing C = (0 | … | 0) and C = (C_{0,fix} | 0 …) for A = p{A}, we get, according to Proposition 2.1, the forms Q(C) = A and Q(C) = 1_{2}^{−}. From this and from decomposition (3.1) we obtain

Proposition 3.1. The number of orbits {b} of the solutions of (1.1) with A = p{A} is equal to the weighted number of decompositions Q ∼ X ⊕ Y, where the forms X have the following appearance and weights:

pX_{1} = p{A}, w_{1} = 1, X_{20} = 1_{2}^{−}, w_{20} = 1.

Remark. The “weighted” number means that every decomposition is counted the number of times corresponding to w. In the next items we shall encounter nonidentity weights, and this explains the use of this terms (for the sake of uniformity).

3.2. If A = p^{2}{A}, then the encircling vectors C of form (2.4) are added:

C_{fix} = pC′_{fix} = (0 | … | pC′_{α} | 0 …)

with the block C′_{α} = (1 0 … 0) and α ≥ 2.

Lemma 3.1. For the fixed form G the enumerated vectors give a complete system of representatives of all diferent orbits {C} of the solutions C mod p^{2} of the congruence E[C] ≡ −G (mod p^{2}).

Proof. According to (2.4), we have reprsentatives C of all orbits, and, consequently, it remains to show that these orbits are different. For G of dimension k_{0} = 1 we have one vector C_{0,fix}, and there is nothing to prove. Let k_{0} = 0 (k_{1} = 0 always). According to [1, (10.7)], in the orthogonal group O(G) any matrix σ = (x_{ij}) with the partition into blocks consistent with the partition into blocks of the form

G = ⨁_{α≥2} p^{α}G_{p^{α}}, |G_{p^{α}}| ≢ 0 (mod p),

can be written as

σ = ( \begin{matrix}x_{22} & & px′_{23} & & & … & & & p^{α−2}x′_{2α} & & & … \\ x_{32} & & x_{33} & & & … & & & p^{α−3}x′_{3α} & & & … \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ x_{α2} & & x_{α3} & & & … & & & x_{αα} & & & … \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \end{matrix} ) .  (3.2)

All blocks x_{ij}, x′_{ij} have p-integer coefficients and the diagonal blocks x_{ii} belong to the groups O(G_{p^{i}}/ mod p). Since (3.2) yields the congruence

C · σ = (… | pC′_{α}x_{αα} | 0 …) (mod p^{2}),  (3.3)

α and c_{α} mod p, where G_{p^{α}}^{−1}[^{t}C′_{α}] = c_{α}, parametrize the orbits {C}. We have proved the lemma.

From Lemma 3.1 and Proposition 2.2 we obtain

Proposition 3.2. Equation (1.1) with A = p^{2}{A} has the number of orbits {b} which is equal to the weighted number of decompositions Q ∼ X ⊕ Y, where the forms X have the following appearance and weights:

(1) one-dimensional forms

p^{2}X_{1} = p^{2}{A}, w_{1} = 1;

(2) two-dimensional forms

X_{20} = 1_{2}^{−}, pX_{20} = p · 1_{2}^{−}, w_{20} = 1,
X_{22}^{η} = 1^{η_{1}} ⊕ p^{2} · 1^{η_{2}}, w_{22}^{η} = (η),
X_{2α} = \{A\} ⊕ p^{α} · 1^{±}  for α > 2,  w_{2α} = \frac{p − 1}{2};

(3) three-dimensional forms

\begin{matrix}X_{32} = 1_{2}^{−} ⊕ p^{2} · \{A\},  w_{32} = 1, \\ X_{3α} = \{A\} ⊕ p^{α} · 1_{2}^{−}  for α ≥ 2,  w_{3α} = 1. \end{matrix}

3.3. For A = p^{3}{A}, in addition to vectors C of types 1 and 2

(C_{0,fix} | 0 …) and (0 | … | 0), (0 | … | p^{2}C′_{α} | 0 …)

with α ≥ 3 in the case of the form

G = ⨁_{α≥2} p^{α}G_{p^{α}},

where k_{0} = k_{1} = 0, k_{2} = 1, and G_{p^{2}} ∼ −E, there appear the vectors of the third type

(C_{2} | 0 …)  with  C_{2} = p(C′_{2,fix} + pC′′_{2}),  C′′_{2} mod p,  (3.4)

and

(C_{2} | … | p^{2}C′_{α} | 0 …)  with  C′_{α} = (1 0 … 0) and α ≥ 4.  (3.5)

Here the value α = 3 is excluded. Obviously, orbits of different types are different themselves since every type has its own form G. The reasoning for vectors of the first two types repeats the reasoning from item 3.2 for the case A = p^{2}{A}.

According to (3.2), the matrices σ from the orthogonal group O(G) transform the vector C (3.5) into the vector C · σ of the form

(pC′_{2}x_{22} + p^{2}C′_{α}x_{α2} | … | p^{α−1}C′_{α}x′_{2α} + p^{2}C′_{α}x_{αα} | p^{α}C′_{2}x′_{2,α+1} + p^{3}C′_{α}x_{α,α+1} | …).

Therefore, if we a priori assume the fulfillment of the condition α > 3, then the number α of the last, mod p nondegenerate block C′_{α} is an invariant of the orbit {C}. Since, in addition, the congruence of α-blocks C · σ ≡ p^{2}C′_{α}x_{αα} (mod p^{3}) is fulfilled for x_{αα} ∈ O(G_{p^{α}}/ mod p), it follows that c_{α} mod p is also an invariant of the orbit. Relation (2.3) show that in the orbit {C} the block C′_{2} of the vector C is defined only mod p, and, hence, we can take C′_{2} = C′_{2,fix}. Consequently, for α ≥ 4 every form Q(C) = E_{c} from decomposition (2.13) writen as 1_{2}^{−} ⊕ p^{α−1} · 1^{±} has the weight (p − 1)/2, and the form Q(C) = E_{0} from decomposition (2.14) written as 1_{2}^{−} ⊕ p^{α−1} · 1_{2}^{−} has the identity weight.

Now if α = 3, then, using the defining relationships (2.3) between the blocks of orthogonal matrices, we can show that there exists a matrix σ from the group O(G) with the block x′_{23}: 3block C · σ ≡ 0 (mod p^{3}). In this case, the orbit {C} contains a vector C of form (3.4). Suppose that the congruence

C · σ ≡ (pC′_{2}x_{22} | p^{2}C′_{2}x′_{23} | 0 …) ≡ C (mod p^{3})

is satisfied for this vector and for the arbitrary matrix σ (3.2) from O(G). Then x′_{23} ≡ 0 (mod p) by virtue of the condition C′_{2} ≢ 0 (mod p). From this and from the first relation of (2.3) we have the congruence

G_{p^{2}}[x_{22}] ≡ G_{p^{2}} (mod p^{2}),

and, hence, the block x_{22} ≡ ±1 (mod p^{2}) since the form G_{p^{2}} is one-dimensional. Therefore, for all σ of this kind, the 2-block of the transformed vector C · σ satisfies the congruence 2-block C · σ ≡ ±pC′_{2} (mod p^{3}). This proves that the vectors C of form (3.4) belong to different orbits and the corresponding forms Q(C) = E_{C} from decomposition (2.12) written as 1 ⊕ p^{2} · (−1) and −1 ⊕ p^{2} · 1 have the weight (p − 1)/2 each. We have proved

Proposition 3.3. The number of orbits {b} of the solutions of (1.1) in the case A = p^{3}{A} coincides with the weighted number of decompositions Q ∼ X ⊕ Y, where the forms X have the following appearance and weights:

(1) one-dimensional forms

p^{3}X_{1} = p^{3}{A}, w_{1} = 1;

(2) two-dimensional forms

X_{20} = 1_{2}^{−}, pX_{20}, p^{2}X_{20}, w_{20} = 1,

X_{22}^{1} = ±1 ⊕ p^{2}(∓1), w_{22}^{1} = p − 1,

pX_{22}^{η} = p(1^{η_{1}} ⊕ p^{2} · 1^{η_{2}}), w_{22}^{η} = (η),

X_{23} = 1^{±} ⊕ p^{3}\{A\}, w_{23} = \frac{p − 1}{2},

pX_{2α} = p(\{A\} ⊕ p^{α} · 1^{±})  for α ≥ 3,  w_{2α} = \frac{p − 1}{2};

(3) three-dimensional forms

\begin{matrix}pX_{32} = p(1_{2}^{−} ⊕ p^{2}\{A\}),  w_{32} = 1, \\ X_{33} = 1_{2}^{−} ⊕ p^{3}\{A\},  w_{33} = 1, \\ pX_{3α} = p(\{A\} ⊕ p^{α} · 1_{2}^{−})  for α ≥ 2,  w_{3α} = 1, \\ X_{3α} = 1_{2}^{−} ⊕ p^{α} · 1^{±}  for α ≥ 3,  w_{3α} = \frac{p − 1}{2}; \end{matrix}  (3.6)

(4) four-dimensional forms

X_{4α} = 1_{2}^{−} ⊕ p^{α} · 1_{2}^{−}  for α ≥ 2,  w_{4α} = 1.

Remark. In the case of primitive representations of cubes A = p^{3}{A} a new phenomenon appears. The three-dimensional quadratic forms Q = Q(C), which are equivalent to the form X_{33} = 1_{2}^{−} ⊕ p^{3}{A} from (3.6), admit two types of the so-called minimal primitive representations of the numbers A = p^{3}{A}, corresponding, respectively, to the forms G from item 1.1 which can be written as

G ∼ p^{3} · 1_{2}^{−}  and  G ∼ p^{2}(−{a}{A}) ⊕ p^{4}{a}{A}.

The total number of all orbits {b} of solutions of equation Q[b] = p^{3}{A} is, according to Proposition 3.3, equal to p + 2. Out of this number, 1 + (p − 1)/2 = (p + 1)/2 orbits {b} belong to the minimal representations and (p + 3)/2 orbits belong to the other types.

ACKNOWLEDGMENTS

The work was supported by the Russian Foundation for Fundamental Research (project no. 96-01-00502).

REFERENCES

1. Zhuravlev, V. G., Representing of the Form by the Genus of Quadratic Forms, Algebra i Analiz, 1996, vol. 8, no. 1, pp. 21–112.

2. Conway, J. H. and Sloane, N. J. A., Sphere Packings, Lattices and Groups, Springer, 1988.

3. Artin, E., Geometric Algebra, Princeton Univ., 1957.

4. Cassels, J., Rational Quadratic Forms, London Academic, 1978.

5. Vinogradov, I. M., Osnovy teorii chisel (Foundations of Number Theory), Moscow: Nauka, 1953.

Translated by I. Aleksandrova