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A character of the Siegel modular group of level 2 from theta constants.

1. Introduction. The theta function of a characteristic m of degree g is the series

[mathematical expression not reproducible],

where [mathematical expression not reproducible] is the Siegel upper half-plane, m' and m" denote vectors in [Z.sup.g] determined by the first and last g coefficients of m. If we put z = 0, we get the theta constant [[theta].sub.m]([tau]) = [[theta].sub.m]([tau], 0). The study of theta functions and theta constants has a long history, and they are very important objects in arithmetic and geometry. They can be used to construct modular forms and to study geometric properties of abelian varieties. Farkas and Kra's book [1] contains very detailed descriptions for the case of degree one. In [3], [4], and [5], Matsuda gives new formulas and applications. It is Igusa in [2] who began to study the cases of higher degrees. He used [[theta].sub.m]([tau])[[theta]9.sub.n]([tau]) to determine the structure of the graded rings of modular forms belonging to the group [[GAMMA].sub.g](4, 8).

In this note, we will define a character of the group [[GAMMA].sub.g](2), the principal congruence group of degree g and of level 2. We obtained its computation formula. Using our results, Igusa's key Theorem 3 in [2] can be recovered.

2. The Siegel modular group of level 2. The Siegel modular group Sp(g, Z) of degree g is the group of 2g x 2g integral matrices M satisfying

[mathematical expression not reproducible],

in which [sup.t]M is the transposition of M, [I.sub.g] is the identity of degree g. If we put [mathematical expression not reproducible], the condition for M in Sp(g, Z) is [a.sup.t]d - [b.sup.t]c = [I.sub.g], [a.sup.t]b and [c.sup.t]d are symmetric matrices. In fact, if M is in Sp(g, Z), then [sup.t]bd and [sup.t]ac are also symmetric, see [6, p. 437]. In this paper, we discuss two special subgroups of the Siegel modular group. The first is the principal congruence subgroup [[GAMMA].sub.g](2) of degree g and of level 2 which is defined by M [equivalent to] [I.sub.2g] (mod 2). The second is the Igusa modular group [[GAMMA].sub.g](4, 8), which is defined by M [equivalent to] [I.sub.2g] (mod 4) and ([a.sup.t]b)0 [equivalent to] [([c.sup.t]d).sub.0] [equivalent to] 0 (mod 8). If s is a square matrix, we arrange its diagonal coefficients in a natural order to form a vector [(s).sub.0]. The Siegel modular group Sp(g, Z) acts on [H.sub.g] by the formula

M[tau] = (a[tau] + b)[(c[tau] + d).sup.-1],

where [mathematical expression not reproducible]. An element m [member of] [Z.sup.2g] is called a theta characteristic of degree g. If n is another characteristic, then we have

[mathematical expression not reproducible].

Since

[mathematical expression not reproducible],

m is called even or odd according as [sup.t]m'm" is even or odd. Only for even m, theta constants are none zero. Given a characteristic m and an element M in the Siegel modular group, we define

[mathematical expression not reproducible].

This operation modulo 2 is a group action, i.e. [M.sub.1] [omicron] ([M.sub.2] [omicron] m) [equivalent to] ([M.sub.1][M.sub.2]) [omicron] m (mod 2). Next we explain the transformation formulas for theta constants: for any m [member of] [Z.sup.2g] and M [member of] Sp(g, Z), we put

[mathematical expression not reproducible],

then we have

[mathematical expression not reproducible], in which a(M) is an eighth root of unity depending only on M and the choice of square root sign for det[(c[tau] + d).sup.1/2], and [mathematical expression not reproducible]. From now on, we always discuss the group [[GAMMA].sub.g](2), unless specified. Hence, we can write

[mathematical expression not reproducible].

3. Main theorems and proofs. The character mentioned in the abstract is as follows:

Definition. Let m [member of] [Z.sup.2g], M [member of] [[GAMMA].sub.g](2), we define [[chi].sub.m](M) by

[mathematical expression not reproducible],

where a(M) comes from the transformation formulas of theta constants.

Theorem 3.1. For a fixed m, [[chi].sub.m](M) is a character of [[GAMMA].sub.g](2).

The proof of Theorem 3.1 needs two lemmas, in which [[GAMMA].sub.g](2) is essential.

Lemma 3.2. If m, n [member of] [Z.sup.2g] and m [equivalent to] n (mod 2), then [[PHI].sub.m](M) [equivalent to] [[PHI].sub.n](M) (mod 1).

Proof. Let m = n + 2[DELTA], then

[mathematical expression not reproducible],

since b [equivalent to] 0 (mod 2) and [sup.t]bd is symmetric, the last two terms are equal. Similarly, [sup.t]ac is symmetric which implies that [sup.t]m"[sup.t]acm" [equivalent to] [sup.t]n"[sup.t]acn" (mod 8). Moreover, [2.sup.t][([a.sup.t]b).sub.0]dm' [equivalent to] [2.sup.t][([a.sup.t]b).sub.0]dn' (mod 8) is trivial. By the definition of [[PHI].sub.m](M), Lemma 3.2 is true.

Lemma 3.3. For M, M' [member of] [[GAMMA].sub.g](2), we have [[PHI].sub.M'[omicron]m](M) [equivalent to] [[PHI].sub.m](M) (mod 1).

Proof. This lemma can be proved from the definition of M' [omicron] m with M' [member of] [[GAMMA].sub.g](2) and Lemma 3.2.

Proof of Theorem 3.1. We firstly give a formula for [[chi].sub.m](M). By the definition of the operation [omicron], we can find a unique n in [Z.sup.2g] with M [omicron] n = m. Define [DELTA] [member of] [Z.sup.2g] by m + 2[DELTA] = n, then by Lemma 3.2 and Lemma 3.3, we have

[mathematical expression not reproducible].

Hence,

(1) [mathematical expression not reproducible].

To prove Theorem 3.1 is equivalent to prove [[chi].sub.m]([M.sub.1])[[chi].sub.m]([M.sub.2]) = [[chi].sub.m]([M.sub.1][M.sub.2]) for any [M.sub.1], [M.sub.2] [member of] [[GAMMA].sub.g](2). Now fix [M.sub.1], [M.sub.2] and m, define [[DELTA].sub.1], [[DELTA].sub.2] by m + 2[[DELTA].sub.1] = [n.sub.1], [M.sub.1] [omicron] [n.sub.1] = m; m + 2[[DELTA].sub.2] = [n.sub.2], [M.sub.2] [omicron] [n.sub.2] = m. Write

[mathematical expression not reproducible].

by (1), we have

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

In order to compute [[chi].sub.m]([M.sub.1][M.sub.2]), we write [mathematical expression not reproducible], and define [[DELTA].sub.3] by m + 2[[DELTA].sub.3] = [bar.n] with [M.sub.1] [omicron] [n.sub.1] = m, [M.sub.2] [omicron] [bar.n] = [n.sub.1]. Then by Lemmas 3.2 and 3.3, we have

[mathematical expression not reproducible],

in which, we use c[tau] + d = ([c.sub.1][tau]' + [d.sub.1])([c.sub.2][tau] + [d.sub.2]) and a(M) = a([M.sub.1])a([M.sub.2]), the latter is implied by Igusa's Theorem [member of] in [2]. Hence,

[mathematical expression not reproducible].

Now we compute

[mathematical expression not reproducible].

From [M.sub.1] [omicron] [n.sub.1] = m, we get

(2) [mathematical expression not reproducible].

Therefore, from m + 2[[DELTA].sub.1] = [n.sub.1], we have

[mathematical expression not reproducible],

by noting that [b.sub.1] [equivalent to] [c.sub.1] [equivalent to] 0 (mod 2) and [a.sub.1] [equivalent to] [d.sub.1] [equivalent to] [I.sub.g] (mod 2). So

[mathematical expression not reproducible],

because [sup.t][b.sub.1]/2 (mod 2) is symmetric, which follows from the fact a[sup.t][b.sub.1] is symmetric. Similarly,

[mathematical expression not reproducible].

The computation for [[DELTA]".sub.3] is more complicated. Recall that [M.sub.1] [omicron] [n.sub.1] = m, [M.sub.2] [omicron] [bar.n] = [n.sub.1], m + 2[[DELTA].sub.3] = [bar.n] and (2), we have

[mathematical expression not reproducible].

From [M.sub.2] [omicron] [bar.n] = [n.sub.1], we get

[mathematical expression not reproducible].

Therefore,

[mathematical expression not reproducible].

By the definition of [[DELTA].sub.3], we have

[mathematical expression not reproducible],

and

[mathematical expression not reproducible],

by using the expansions of quadratic forms and the fact that [sup.t][b.sub.2]/2, [sup.t][b.sub.1]/2 are symmetric modulo 2. Finally, the verification of

[sup.t]m'[[DELTA]".sub.3] [equivalent to] [sup.t]m'[[DELTA]".sub.1] + [sup.t]m'[[DELTA]".sub.2] (mod 2)

is easy, which comes from the simple fact

([sup.t][d.sub.2] - [I.sub.g])([sup.t][d.sub.1] - [I.sub.g]) [equivalent to] 0 (mod 4).

This completes the proof of Theorem 3.1.

In [2], Igusa gave the generators [A.sub.ij], [B.sub.ij], [C.sub.ij] of [[GAMMA].sub.g](2), where

(a) [mathematical expression not reproducible], a is obtained by replacing (i, j)-coefficient in [I.sub.g] by 2;

(b) [mathematical expression not reproducible], a is obtained by replacing (i, i)-coefficient in [I.sub.g] by -1;

(c) [mathematical expression not reproducible], b is obtained by replacing (i, j)- and (j, i)-coefficients in 0 by 2;

(d) [mathematical expression not reproducible], b is obtained by replacing (i, i)-coefficient in 0 by 2;

(e) 1 [less than or equal to] i [less than or equal to] j [less than or equal to] g, [C.sub.ij] = [sup.t][B.sub.ij].

By noting the computation of A, which depends on m and M, we find [mathematical expression not reproducible] for M = [B.sub.ij] or [C.sub.ij], because in these cases, d = [I.sub.g], hence [sup.t]m'[DELTA]" = 0. If M = [A.sub.ij], it is easy to find that [sup.t]m'[DELTA]" = -[m'.sub.i][m".sub.j] = [m'.sub.i][m".sub.j] (mod 2). We can easily compute [[PHI].sub.m](M) for M = [A.sub.ij], [B.sub.ij] and [C.sub.ij]. Now the values of [[chi].sub.m](M) for the generators are

[mathematical expression not reproducible].

Using the definitions of [[PHI].sub.m](M) and [sup.t]m'[DELTA]", it is easy to prove [[chi].sub.m](M) = 1 for M in the Igusa modular group [[GAMMA].sub.g](4, 8). Hence, by the computations above, we get

Theorem 3.4. Write M in the form

[mathematical expression not reproducible]

with [p.sub.ij], [q.sub.ij], [r.sub.ij] [member of] Z and M' is in the commutator subgroup of [[GAMMA].sub.g](2), which is in [[GAMMA].sub.g](4, 8), then

[[chi].sub.m](M) = [(-1).sup.A]e(-[B/4])

with

[mathematical expression not reproducible].

4. Applications. If we define [psi]([tau]) = [[theta].sub.m]([tau])[[theta].sub.n]([tau]), then for M [member of] [[GAMMA].sub.g](2),

[mathematical expression not reproducible],

we find [[chi].sub.m](M)[[chi].sub.n](M) is exactly the character defined by Igusa in [2], hence our theorems can recover Igusa's Theorem 3 in [2]. Our character [[chi].sub.m](M) is more fundamental, moreover we can see the relations between [[chi].sub.m](M) and [[PHI].sub.m](M).

We can use our results to give a more transparent proof of the key part of Theorem 5 in Igusa's paper [2]. The key part of Theorem 5 in that paper is from the invariant condition that [[theta].sub.m](M[tau])/[[theta].sub.n](M[tau]) = [[theta].sub.m]([tau])/[[theta].sub.n]([tau]) holds for all even m, n to infer M is in [[GAMMA].sub.g](4, 8), where M is in [[GAMMA].sub.g](2). By the definition of [[chi].sub.m](M), this is equivalent to the congruence [[chi].sub.m](M) = [[chi].sub.n](M) holds for all even m, n, i.e.

[mathematical expression not reproducible]

is equal to

[mathematical expression not reproducible].

Let n' = n" = 0, we get for any even m,

[sup.t]m'([sup.t]d - [I.sub.g])m"/2 [equivalent to] 0 (mod 2)

and

[sup.t]m'[sup.t]bdm' + [sup.t]m"[sup.t]acm" - [2.sup.t][([a.sup.t]b).sub.0]dm' [equivalent to] 0 (mod 8).

The first congruence implies d [equivalent to] [I.sub.g] (mod 4), from [a.sup.t]d - [b.sup.t]c = [I.sub.g], and we get a [equivalent to] [I.sub.g] (mod 4). In the second congruence, let m' = 0, we get [sup.t]m"[sup.t]acm" [equivalent to] 0 (mod 8), this implies [sup.t]tac [equivalent to] 0 (mod 4), hence c [equivalent to] 0 (mod 4), and [([sup.t]ac).sub.0] [equivalent to] 0 (mod 8). If we write [sup.t]a = [I.sub.g] + 4[bar.a], then we have

[([sup.t]ac).sub.0] = [(([I.sub.g] + 4[bar.a])c).sub.0] [equivalent to] [(c).sub.0] [equivalent to] 0 (mod 8).

Let m" = 0, we get the congruence

[sup.t]m'[sup.t]bdm' - [2.sup.t][([a.sup.t]b).sub.0]dm' [equivalent to] 0 (mod 8),

which is equivalent to the congruence

(3) [sup.t]m'[sup.t]bdm' - [2.sup.t][(b).sub.0]m' [equivalent to] 0 (mod 8).

Write [sup.t]b = (2[bar.[b.sub.ij]]) and d = [I.sub.g] + 4[bar.d], then

[mathematical expression not reproducible]

by taking [sup.t]m' = ([m'.sub.1], 0, 0, ..., 0). Hence (3) implies [bar.[b.sub.11]] [equivalent to] 0 (mod 4). Similarly, we can prove [bar.[b.sub.ii]] [equivalent to] 0 (mod 4) holds for each 1 [less than or equal to] i [less than or equal to] g. Therefore, we have [(b).sub.0] [equivalent to] 0 (mod 8). Combining it with (3), we find the congruence [sup.t]m'[sup.t]bdm' [equivalent to] 0 (mod 8) holds for any even m', which implies [sup.t]bd [equivalent to] 0 (mod 4), hence b [equivalent to] 0 (mod 4). The analysis above shows that M is in [[GAMMA].sub.g](4) and [(b).sub.0] [equivalent to] [(c).sub.0] [equivalent to] 0 (mod 8). The observation of Igusa in [2, p. 222, line 4-line 6] shows M is in [[GAMMA].sub.g](4, 8).

doi: 10.3792/pjaa.93.77

Acknowledgments. The author would like to thank the referee for his/her helpful corrections and suggestions. This work was supported by the National Natural Science Foundation of China (No. 11101238).

References

[1] H. M. Farkas and I. Kra, Theta constants, Riemann surfaces and the modular group, Graduate Studies in Mathematics, 37, American Mathematical Society, Providence, RI, 2001.

[2] J. Igusa, On the graded ring of theta-constants, Amer. J. Math. 86 (1964), 219-246.

[3] K. Matsuda, Generalizations of the Farkas identity for modulus 4 and 7, Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 10, 129-132.

[4] K. Matsuda, The determinant expressions of some theta constant identities, Ramanujan J. 34 (2014), no. 3, 449-456.

[5] K. Matsuda, Analogues of Jacobi's derivative formula, Ramanujan J. 39 (2016), no. 1, 31-47.

[6] R. Salvati Manni, Thetanullwerte and stable modular forms, Amer. J. Math. 111 (1989), no. 3, 435-455.

By Xinhua XIONG

Department of Mathematics, College of Science, China Three Gorges University, Daxue Road No. 8, Yichang, Hubei Province, 443002, P. R. China

(Communicated by Shigefumi MORI, M.J.A., Sept. 12, 2017)
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