On the genera of X_{0}(N)

János A. Csirik

AT&T Shannon Lab, 180 Park Ave, Florham Park, NJ 07932–0971 E-mail: janos@research.att.com

and

Joseph L. Wetherell ^{*}

Department of Mathematics, University of Southern California, Los Angeles, CA

90089–1113 E-mail: jlwether@alum.mit.edu

and

Michael E. Zieve ^{∗}

Center for Communications Research, 29 Thanet Rd., Princeton, NJ 08540–3699 E-mail: zieve@idaccr.org

Proposed running head: Genera of X_{0}(N)

Address for correspondence: Michael Zieve Center for Communications Research 29 Thanet Rd. Princeton, NJ 08540–3699 e-mail: zieve@idaccr.org phone: 609-924-4600 fax: 609-924-3061

Let g_{0}(N) be the genus of the modular curve X_{0}(N). We record several properties of the sequence {g_{0}(N)}. Even though the average size of g_{0}(N) is (1.25/π^{2})N, a random positive integer has probability zero of being a value of g_{0}(N). We give bounds (in terms of x) on the size of the set of values of g_{0}(N) below some given x. Finally, we investigate the distribution of g_{0}(N) as N varies modulo a fixed prime ℓ.

1. INTRODUCTION

For each positive integer N, the modular curve X_{0}(N) parametrizes elliptic curves together with a cyclic subgroup of order N (for more details, see [10] and [13]). The genus g_{0}(N) of X_{0}(N) tends to infinity as the level N increases. We will examine the sequence {g_{0}(N)} more closely. Among other things, we will show that the average size of g_{0}(N) is 1.25N/π^{2}; however, the values of g_{0}(N) form a density-zero subset of the integers. These two properties imply that there is much collapsing under the map N ↦ g_{0}(N): for instance, there are integers whose preimage under this map is arbitrarily large.

The results of this note are:

1. Upper and lower bounds on g_{0}(N), including the asymptotic results lim inf g_{0}(N)/N = 1/12, and lim sup g_{0}(N)/(N log log N) = e^{γ}/(2π^{2}).

2. Average behavior: lim_{B→∞}(1/B) ∑_{N≤B} g_{0}(N)/N = 1.25/π^{2}.

3. Natural density: {g_{0}(N)} is a density zero subset of the integers. More specifically, let

S(x) = {n ∈ ℤ: n ≤ x and n = g_{0}(N) for some N}.

We then prove

Theorem 1.1 (Theorem One). For x ≥ 3,

\frac{x}{log x}e^{a(log log log x)^{2}} ≪ #S(x) ≪ \frac{x}{(log x)^{b}(log log x)^{c}} .

(For positive f(x), g(x), the notation f(x) ≪ g(x) is equivalent to f(x) ≤ O(g(x)).) Here a is any constant less than a_{0}, where the constants a_{0} = 0.8168146... and b = 0.2587966... and c = 0.2064969... are specified precisely in Section 6.

*Supported in part by NSF Mathematical Sciences Postdoctoral Research Fellowships.

GENERA OF X_{0}(N) 3

4. Non-uniformity of g_{0}(N) modulo a fixed prime p: for instance, g_{0}(N) is odd with probability 1, and (for a fixed odd prime p) the probability that g_{0}(N) ≡ 1 (mod p) is much less than 1/p. For more details, see Theorem Two in Section 7.

This note arose from a question about curves over finite fields. Typically one estimates the number of points on such a curve by means of the Weil (upper) bound. However, in various situations this bound can be improved, which leads to the converse question of producing curves with many points (of a given genus, over a given field) in order to see how far the Weil bound can be improved. It is known that, if the prime p does not divide N, then X_{0}(N) has many rational points over the finite field 𝔽_{p^{2}}. Covers of these curves have been used to show that there are curves with many points (over 𝔽_{p^{2}}) in every genus [1]. It is natural to ask whether covers are needed to prove the latter result, or whether the genera of X_{0}(N) already achieve all sufficiently large integer values. The present note grew from our proof that infinitely many positive integers do not occur in the sequence {g_{0}(N)}.

After circulating an earlier draft of this paper, the authors found out about [14], where S. Wong independently proved a subset of the results presented here.

2. THE GENUS OF X_{0}(N)

For a positive integer N, let g_{0}(N) denote the genus of X_{0}(N). By [13, Prop. 1.43], we have

g_{0}(N) = 1 + \frac{μ}{12} − \frac{ν_{2}}{4} − \frac{ν_{3}}{3} − \frac{ν_{∞}}{2}

where ν_{2} and ν_{3} are the numbers of solutions in ℤ/Nℤ of the equations x^{2} + 1 = 0 and x^{2} + x + 1 = 0, respectively; ν_{∞} = ∑_{d} φ(gcd(d, N/d)); and μ = N ∑_{a} 1/a. Here φ denotes Euler’s totient function, d ranges over the positive divisors of N, and a ranges over the squarefree positive divisors of N. Likewise, let p range over the prime divisors of N, and write N = ∏_{p} p^{r_{p}}. Then μ = ∏_{p}(p + 1)p^{r_{p}−1}. Also ν_{2} is 0 if 4 | N or if some p ≡ 3 (mod 4), and otherwise ν_{2} = 2^{s} where s is the number of p ≡ 1 (mod 4); similarly, ν_{3} is 0 if 9 | N or if some p ≡ 2 (mod 3), and otherwise ν_{3} = 2^{t} where t is the number of p ≡ 1 (mod 3). Finally, ν_{∞} = ∏_{p} θ(p, r_{p}), where we define θ(p, 2R + 1) = 2p^{R} and θ(p, 2R) = (p + 1)p^{R−1}.

3. BOUNDS ON g_{0}(N)

We now give general upper and lower bounds on g_{0}(N). For the lower bound we show that

g_{0}(N) ≥ (N − 5\sqrt{N} − 8)/12,

with equality if and only if N = p^{2} where p is a prime congruent to 1 (mod 12). For the upper bound we show the asymptotic result

lim sup_{N→∞} \frac{g_{0}(N)}{N log log N} = \frac{e^{γ}}{2π^{2}}

and the explicit bound

g_{0}(N) < N \frac{e^{γ}}{2π^{2}} (log log N + 2/ log log N) for N > 2,

where γ = 0.5772... is Euler’s constant.

We start by proving the lower bound g_{0}(N) ≥ (N − 5\sqrt{N} − 8)/12. If N is a prime power (or N = 1), it is easy to prove this bound and to show that equality occurs precisely when N is the square of a prime congruent to 1 (mod 12). It is also easy to check the lower bound for N less than 2000, so we may assume in the sequel that N ≥ 2000.

Suppose that N has exactly two distinct prime factors p and q. Then μ − N > N/p + N/q > 2\sqrt{N}. Since ν_{∞} ≤ μ/\sqrt{N} and ν_{2}, ν_{3} ≤ 4, we have g_{0}(N) ≥ 1 + μ/12 − 4/4 − 4/3 − μ/(2\sqrt{N}). Thus,

12g_{0}(N) ≥ μ(1 − 6/\sqrt{N}) − 16
> (N + 2\sqrt{N})(1 − 6/\sqrt{N}) − 16
> N − 5\sqrt{N} − 8.

Finally, consider the case where N has at least 3 distinct prime factors. In this case we observe that μ − N > 3N^{2/3} and ν_{2}, ν_{3} ≤ ν_{∞}. It follows that g_{0}(N) ≥ 1 + μ/12 − (1/4 + 1/3 + 1/2)ν_{∞}; hence,

12g_{0}(N) ≥ 12 + μ − 13μ/\sqrt{N}
> (N − 5\sqrt{N} − 8) + (μ − N)(1 − 13/\sqrt{N}) − 8\sqrt{N} + 20
> (N − 5\sqrt{N} − 8) + 3N^{4/6} − 39N^{1/6} − 8N^{3/6} + 20.

The largest real root of 3x^{4} − 8x^{3} − 39x + 20 is 3.5495... < 2000^{1/6}. The lower bound on g_{0}(N) follows.

We now prove the limsup result. We have already noted that ν_{2}, ν_{3}, and ν_{∞} are each bounded by μ/\sqrt{N}; thus, it suffices to show that

lim sup_{N→∞} \frac{μ}{N log log N} = \frac{6}{π^{2}} e^{γ}.

For any x ≥ 2 let N_{x} = ∏_{p≤x} p. Among all N with N_{x} ≤ N < N_{x+1} (for fixed x), the case N = N_{x} maximizes μ/N and minimizes log log N. Thus, we need only consider values N = N_{x}. For these N, Mertens showed in 1874 that μ/N is asymptotic to (log x)e^{γ}6/π^{2} [6, p. 429]; the limsup result follows, since log log N_{x} is asymptotic to log x [6, p. 341].

The upper bound on g_{0}(N) is easily verified for N < 210, so we now assume N ≥ 210. Note that g_{0}(N) ≤ μ/12, so our upper bound follows from the inequality μ/N < (log log N + 2/ log log N)e^{γ}/(2π^{2}). As above, it suffices to prove this inequality when N = N_{x} (since N ≥ 210 = N_{7}). But in this case the inequality follows easily from results in [11].

4. DATA FOR SMALL N

We now determine the first few positive integers n which do not occur as g_{0}(N) for any N. The lower bound of the previous section implies that, if n = g_{0}(N), then N < 12n + 18\sqrt{n} + 40. So, to determine whether n occurs as g_{0}(N), we just check all levels N up to this bound. Doing this by computer, we find the first few missed values (i.e., positive integers n not of the form g_{0}(N)): 150, 180, 210, 286, 304, 312, ... Note that all of these are even; in fact, the first several thousand missed values are even. It is easy to describe all positive integers N for which g_{0}(N) is even. They are given in the following list, where p denotes a prime and r denotes a positive integer:

1. N = 1, 2, 3, 4, 8 or 16;

2. N = p^{r} where p ≡ 5 (mod 8);

3. N = p^{r} where p ≡ 7 (mod 8) and r odd;

4. N = p^{r} where p ≡ 3 (mod 8) and r even;

5. N = 2p^{r} where p ≡ ±3 (mod 8);

6. N = 4p^{r} where p ≡ 3 (mod 4) and r odd. It follows that, if N is a randomly chosen positive integer, then g_{0}(N) is odd with probability 1. This shows that a randomly chosen even positive integer has probability zero of occurring as a value of g_{0}(N).

Looking further in the list of integers not of the form g_{0}(N), we do eventually find some odd values, the first one occuring at the 3885th position. There are four such up to 10^{5} (out of 9035 total missed values), namely 49267, 74135, 94091, 96463. In Section 6 we will show that the paucity of odd missed values is not a general phenomenon, but instead an accident caused by the fact that ‘small’ numbers do not have enough prime factors. In particular, we will see that there are infinitely many positive odd integers not of the form g_{0}(N), and in fact the set of integers of the form g_{0}(N) has density zero in the set of all nonnegative integers.

5. AVERAGE SIZE OF g_{0}(N)

We have shown that g_{0}(N) is sometimes as small as N/12, and sometimes as large as cN log log N. We now determine the average behavior. More precisely, we show that

\frac{1}{B} ∑_{N=1}^{B} g_{0}(N) = \frac{5}{8π^{2}} B + o(B).

By Abel’s lemma [12, VI. §2.], this result is equivalent to the following (for the forward or backward implication, set (a_{n}, b_{n}) to (g_{0}(n), 1/n) or (g_{0}(n)/n, 1), respectively):

lim_{B→∞} \frac{1}{B} ∑_{N=1}^{B} \frac{g_{0}(N)}{N} = \frac{5}{4π^{2}} = 0.12665...  (1)

In Section 3, we showed that g_{0}(N)/N = (1/12) ∑_{a|N}(1/a) + o(1), and we can ignore the error term since it contributes nothing to the limit. The following computation proves (1) (for notational simplicity, we let the sums involving a run over square-free integers only):

lim_{B→∞}(1/B) ∑_{N=1}^{B} ∑_{a|N} 1/a = lim_{B→∞}(1/B) ∑_{a≤B}(B/a)(1/a)

= ∑_{a<∞} 1/a^{2} = ζ(2)/ζ(4) = 15/π^{2}.

6. DENSITY OF {g_{0}(N)}

In this section we consider the (natural) density of the set {g_{0}(N)} as a subset of the non-negative integers. We have already seen that g_{0}(N) is almost never even, so this density (if it exists) is at most 1/2.

Here is a quick proof that lim_{x→∞}(#S(x))/x = 0, (with S(x) as defined in the Introduction). Let N be a positive integer and suppose that N is divisible by at least s > 2 distinct odd primes. The formulas in Section 2 imply that 2^{s−1} divides ν_{3} and 2^{s} divides each of μ, ν_{2}, and ν_{∞}. It follows that g_{0}(N) ≡ 1 (mod 2^{s−2}).

We now show that #S(x) ≤ x/2^{d} +o_{d}(x) for each positive integer d; this implies that #S(x) = o(x), as desired. Fix a positive integer d. Clearly the number of n ∈ S(x) with n ≡ 1 (mod 2^{d}) is less than x/2^{d} + 1. It remains to show that the number of n ∈ S(x) with n ≢ 1 (mod 2^{d}) is o(x). Each such n has the form g_{0}(N) where N has at most d + 1 distinct odd prime divisors and N < 12x + 18\sqrt{x} + 40. And the number of such N (hence also the number of such n) is well-known to be o(x) [6, p. 356].

Now we can prove:

Theorem 6.1 (Theorem One). For x ≥ 3,

\frac{x}{log x}e^{a(log log log x)^{2}} ≪ #S(x) ≪ \frac{x}{(log x)^{b}(log log x)^{c}} .

(For positive f(x), g(x), the notation f(x) ≪ g(x) is equivalent to f(x) ≤ O(g(x)).) Here a is any constant less than a_{0}, where the constants a_{0} = 0.8168146... and b = 0.2587966... and c = 0.2064969... are defined as follows. Let B be the unique root of 1/B + log B = 1 + log 2 in the interval (0, 1), let A be the unique root of ∑_{n=1}^{∞} A^{n}((n+1) log(n+1)−n log n−1) = 1 in the interval (0, 1), and put a_{0} = −1/(2 log A) and b = B log 2 and c = (B log 2)/(2 − 2B).

Proof of Theorem One, lower bound. The lower bound is proved by considering numbers N which are products of a fixed number k of distinct primes, the least of which is 11; for such N we have g_{0}(N) = 1 − 2^{k−1} + (1/12) ∏_{p|N}(p + 1). Each of these g_{0}(N) are distinct, and thus we obtain the lower bound by pluggin in the results of [7].

Proof of Theorem One, upper bound. The upper bound is proved by optimizing the choice of d in our density-zero proof, after one has modified that proof by replacing the o(x) bound from [6, p. 356] by the more precise bound from [5].

Let W_{j}(x) denote the number of positive integers less than x which have exactly j distinct prime factors. Lemma B of Hardy–Ramanujan is: there are positive constants G and H such that for all x > 2 and all positive j,

W_{j}(x) ≤ Gx(H + log log x)^{j−1}/[(log x)(j − 1)!]

For fixed x and d, the number of elements of S(x) congruent to 1 mod 2^{d−1} is less than 1 + x/2^{d−1}, and any other element of S(x) must be g_{0}(N) where N ≤ 12x+18\sqrt{x}+40 and N has at most d+1 distinct prime factors; thus, #S(x) ≤ x/2^{d−1} + ∑_{j<d+2} W_{j}(12x + 18\sqrt{x} + 40).

Put y = 12x+18\sqrt{x}+40. Let D = B(H+log log y)+B/(2−2B) log log log y, and let d be the least integer >= D, where B = 0.373364... satisfies 1/B + log B = 1 + log 2. Throughout this argument, we assume y is larger than any convenient absolute constant – so to start with, y is big enough so that log log log y is defined and positive. Also c′, c′′, ... denote positive absolute constants.

We give a preliminary upper bound on ∑_{j<d+2} W_{j}(y): for 1 ≤ j ≤ d+2, we have j ≤ (H + log log y)/2 (for large y), so

Gy(H + log log y)^{j−1}/((log y)(j − 1)!) ≤ (1/2)Gy(H + log log y)^{j}/((log y)j!)
≤ 1/2^{d+1−j}Gy(H + log log y)^{d}/((log y)d!)

Applying Hardy–Ramanujan, we get ∑_{j<d+2} W_{j}(y) < 2Gy(H+log log y)^{d}/((log y)d!).

Next we bound this last expression:

∑_{j<d+2} W_{j}(y) < 2G(y/ log y)(H + log log y)^{d}/((d/e)^{d}\sqrt{2πd})
= c′(y/ log y)(e/d ∗ (H + log log y))^{d}/\sqrt{d}

Note that (e/d)(H + log log y) ≤ e/B/(1 + (log log log y)/(2 − 2B)/(H + log log y)) is bounded (e.g., by 2e/B) when y is sufficiently large; thus, replacing the exponent d by D only increases our bound by a constant factor:

((e/d)(H + log log y))^{d} ≤ c′′(e/d(H + log log y))^{D} ≤ c′′(e/B)^{D}(1 + (log log log y)/(2 − 2 ∗ B)/(H + log log y))^{(} − D)
= c′′(e/B)^{D} exp(−D ∗ log(1 + (log log log y)/(2 − 2 ∗ B)/(H + log log y)))
= c′′(e/B)^{D} exp(−B/(2 − 2B)(log log log y) + o(1))
= c′′ exp((B − B log B)(H + log log y) − (B log B)/(2 − 2B)(log log log y) + o(1))
≤ c′′′ exp((B − B log B)(log log y) − (B log B)/(2 − 2B)(log log log y))
= c′′′(log y)^{B−B log B}(log log y)^{(−B log B)/(2−2B)}.

Also we have 1/sqrtd ≤ 1/sqrtD < (1/sqrtB)(log log y)^{(} − 1/2). Putting these bounds together gives

∑_{j<d+2} W_{j}(y) < c′′′′y(log y)^{−1+B−B log B}(log log y)^{−1/2−B(log B)/(2−2B)}
= c′′′′y(log y)^{−B log 2}(log log y)^{(−B log 2)/(2−2B)}.

Next we bound x/2^{d−1}.

x/2^{d−1} < c′′′′′x/2^{D}
= c′′′′′x/ exp(D log 2)
< c′′′′′x/ exp((B log 2)(log log y) + (B log 2)/(2 − 2B)(log log log y))
= c′′′′′x(log y)^{−B log 2}(log log y)^{(−B log 2)/(2−2B)}

Combining our last two bounds gives #S(x) < c′′′′′′y log y)^{−B log 2}(log log y)^{(−B log 2)/(2−2B)}. Finally, y/x is bounded, so we can replace y by x in the above (as long as

we increase the constant by the appropriate factor). This gives the desired bound.

Remark. Similar bounds have been proved for the number #V (x) of distinct values of Euler’s φ-function not exceeding x. In this setting, our upper bound is essentially an optimization of the paper [9]. In [7], upper and lower bounds are proved for #V (x), both of which have the same shape as the above lower bound (but the upper bound is for any b > b_{0}). The precise order of magnitude of #V (x) is determined in the excellent paper [2], which (together with [3]) also contains several other interesting results whose analogues would be interesting to study in our situation. However, the multiplicativity of φ(n) plays a crucial role in all proofs giving better upper bounds for #V (x) than our upper bound for #S(x) above. Since g_{0}(N) (and g_{0}(N) − 1) is not multiplicative, it seems that these methods do not apply to #S(x). We do not know which of our upper and lower bounds is closer to the truth.

7. DISTRIBUTION OF g_{0}(N) MODULO PRIMES

We now study the distribution of g_{0}(N) modulo a fixed prime ℓ as N varies. We will see that some residue classes occur more often than others. Let ℓ be a fixed prime. We may restrict to levels N having a prime factor p congruent to −1 modulo 12ℓ, since the set of such N’s has density 1 in the set of positive integers. Our assumption on N forces ν_{2} = ν_{3} = 0 and 12ℓ | μ, so

g_{0}(N) ≡ 1 − ν_{∞}/2 (mod ℓ).

First consider the case ℓ = 2: for our restricted class of N’s, we have g_{0}(N) ≡ 1 (mod 2) unless N is either p^{2r} or 4p^{2r}, so certainly g_{0}(N) is odd with probability 1 (as was observed in Section 4).

We have seen that the sequence of residues mod 2 of g_{0}(N) is biased. We now show a similar result modulo other primes ℓ.

Theorem 7.1. Theorem 2 Let ℓ > 2 be a prime. Then the density of N such that g_{0}(N) ≡ 1 (mod ℓ) is

p_{ℓ} = (1 − 1/ℓ^{3}) ∏(1 − 1/(s^{2} + s)).

Proof.. As above, we may assume N has a prime factor p with 12ℓ | (p + 1); then ℓ | (g_{0}(N) − 1) is equivalent to ℓ | ν_{∞}, and this occurs precisely when either ℓ^{3} | N or some prime congruent to −1 mod ℓ divides N with (positive) even multiplicity. By a standard argument (similar to [8, Thm. 2.18]), the probability that N does not satisfy either of these conditions is

(1 − 1/ℓ^{3}) ∏_{s≡−1 (mod ℓ) s prime}(1 − 1/s^{2} + 1/s^{3} − 1/s^{4} + 1/s^{5} − …),

which equals p_{ℓ}.

The following table gives upper bounds for the probability P (ℓ) that g_{0}(N) ≡ 1 (mod ℓ):

\begin{matrix}ℓ & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 \\ P (ℓ) < & 1/4 & 1/78 & 1/105 & 1/653 & 1/1542 & 1/1793 & 1/978 & 1/5821 \end{matrix}

In these first few cases, we see that the probability is much less than 1/ℓ. More generally, it is easy to see that for every ℓ we have P (ℓ) < 3/ℓ^{2}.

Also note that other cosets mod ℓ will have special behavior as well. If N is squarefree, ν_{∞} is a power of 2, so that g_{0}(N) is congruent modulo ℓ to a number of the form 1 − 2^{k}. Recall that N is squarefree with probability 6/π^{2} = 0.6079... [6, p. 269]. Thus, if 2 is not a primitive root mod ℓ, then the residue classes (mod ℓ) of the integers 1−2^{k} will occur more frequently than the other residue classes. For example, for ℓ = 7, the classes 0, 4, 6 occur much more frequently than the classes 2, 3, 5.

Using the fact that the number of prime factors of square-free integers are distributed uniformly modulo ℓ, we can go further.

Claim 7.1. Let λ ∈ (ℤ/ℓℤ)^{×}. The density of N such that

g_{0}(N) ≡ 1 − λ (mod ℓ)

depenends only on the orbit of λ under the multiplicative action of 2.

As we have seen above, the orbit with 1 in it is always the one corresponding to the highest density.

The above Claim, combined with Theorem Two, immediately yields

Corollary 7.1. Let ℓ be a prime such that 2 is a primitive root modulo ℓ (and thus the action of 2 on (ℤ/ℓℤ)^{×} is transitive). Then for all 1 ≠ μ ∈ ℤ/ℓℤ, the density of N such that

g_{0}(N) ≡ μ (mod ℓ)

is (1 − p_{ℓ})/(ℓ − 1).

The proof of the claim, which will be given elsewhere, proceeds by writing N = ab, where a is powerful, b is square-free, and (a, b) = 1. Noting that ν_{∞}(N) = ν_{∞}(a)ν_{∞}(b), and fixing a and letting b vary, we obtain the proof.

ACKNOWLEDGMENTS

The first author thanks H. Zhu for bringing this topic to his attention. The third author thanks A. Granville, C. Pomerance, and J. Vanderkam for helpful correspondence. The authors used the computer package GP/PARI for various computations related to this note. The authors thank the referee for useful suggestions.

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