C. R. Math. Rep. Acad. Sci. Canada Vol. 41 (1) 2019, pp. 1–6

Transcendence of Zeros of Automorphic
Forms for Cuspidal Triangle Groups
Paula Tretkoff
Presented by Pierre Milman, FRSC
Abstract. We extend some results of [4] on elliptic modular forms.
We take any Fuchsian triangle group with a cusp and look at power series
expansions in a natural parameter around that cusp. Consider the automorphic forms for such a triangle group whose power series expansions in
the natural parameter have algebraic coefficients. We show that the zeros ofsuch forms are either transcendental, or are “CM.” By “CM,” we mean theycorrespond to abelian varieties with complex multiplication. This result isthe first of its kind in the case of non-arithmetic groups.
Résumé. Nous étendons certains résultats de [4] sur les formes modulaires elliptiques. Nous prenons un groupe fuchsien triangulaire quelconqueavec une pointe et examinons les développements en série de puissance dans
un paramètre naturel autour de cette pointe. Considérons les formes automorphes pour un tel groupe triangulaire dont les développements en série depuissance dans le paramètre naturel ont des coefficients algébriques. Nous
montrons que les zéros de telles formes sont soient transcendants soient“CM”. Par “CM,” nous voulons dire qu’ils correspondent à des variétés
abéliennes à multiplication complexe. Ce résultat est le premier du genre
au cas des groupes non-arithmétiques.

A point τ in the upper half plane H such that [Q(τ ) : Q] = 2 is called
an imaginary quadratic point of H. It is also called a complex multiplication(CM) point since there are endomorphisms of the lattice Lτ = Z + τ Z given
by multiplications by complex numbers that are not real. The fractional linear
action of PSL(2, Z) on H replaces Lτ by a lattice λLτ , for a suitable complex
number λ 6= 0, which clearly also has complex multiplications.
Let p, q be integers or infinity with 2 ≤ p ≤ q ≤ ∞ and p−1 +q −1 < 1. A resultof Takeuchi [10] says that there are exactly nine signatures (p, q, ∞) such that any(Fuchsian) triangle group with that signature is arithmetic. A suitable conjugate
in PSL(2, R) of any such arithmetic group is commensurable with PSL(2, Z),
which has signature (2, 3, ∞). There are, clearly, infinitely many signatures(p, q, ∞) such that triangle groups with that signature are non-arithmetic.
Let f be a non-zero modular form for PSL(2, Z), with Fourier coefficientsalgebraic numbers. By [4], Theorem 1, any zero of f is either transcendental
Received by the editors on May 21, 2018; revised September 18, 2018.
The maiden name of the author is Paula B. Cohen. She authored [1] and coauthored [2].
AMS Subject Classification: Primary: 11J81; secondary: 11J91,20H10.
Keywords: Transcendence, automorphic forms, modular embedding, non-arithmetic group.
Royal Society of Canada 2019.

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Paula Tretkoff

or imaginary quadratic. We extend this result in §3, Theorem 3, to zeros of
non-zero automorphic forms for any Fuchsian triangle group with a cusp. In
the non-arithmetic case, we need to replace the usual Fourier expansion by an
expansion in another natural parameter at the cusp at i∞. We also need asuitable generalization of the notion of “CM point,” see §2. Overall, the modular
embedding techniques of [2] are what allow us to relax the assumption that thetriangle group is arithmetic. Our result on zeros of automorphic forms seems tobe the first of its nature for non-arithmetic groups.
1. Automorphic Forms and Hauptmodulen We begin with some remarks, some of which are classical facts, whereas others have their origin in [3]and the older reference [11]. We will not directly quote [11], but refer the reader
to [3] for appropriate acknowledgements. The Fuchsian groups considered fromnow on are triangle groups with cusps. Assume that p, q are either integers
or infinity, with 2 ≤ p ≤ q and p1 + 1q < 1. A (Fuchsian) triangle group withsignature (p, q, ∞) is by definition a subgroup of PSL(2, R) with presentation interms of generators and relations given by:
hM1 , M2 , M3 : M1p = M2q = M1 M2 M3 = 1i.
Let H be the upper half plane, that is the set of complex numbers τ withpositive imaginary part. A Fuchsian triangle group acts on H by fractional linear
transformations. The presentation determines the group up to conjugacy, andsuch a group has a fundamental region whose closure consists of two contiguous
1= 0). We
hyperbolic triangles with vertex angles πp , πq , 0 (here, we take ∞fix the triangle group by specifying that its fundamental region must containthe triangle with vertices ζ1 = − exp(−πi/p), ζ2 = exp(πi/q), ζ3 = i∞ (again1∞ = 0). Let Γ = Γ(p, q, ∞) be this triangle group with ζi the fixed point of Mi ,i = 1, 2, 3. The transformation M3 is the translation τ 7→ τ + h3 , whereh3 = 2 cos(π/p) + 2 cos(π/q).
The number h3 is called the cusp width (of the cusp ζ3 ).
Let f = f (τ ) be a meromorphic function on H, and let k be a positive
even


a b
integer. For γ in PSL(2, R) = SL(2, R)/{±Id2 }, with representative
inc dSL(2, R), define(1)(f |k γ)(τ ) := (cτ + d)−k faτ + bcτ + d,for all τ ∈ H.

A holomorphic automorphic form of weight k > 0 with respect to Γ(p, q, ∞) is
defined as follows. It is a function f on H such that(2)

(f |k γ)(τ ) = f (τ ),for all τ ∈ H

and all γ ∈ Γ(p, q, ∞),Transcendence of Zeros of Automorphic Forms

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which is holomorphic on H and at all the cusps of Γ(p, q, ∞). See [3], §2.1, forthe definition of meromorphic and of holomorphic at a cusp. As f is invariant
under the transformation τ 7→ τ + h3 , τ ∈ H, it has a Fourier expansion of the
form
(3)

f (τ ) =X

an exp(2πinτ /h3 ).

n∈Z

As it is holomorphic at ζ3 = i∞, we have an = 0 for all negative integers n. If
we only require f to be meromorphic at ζ3 = i∞, we allow it to have finitely
many non-zero Fourier coefficients an with n negative.
Let f be holomorphic on H, meromorphic at the cusps of Γ(p, q, ∞), andinvariant under the action of Γ(p, q, ∞) by fractional linear transformations,so that k = 0 in (2). We call f an automorphic function. The automorphic
functions for Γ(p, q, ∞) form a field generated by any one such function, calleda Hauptmodul. A Hauptmodul for PSL(2, Z), which has signature (2, 3, ∞), is
given by the classical elliptic modular function j. The choice of Hauptmodul is
unique once we specify a suitable normalization. Following [3], we choose the
Hauptmodul JΓ , Γ = Γ(p, q, ∞), uniquely determined by the conditionsJΓ (ζ1 ) = 1,JΓ (ζ2 ) = 0,JΓ (ζ3 ) = ∞.

(For Γ(2, 3, ∞), this gives j/1728.) Wolfart [11] was the first to investigate
arithmetic properties of expansions around elliptic fixed points, and Fourier expansions about cusps, of automorphic forms and functions with respect to all
Fuchsian triangle groups. We focus on Fourier expansions around ζ3 = i∞.
Wolfart proved that certain transcendence properties of the Fourier coefficientsan in (3) reflect whether or not the group Γ(p, q, ∞) is arithmetic. Namely, thenumber an is algebraic if and only if this group is arithmetic. More precisely,
the number an is of the form bn αn , with bn algebraic and α transcendental, if it
is non-arithmetic, see [3]. Let α3 be given by the closed formula in [3], Theorem
1, see also [11]. Then α3 is algebraic if Γ(p, q, ∞) is arithmetic, and α3 /α isalgebraic if Γ(p, q, ∞) is non-arithmetic. Let qe3 = α3 exp(2πiτ /h3 ). Again by[3], Theorem 1, the Fourier expansion of JΓ about ζ3 = i∞ is of the form
qe3−1 +∞
X

ck qe3k ,k=0with the ck polynomials (with signature independent coefficients) in Q[ p1 , 1q ]. In
particular, the above Fourier expansion lies in qe3−1 + Q[[eq3 ]].
Let L be the least common multiple of p and q if both these integers are
finite. If p is finite and q = ∞, let L equal p, and if both p and q are infinite,
let L = 1. By [3], Theorem 2, there is a closed formula for a holomorphic
automorphic form ∆2L of weight 2L which is the analogue of the discriminant4

Paula Tretkoff

form of weight 12 = 2lcm(2, 3) for the classical modular group PSL(2, Z) of
signature (p, q, ∞) = (2, 3, ∞). By [3], Theorem 2, equation (2.17), we haveLdd L e−Ld L e−L∆2L (τ ) = (−1)L qe3(JΓ ) q(JΓ − 1) p
JΓ
de
q3

= qe3n∆ + O qe3n∆ +1 ,(4)

where the Fourier expansion on the right hand side of the above equation has
coefficients rational numbers and n∆ = L(1 − p1 − 1q ). The automorphic form
∆2L has no zeros in H nor at the Γ-orbits of the cusps not in the orbit Γ(i∞).
2. Modular Embedding To state our main results in §3, we need the analog, for arbitrary signature (p, q, ∞), of the classical notion of a complex multiplication (CM) point for the signature (2, 3, ∞). For this, and for the proofsof our results in §3, we use the modular embedding techniques of [2]. For thenine arithmetic triangle groups of signature (p, q, ∞), the CM points we define
shortly all correspond to abelian varieties that are isogeneous to powers of asingle elliptic curve with complex multiplication. This fact is not surprising, as
these arithmetic groups are all commensurable with a subgroup of a trianglegroup with signature (2, 3, ∞). They are called modular triangle groups in [3].
By the techniques of [2], for all Γ(p, q, ∞) there is a positive integer g, a totallyreal number field F of degree g over Q, and an order O in F with the following
property. There is a complex analytic embeddingϕ : H ,→ Hg
and a group embeddingι : Γ(p, q, ∞) ,→ ΓFsuch thatϕ(γ(τ )) = ι(γ)ϕ(τ ),for all τ ∈ H,γ ∈ Γ(p, q, ∞).

Here, the group ΓF is a suitable subgroup of PSL(2, F) commensurable with
PSL(2, O). We can, and do, assume that ϕ1 (τ ) = τ , for all τ ∈ H, where πi isthe projection onto the ith factor of Hg , and ϕi (τ ) := (πi ◦ ϕ)(τ ), i = 1, . . . , g.
We call ϕ a modular embedding, as the quotient space ΓF \Hg is the coarse moduli
space for an analytic family of complex abelian varieties of dimension g whose
endomorphism algebra contains a subfield isomorphic to F, see [8].
We call z = (z1 , . . . , zg ) ∈ Hg a CM point if the ΓF -orbit of z corresponds to an
isomorphism class of abelian varieties with CM. Recall that an isotypic abelianvariety B of dimension n has CM (complex multiplication) if and only if there isa number field L contained in its endomorphism algebra End0 (B) = End(B) ⊗ Q

Transcendence of Zeros of Automorphic Forms

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with [L : Q] = 2n. The field L is then a CM field, that is, a totally imaginary
quadratic extension of a totally real number field. We call τ ∈ H a CM point if
ϕ(τ ) is a CM point.
The analytic space ΓF \Hg is isomorphic to the complex points V (C) of aquasi-projective variety V defined over Q, called a Hilbert modular variety. We
assume that V is embedded into a suitable projective space. There is a ΓF invariant holomorphic map J : Hg → V (C) that takes values in V (Q) at allcomplex multiplication points of Hg .
From the joint results of [1] and [6] (see, for example Proposition 4, §4, of thelatter reference), we have the following criterion for a point of Hg to be CM.
Proposition 1. A point z = (z1 , . . . , zg ) ∈ Hg is a CM point if and only if wehave both zi ∈ Q for at least one i = 1, . . . , g and J(z) ∈ V (Q).
Applying, from [2], the Theorem in §2, the construction of §3, Part 3, (seealso Proposition 1, §4), we can choose projective coordinates for J(ϕ(τ )), τ ∈ H,
that are polynomials, not all constant and with algebraic coefficients, evaluatedat JΓ (τ ).
The discussion of this section yields the following characterization of CM
points τ ∈ H. It generalizes the well-known result of Th. Schneider [5] on thevalues of the elliptic modular function.
Proposition 2. The point τ ∈ H is a CM point if and only if we have bothτ ∈ Q and JΓ (τ ) ∈ Q.
Proof. Let τ ∈ H be a CM point. Then, by well-known facts from thetheory of complex multiplication, see [9], we have ϕi (τ ) ∈ Q, for all i = 1, . . . , g.
Therefore ϕ1 (τ ) = τ ∈ Q. We also have J(ϕ(τ )) ∈ V (Q). By the comments
preceding this proposition, we deduce that P (JΓ (τ )) ∈ Q for some non-constantpolynomial P (x) ∈ Q[x]. It follows that JΓ (τ ) ∈ Q.
Conversely, suppose τ = ϕ1 (τ ) ∈ Q and JΓ (τ ) ∈ Q. By the Theorem in [2],§2, we have J(ϕ(τ )) ∈ V (Q). From Proposition 1, we deduce that ϕ(τ ) is a CM
point. Therefore, by definition, the point τ is CM.

3. Statement and Proof of Main Results We can now state and prove
our main results. They generalize to arbitrary triangle groups with cusps theresults for Γ(2, 3, ∞) given by Theorem 1 and Theorem 2 of [4].
Theorem 3. Let 2 ≤ p ≤ q ≤ ∞ with p1 + 1q < 1. Let f be a non-zero
holomorphic automorphic form of weight k > 0 with respect to Γ(p, q, ∞) whose
Fourier expansion lies in the ring Q[[eq3 ]]. Then any zero of f is either a CM
point or is a transcendental number.
Proof. Let f be a non-zero holomorphic automorphic form of weight k > 0
with respect to Γ(p, q, ∞) whose Fourier expansion lies in the ring Q[[eq3 ]]. We
defined ∆2L in (4). Let g be the function defined byg(τ ) =

f 2L (τ ),∆k2L (τ )τ ∈ H.

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Paula Tretkoff

Then g is an automorphic function (weight 0) with no poles in H, nor at theΓ(p, q, ∞)-orbits of the cusps not in Γ(p, q, ∞)(i∞). The function g has a Fourier
expansion in powers of qe3 (where we allow negative integer powers) and withcoefficients in Q. Therefore g = P (JΓ (τ )) for some P (x) ∈ Q[x] which is not
identically zero. If α is a zero of f , we have g(α) = P (JΓ (α)) = 0, and it follows
that JΓ (α) is an algebraic number. From Proposition 2, it follows that α is either
a CM point or is a transcendental number, as required.

Following [4], we define two automorphic forms f1 , f2 with respect to thegroup Γ(p, q, ∞) to be equivalent, denoted f1 ∼ f2 , if there are natural numbers
k1 and k2 such that we have the equality of functions f k1 = λf k2 , for someλ ∈ Q, λ 6= 0. We have the following generalization of [4], Theorem 2.
Theorem 4. Let f be a non-zero holomorphic automorphic form of weightq3 ]].
k > 0 with respect to Γ(p, q, ∞) whose Fourier expansion lies in the ring Q[[eSuppose that f is not equivalent to ∆2L . If α ∈ H ∩ Q and f 2L (α)/∆k2L (α) ∈ Q,
then α is a CM point.
Proof. As in the proof of Theorem 3, we deduce that f 2L (α)/∆k2L (α) is a
non-constant polynomial in JΓ (α) with algebraic coefficients and therefore thatJΓ (α) is algebraic. By Proposition 2, we deduce that α is a CM point.

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Department of Mathematics, Texas A & M University, College Station, TX 77843-3368 USA;
CNRS, UMR 8524, Université de Lille 1, Cité Scientifique, 59655 Villeneuve d’Ascq, France
e-mail: paulatretkoff@tamu.edu