INTERNATIONAL JOURNAL OF GEOMETRYVol. 8 (2019), No. 1, 46-55

HYPERELLIPTIC CURVES AND ARITHMETIC FUNCTIONS

SIMON DAVIS

Abstract. The precise form of the correspondence between Dirichletseries, modular forms and Riemann surfaces is given. An upper boundfor the prime p following from maximum value of the number of pointsof an unramified algebraic curve of finite genus over Fq is derived. Therational characters of absolute invariants of arithmetic subgroups of themodular group are verified and their role in rational conformal fieldtheories is elucidated. The transcendental limit at infinite genus is ex-amined.

1. Introduction

The analytic description of Riemann surfaces begins with functions thatare invariant under the uniformizing groups, which are finite-index sub-groups of PSL(2;Z) generally. Modular invariant functions are known forelliptic curves, for example, and the integrality of the coefficient in theFourier series expansion of j(τ), which is a modular form of weight zero,and all modular functions can be written as rational functions of j(τ). Thecoefficients of the expansion of the j(t) also satisfy the conditions requiredfor a Dirichlet series which occurs in construction of meromorphic functionswhich have well-defined vanishing properties at infinity and functional rela-tion with a symmetry axis characteristic of L-functions.

Consider the arithmetic subgroups Γ0 of Γ1 = PSL(2;Z). On a Rie-mann surface with cusps, Σ0 = H/Γ0 + cusps, the class of invariant func-tions would have proper rational characters at the corners.————————————–

Keywords and phrases:Hyperelliptic Curves, Modular Groups, Arith-metic Subgroup

(2010)Mathematics Subject Classification: 11F41, 11G30, 20H05Received: 29.12.2018. In revised form: 20.03.2019. Accepted: 04.09.

2019

46

47

More generally, Σ = H/Γ + cusps, where Σ is a d-fold cover of Σ0,such that K(Σ) is an extension of K(Σ0) of degree d and relations existbetween different invariant functions. The congruence group Γ0(p) is thestability group of j(pω) in the modular group and it has index p + 1. Thegenus of the corresponding Riemann surface Σ0 can be found through theRiemann-Hurwitz formula [1]. The fundamental region of Γ0(p) may beviewed as p = 1 copies of F0 + i∞ produced by the substitutions ω → − 1

ωand ω → ω + q, 0 ≤ q ≤ p. Since the ramification is equal to twice thecombination (sheet number + genus− 1), the genus equals

g = −p+1

2× ramification index(1)

= −p+1

2

p+1

2(p− 1− n− n1) +

2

3(p− n− n2)

= −p+1

2

13

6p+

1

6− 1

2n1 −

2

3n2

=p+ 1

12− 1

4n1 −

1

3n2

The following values of (p;n, n2; g) are listed: (2; 1, 0; 0), (3; 0, 1; 0), (5; 2, 0; 0), (7; 0, 2; 0),

(11; 0, 0; 1), (13; 2, 2; 0). The surface H/Γ0(11) + cusps can be identifiedwith the elliptic curve y2 = x(x3−20x2+56x−44) and the class of invariantfunctions of Γ0(11) is rationally expressible in terms of x and y [1].

The correspondence between Dirichlet sums, the modular-invariant func-tions and the Riemann surfaces will be examined in the second section. Thezeta function ζ(s) is an example of a meromorphic function with functionalrelation having an axis of symmetry. On an algebraic curve C over a finitefield Fq, this symmetry follows from the location of the zeros of the zetafunction Z(C, t) on a circle of radius 1√

q . The Riemann-Hurwitz formula

relating the genus to the congruence index p of the modular subgroup yieldsan upper bound for p such that the standard limits on the number of rationalpoints on an unramified curve are preserved with a uniform angular distri-bution of zeros of Z(C, t). This result can be extended to ramified curves.Given the degree of the mappings between the classes,the extension of thecorrespondence to transformations of Dirichlet sums and infinite genus sur-faces is developed in the third and fourth sections. The rationality of thecharacters of the arithmetic groups uniformizing the surfaces are consideredwith respect to invariant functions. The effect of the infinite-genus limit onthe form of the invariant function then is related to the congruence subgroupproblem.

2. The Action of the Group G(Q/Q)

The following discussion is given in most standard treatises on the actionof discrete groups on modular functions [2]. Consider the action of the groupPSL(2;Z), τ → aτ+b

cτ+d , a, b, c, d ∈ Z, ad − bc = 1. The term 1(cτ+d)2m

in

48

the sum representing automorphic forms on the complex plane is invariantunder the action of the subgroup

(2) Γ0 =

(

1 b0 1

)∣

∣

∣

∣

b ∈ Z

because the elements c and d are left unchanged. Under the elements of

(3) Γ00(2) =

(

a bc d

) ∣

∣

∣

∣

b ≡ c ≡ 0 (mod 2)

,

(4) f(τ) =∑

Γ0

0(2)/Γ0

1

(cτ + d)2m

is transformed to

(5) f [γ]m(τ) = (cτ + d)−2mf

(

aτ + b

cτ + d

)

.

Suppose that ∆ is a normal subgroup of Γ00(2). A modular form can be

defined for this subgroup through

(6) f∆(τ) =∑

∆/Γ0

1

(cτ + d)2m.

Amongst the subgroups which admit these forms are

(7) Γ0(n) =

(

a bc d

) ∣

∣

∣

∣

c ≡ 0 (mod n)

and Γ0 ⊂ Γ0(n) through the specification of the values a = d = 1 and c = 0.The hyperelliptic curves have been derived through consideration of thesubgroups Γ0(p), where p is prime, whereas the only such class of subgroupsof Γ0(4) ≡ Γ0

0(2) is Γ0(n)|4|n. It may be recalled that ord(Γ1/Γ0(n)) =

µ0(n) = n∏

p|n

(

1 + 1p

)

and (Γ1 : Γ0(4)) = 6, the order of the symmetric

group S3.It may be established that Γ/∆ = G(K∆/C(λ)), where C(λ, µ) is an

extension of C(λ) defined by f(λ, µ) = 0 for some function having the re-quired periodicity properties, The Galois group G(Q/Q) acts on the set ofsubgroups of Γ0

0(2) since f σ generates another function field K∆σ ⊆ Γ00(2).

Then G∆ = σ ∈ G( ¯Q/Q) acts as an automorphism group of Γ00(2)/∆ [2].

It has been suggested that there is a one-to-one correspondence betweenmeromorphic functions, that satisfy a relation with a symmetry axis with aconstant real value and have a maximum of two symmetrically placed simplepoles, and modular forms transforming under PSL(2;Z). A 2-1 mappingbetween these spaces may be derived.

Theorem 2.1. There is a 2-1 correspondence between meromorphic func-tions in the complex plane, that are symmetric about a fixed axis and de-crease exponentially to zero at infinity, and modular forms transformingunder PSL(2;Z).

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Proof.The proof begins with the representation [2]

(8) Φ(s) =

(

2π

λ

)−s

Γ(s)ϕ(s) =∞∑

n=0

an

∫ ∞

0

(

2πn

λ

)−s

t−se−tdt,

where Φ(s) =(

2πλ

)−sΓ(s)ϕ(s), ϕ(s) =

∑∞n=1

anns , and an = O(nc), since

(9)∞∑

n=1

ane−2πny

λ =∑

n

1

2πi

∫

Re s=σ0

Γ(s)

(

2πny

λ

)−s

ds =1

2πi

∫

Re s=σ0

Φ(s)y−s ds

with σ0 > c+ 1 because Ress=−nΓ(s) =(−1)n

n! . Integrating over y(10)1

2πi

∫

Re s=σ0

Φ(s)

∫

γyt−1y−s dy ds =

1

2πi

∫

Re s=σ0

Φ(s)·2πiδ(t−s) ds = Φ(t)

with the contour γ given by a circle of radius ǫ around the origin and the line

from ǫ to ∞ in the limit ǫ → 0. Defining f(t) =∑∞

n=0 ane2πint

λ , it followsthat

Φ(s) =

∫ ∞

0ts−1(f(it)− a0) dt =

∫ ∞

1ts−1(f(it)− a0) dt+

∫ 1

0ts−1(f(it)− a0) dt

(11)

=

∫ ∞

1ts−1(f(it)− a0) dt+

∫ ∞

1t−s−1f

(

i

t

)

dt.

Suppose that f(t) a modular function of weight k satisfying

(12) f

(

−1

t

)

= C

(

t

i

)k

f(t).

Then

∫ ∞

1t−s−1f

(

i

t

)

dt =

∫ ∞

1Ctk−s−1f(it) dt

(13)

= C

∫ ∞

1tk−s−1(f(it)− a0) dt+ C

∫ ∞

1tk−s−1a0 dt

= C

∫ ∞

1tk−s−1(f(it)− a0) dt− C

a0k − s

if s > k + 1. Substituting this formula into the sum of the integrals,

(14) Φ(s)+a0s+C

a0k − s

=

∫ ∞

1[ts−1(f(it)− a0)+Ctk−s−1(f(it)− a0)] dt,

and interchanging s with k − s,

(15) Φ(k−s)+Ca0s+

a0k − s

=

∫ ∞

1[Cts−1(f(it)−a0)+tk−s−1(f(it)−a0)] dt.

Subtracting the two equations yields

(16) CΦ(k − s)− Φ(s) + (C2 − 1)a0s

=

∫ ∞

1(C2 − 1)ts−1(f(it)− a0) dt.

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The integral does not converge for Re s ≥ 0. However, if C2 = 1, it vanishesand

(17) Φ(s) = CΦ(k − s)

with C = 1 or C = −1.

The symmetry group of the modular form is generated by t → t + λ

and t → −1t . The matrix

(

1 λ0 1

)

belongs to Γ00(λ) =

(

a bc d

)

∈

PSL(2;Z)

∣

∣

∣

∣

b ≡ c ≡ 0 (mod λ)

. However, the matrix

(

0 −11 0

)

does not

belong to this group. Nevertheless, the second transformation also has the

form t → −λλt , which has the matrix representation

(

0 −λλ 0

)

, with the

determinant λ2.

Lemma 2.2. The correspondence between the hyperelliptic curves y2 =4x3 − g2x− g3, where g2 and g3 are Eisenstein series, and the elliptic mod-ular functions is 1:3 for λ(ω), invariant under Γ0

0(2), and 2:1 for J(ω),invariant under Γ1.

Proof.An elliptic modular function invariant under Γ0

0(2) can be constructed

from theWeierstrass function F (u,w1, w2) =1u2+

∑

06=Ω∈Zw1+Zw2

′(

1(u−Ω)2

− 1Ω2

)

,

which depends only on the lattice Zw1 + Zw2 and satisfies the equationg2 = 60

∑

(m,n) 6=(0,0)(mw1+nw2)−4 and g3 = 140

∑

(m,n) 6=(0,0)(mw1+nw2)−6.

Given the roots of 4x3−g2x−g3 = 0, e1 = F(

w1

2 , w1, w2

)

, e2 = F(

w2

2 , w1, w2

)

and e3 = F(

w1+w2

2 , w1, w2

)

, let λ(w1, w2) =e1−e3e2−e3

and ω = w2

w1. Then

(18) J(ω) =(λ2 − λ+ 1)3

λ2(λ− 1)2

is invariant under the permutations of e1, e2 and e3 or λ → λ, 1λ , 1 −

λ, 11−λ ,

λλ−1 ,

λ−1λ and J

(

aω+bcω+d

)

= J(ω) for

(

a bc d

)

∈ SL(2;Z). It follows

that λ would be invariant under Γ00(2) and λ(ω) = λ(ω + 2). The modular

functions with this periodicity relation differ from those that are single-valued on the fundamental domain of PSL(2;Z).

Two modular functions λ(ω) and J(ω) are derived, which are invariantunder Γ0

0(2) and Γ1 respectively. The Riemann surface is the hyperellipticcurve y2 = 4x3−g2x−g3. The involution e1 → −e1, e2 → −e2 and e3 → −e3leaves invariant λ(ω) and J(ω), although the branch points in the complexplane have changed. The permutation of e1, e2 and e3 affects λ(ω) and notJ(ω) and keeps invariant the hyperelliptic representation of the algebraiccurve. Therefore, the correspondence between the set of Riemann surfacesand the set of functions λ(ω) is 1 : 3 while the ratio for the set of functionsJ(ω) is 2 : 1.

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Theorem 2.3. There is an n : |If | correspondence between equivalenceclasses of Dirichlet series

∑

nanns and modular curves uniformized by a sub-

group of PSL(2;Z), where |If | is the order of a separate group of involutions

preserving the modular form f =∑

n ane2πint

λ , which reduces to a k : 1 ratiowith k|n if the index is even and divisible by |If |.

Proof.There will be a Dirichlet series for each modular form defined for a normal

subgroup of PSL(2;Z) and a correspondence between Riemann surfaces andDirichlet series through each subgroup, which may be a congruence subgroupΓ0(p) of SL(2,Z) or a noncongruence subgroup. By Lagrange’s theorem, theorder of a finitely generated group is divisible by the order of a subgroup.Involutions of order 2, together generating a group of order |If |, alteringthe Riemann surface and preseving the modular function, would changethe ratio to n : |If |. A recursion relation for the number of subgroupsof SL(2;Z) of index n has been given with the values for n = 1, ..., 9 [3].Since [SL(2;Z) : PSL(2;Z)] = 2, given that the projective transformationsfor I and −I are identical, the index in PSL(2;Z) would be half of theindex in SL(2;Z). When the modular function is invariant under a separateinvolution of order 2, the index of the invariance subgroup in PSL(2;Z)must be even and the index in SL(2;Z) is required to be a multiple of 4 forthe ratio to be k : 1. If |If | | n, the modular functions invariant under asubgroup of PSL(2;Z) will have a k : 1 correspondence with the equivalenceclass of modular functions and set of algebraic curves, where k divides theindex. The degree of the mapping from the Dirichlet series to the Riemannsurface similarly would equal k.

The zeta function for algebraic curves is given by Z(C, t) = exp(∑

k1kNkt

k)

= P (t)(1−t)(1−qt) , where Nk is the number of points in Fqk .

Lemma 2.4. The upper bound for the number of points of an unramifiedalgebraic curve over Fq uniformized by Γ0(p) with a uniform angular distri-

bution of zeros, with P (t) =∏2g

n=1(1 − αnt), |αn| =√q, occurring on the

circle |t| = 1√q , yields the following inequality

(19) p ≤ 12√2

πq

1

4

(

1 +1

32q1

2

+ ...

)−1

− 1.

Proof.

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When the angular distribution of the roots of P (t) is uniform, the in-equality |N1 − (q + 1)| ≤ ⌊2√q⌋g follows if

(20)π

g≥ 1

√2q

1

4

+1

32√2q

3

4

+ ... .

If an unramified Riemann surface is uniformized by the modular group Γ0(p),p = 12g − 1 by Eq.(1.1), an upper bound

(21) p ≤ 12√2

πq

1

4

(

1 +1

32q1

2

+ ...

)−1

− 1

is found. More generally, in a uniform angular distribution, the other rootscompensate for the values of cos θn with phases of lesser magnitude. How-ever, when each of the phases have magnitudes of the order of π

g , the in-

equality for N1 will not be valid for unramified surfaces uniformized by Γ0(p)where p exceeds this bound.

The limit p → ∞, for fixed n1 and n2, yields g → ∞. However, thecongruence condition c ≡ 0 (mod p) tends to c = 0. The subgroup thenwould be presented by the matrices

(22)

(

a b0 1

a

)

3. The Generation of New Surfaces

By the correspondence between modular curves and Dirichlet series inTheorem 2.3, it would follow that a transformation of a series represent-ing a modular form would generate a new surface. Suppose a surface isuniformized by a congruence group Γ0(p). Within the conjugacy class, themetric is related by a conformal transformation and represents the samepoint in moduli space. This correspondence is the composition of the Heckecorrespondence and the automorphic representation of the Galois groupsof the quotients of algebraic extensions of the field of meromorphic func-tions with algebraic singularities equivalent to the fundamental groups ofthe ramified coverings of the punctured Riemann surfaces [4]. This lattercorrespondence is a specific example of the the function field version of theLanglands correspondence. The number theoretical analogue is a general-ization of representation of the abelian extensions of the rational numbersand algebraic number fields.

The transformations of the Dirichlet L-series include the convolution oftwo series and the Rankin-Selberg transformation. It may be recalled thatthe latter convolution of two L-functions

∑∞n=1 af (n)n

−s and Lg(s) =∑∞

n=1 ag(n)n−s,

where af (n) =ag

nk−1

2

and ag(n) =ag

nk−1

2

with f(z) =∑∞

n=1 afe2πinz and

g(z) =∑∞

n=1 age2πinz are cusp forms of weight k and level N , is L(f ⊗ g, s)

= ζN (2s)∑∞

n=1af (n)ag(n)

ns [5]. For two eigenforms with respect to the familyof Hecke operators for Γ0(N), such that 〈f, g〉, L(f ⊗ g, 1) 6= 0 [6]. If thelevels are different, the L-function also may be defined [7] and the traceoperator of the theta series has been given for pairs of Hecke subgroups [8]and subgroups of the symplectic group of different levels [9].

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4. The Transcendental Limit of Representations of ModularGroups

The central charges 1 − 6(p−p′)2

pp′ of the minimal conformal models are

found to be constrained by the integrality of the ramification index [10].The three-point vertex represents the matrix elements of an operator fromthe space to an equivalent cohomological space if the ramification numberis integer.

The procedure of sewing thrice-punctured spheres, together with theproduct of the matrices of the vertex operators, can be used to generalthe conformal blocks to higher genus. The construction of conformal blocksis given as follows. A k-punctured surface of genus g, Σg,k is formed by thesewing of 2g − 2 + k thrice-punctured spheres. There are different sewingdiagrams related by transformations of Σg,k. The fusion of three-point oper-ators V ∗ and V ∗ ·Ψ∗ yields the conformal blocks of a k-point operator, withthe matrix between the k highest-weight vectors being a primary conformalblock [11]. Superconformal models may be formulated on a higher-genussurface with the sewing of super-Riemann surfaces.

The rational character of absolute invariants at cusps is related to theproperties of conformal field theories at the boundaries of moduli space. Itis known that there is a corresponding contribution to coefficients in theeffective action given by modular forms. The rationality of conformal fieldtheory and its central charge is consistent with the rational character of theabsolute invariants at the cusps of the Riemann surface.

The expression of the absolute invariant j as a transcendental functionof the invariants of Γ0(p) does not appear to be possible for finite p, Theexistence of a subgroup of the modular group SL(2;Z) or the symplecticgroup Sp(2g;Z) such that the quotient has infinite genus would appear tobe necessary for such a transcendental relation.

Any continuous space VQ based on the rational numbers can be com-pleted into a continuous space VR based on the real numbers. Therefore,it is not possible to separate the VQ from VR/VQ into two Hausdorff mani-folds. Therefore, all irrational conformal field theories that have characterswhich are the limit points of the sequences of characters of rational confor-mal field theories also must belong to the completion of the space of thesetheories, when the defining relations are given by continuous functions ofthe characters.

5. Conclusion

The hyperelliptic locus represents the set of curves which admit an alge-braic solution to the Schottky problem [12] through vanishing sets for thetafunctions and a solution to the congruence subgroup problem [13]. The setof hyperelliptic surfaces forms a 2g − 1-dimensional subspace of the 3g − 3-dimensional moduli space. At genus 3, M3 = M3\hyperelliptic locus ∪M3\locus of plane quartics with aminimum of one hyperflex and it has been conjectured that Mg is the

54

union of g − 1 open varieties [14]. The characterization of all surfaces ofgenus 4 through the vanishing of theta constants [15] [16] demonstrates,however, that there can exist solutions to the Schottky problem on othersubsets of moduli space. It remains to be established if the finite-index sub-groups of the mapping class groups and outer automorphism groups of thefundamental groups of the surfaces are represented faithfully as congruencesubgroups. The group SL(2,Z) is known not to have this property, althoughit may be demonstrated that all subgroups of finite index containing ±I areVeech groups [17].

The equivalence of the differential and algebraic constraints is not neces-sarily valid for other all other classes of surfaces in Mg, because the charac-ters of the arithmetic subgroups must be rational, which may be achievedfor the congruence subgroups of SL(2;Z). There are modular forms of non-congruence subgroups of SL(2;Z) with coefficients that have unboundeddenominators [18] [19], which would reflect the irrational nature of the char-acters.

By contrast with the absence of stable Schottky forms over the entiremoduli space Mg, the existence of stable equations for the hyperellipticlocus at genus g has been proven [20]. There exists a set of surfaces in thecomplement in Mg\Hg that would not have a stable Schottky form, andthe structure of mapping class group may be considered in connection withthe nonexistence of an isomorphism of a congruence subgroup of symplecticmodular group.

The absolute invariants of the surface, which must have the arithmeticgroup symmetry, do not have rational characters generally in the infinite-genus limit. It would follow that there are uniformizing Fuchsian groupsof this class of surfaces that could be finite-index subgroups without beingcongruence subgroups.

55

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Simon DavisResearch Foundation of Southern California,

8861 Villa La Jolla Drive #13595La Jolla, CA, 92039

Email: sbdavis@resfdnsca.org