# A Proof of the Weierstraß Gap Theorem not Using the Riemann–Roch Formula

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## Abstract

Usually, the Weierstraß gap theorem is derived as a straightforward corollary of the Riemann–Roch theorem. Our main objective in this article is to prove the Weierstraß gap theorem by following an alternative approach based on “first principles”, which does not use the Riemann–Roch formula. Having mostly applications in connection with modular functions in mind, we describe our approach for the case when the given compact Riemann surface is associated with the modular curve \(X_0(N)\).

## Keywords

Weierstraß gap theorem Modular functions## Mathematics Subject Classification

Primary 14H55 11F03 Secondary 11P83## 1 Main Objective

Various topical areas in the theory of partitions, such as congruences for partition numbers, are connected to modular functions for congruence subgroups of \({\mathrm {SL}}_2({\mathbb {Z}})\) as, for instance, \(\Gamma _0(N)\); see Sect. 15 for definitions. Such functions live on compact Riemann surfaces, for instance, on \(X_0(N)\) for \(\Gamma _0(N)\). Number theoretic aspects then relate to properties of certain subalgebras formed by these functions. In cases where the genus of such surfaces is zero like, for instance, for \(X_0(5)\) and \(X_0(7)\), these algebras essentially have a relatively simple structure. For positive genus *g*, for example, in the case of \(X_0(11)\), this changes. One explanation is this: when considering sets of meromorphic functions with poles only at one point *p*, the Weierstraß gap theorem says that one can obtain functions with all possible pole orders at *p* with exactly *g* exceptions.

### Theorem 1.1

*X*be a compact Riemann surface having genus \(g\ge 1\). Then, for each \(p\in X\), there are precisely

*g*integers \(n_j=n_j(p)\) with

*X*which is holomorphic on \(X\backslash \{ p\}\) and which has a pole of pole order \(n_j\) at

*p*.

We want to stress that “precisely” in the theorem means that for any positive integer *n* other than the *g* values \(n_j\), a meromorphic function with a pole of order *n* at *p* exists.

Usually, as in [6, III. 5.3], this theorem is derived as a straightforward corollary of the Riemann–Roch theorem. Our main objective in this article is to prove the Weierstraß gap theorem by following an alternative approach based on “first principles” which does not use the Riemann–Roch formula. Having mostly applications in connection with modular functions in mind, we describe our approach for the case when the given compact Riemann surface *X* is associated with \(X_0(N)\). Some ingredients of our setting are related to ideas from the celebrated paper [3] by Dedekind and Weber; see [2] for an English translation together with an excellent introduction by John Stillwell.

## 2 Introduction

*q*instead of \(q(\tau )\).

### Example 2.1

*q*-series (2.5) and (2.8) are the local

*q*-expansions of \(z_5^*\) at these cusps.

*j*is the modular invariant (the Klein

*j*function). The subset

^{1}

### Example 2.2

By definition (2.5) together with (2.6) and (2.9), \(z_5\in M^{!}(5)\), because \(z_5^*\) has its only pole of pole order 1 at \([0]_5\).

### Example 2.3

*j*function, and \(\alpha \), such that \(\mu _N^\alpha j \in M^\infty (N)\). Let \(\beta :={{\,\mathrm{ord}\,}}_{[\infty ]_N} \mu _N\), then \({{\,\mathrm{ord}\,}}_{[\infty ]_N} \mu _N^\alpha j = \alpha \beta -1.\) In particular

Since we will prove the gap theorem in the version of Theorem 12.2, where \(X=X_0(N)\), and with \(p=[\infty ]_N\), a key issue in our approach concerns the question of finding appropriate representations of \(M^\infty (N)\).

### Example 2.4

*x*with complex coefficients. One also has

*U*-operator:

Thus the minimal pole order of the functions which in the sense of (2.10) generate \(M^\infty (11)\) is 2, not 1. Indeed, the gap at 1 is predicted by the Weierstraß gap theorem, Theorem 1.1, owing to the fact that the compact Riemann surface \(X:=X_0(11)\) has genus 1. A formula for the genus of \(X_0(N)\), if \(N=\ell \) is a prime, for instance, can be found in [5, Exercises 3.1.4(e)]; the genus for general *N* is determined in [5, Sect. 3.9].

In Sect. 12, we prove Theorem 12.2, a version of the gap Theorem 1.1 for the case \(X=X_0(N)\) and with the only pole put at \(\infty \), utilizing only first principles and avoiding the use of the Riemann–Roch formula. In particular, we avoid the use of any differentials. In addition, our approach provides new algebraic insight by consisting in a combination of module presentations of modular function algebras, integral bases, Puiseux series, and discriminants. For example, using our approach to prove the bound \(\le 2g-1\) stated in the Weierstraß gap theorem is reduced to an elementary combinatorial argument, see Sect. 12. Another by-product of our proof of the Weierstraß gap Theorem 12.2 is a natural explanation of the genus \(g=0\) case as a consequence of the reduction to an integral basis.

In view of various constructive aspects involved, we are planning to exploit the algorithmic content of our approach for computer algebra applications, for instance, for the effective computation of suitable module bases for modular functions. As already mentioned, some ideas we used trace back to the celebrated work [3] by Dedekind and Weber; see [2] for an English translation together with an excellent introduction by John Stillwell.

Finally, we remark that the history of Weierstraß’s gap theorem and related topics such as Weierstraß points somehow presents a challenge. The historical account [4] by Andrea Del Centina describes the scientific evolution of the gap theorem up to the 1970s. Concerning its beginnings Centina says, “The history of Weierstraß points is not marked by a precise starting date because it is not clear when Weierstraß stated and proved his *Lückensatz* (or “gap” theorem), but one can argue that probably it was in the early 1860s.”

The rest of our article is structured as follows. In Sect. 3, we introduce order-complete bases of modules over a polynomial ring \(\mathbb {C}[t]\) to describe modular function algebras. In Sect. 4, we describe how such bases can be stepwise modified to obtain an integral basis; i.e., an order-complete basis for the full algebra \(M^\infty (N)\). Under particular circumstances, one can keep track of the total number of such steps, which then gives a proof of the Weierstraß gap Theorem 12.2. To do this bookkeeping, one can use “order-reduction” polynomials discussed in Sect. 5. In Sect. 6, we explain how to obtain order-reduction polynomials computationally; Sect. 7 deals with important special cases. In Sects. 8 and 9, we derive important ingredients of our proof of Theorem 12.2; for example, a factorization property of the discriminant polynomial in Proposition 9.3. In Sects. and 11, we relate discriminant polynomials to order-reduction polynomials associated with integral bases. In Sect. 12, we use these results to prove the Weierstraß gap theorem in the version of Theorem 12.2. To prove the bound \(2g-1\) for the size of the maximal gap, our approach allows a purely combinatorial argument (a gap property of monoids) which we describe in Sect. 13. At various places, we require functions to have the separation property, as defined in Sect. 9. In Sect. 14, we prove the existence of such functions by giving an explicit construction.

The first Appendix Sect. 15 gives a short account on basic modular function facts needed; the second Appendix Sect. 16 recollects some fundamental facts about meromorphic functions on Riemann surfaces.

## 3 Modular Function Algebras as \(\mathbb {C}[t]\)-Modules

*f*has a pole, then the pole order is defined as the negative order at this point, that is

### Definition 3.1

### Example 3.2

The tuple \((1,F_6, F_2, F_3, F_4)\) with \(F_j\in M^\infty (11)\) as in (2.10) is order-complete.

### Example 3.3

*f*, has a representation as a \(\mathbb {C}[1/z_{11}]\)-module as follows:

*t*. This motivates the following definition.

### Definition 3.4

*M*be the \(\mathbb {C}[t]\)-module generated by an order-complete tuple in \(M^\infty (N)\), that is,

*M*over \(\mathbb {C}[t]\). Slightly more generally, any tuple \((1,\beta _1,\ldots , \beta _{n-1})\) which is a reordering, in the sense of (3.1), of an order-complete basis \((1,b_1,\ldots , b_{n-1})\) for

*M*is also called an order-complete basis for

*M*.

### Proposition 3.5

### Proof

## 4 Integral Bases

In Example 3.3, we saw that \((1,f,\ldots ,f^4)\) is an order-complete basis of \(\mathbb {C}[1/z_{11},f]\) which is a proper subalgebra of \(M^\infty (11)\).^{2} In this section, we shall see how such an order-complete basis can be step-wise modified to obtain an order-complete basis for the full algebra \(M^\infty (11)\).

### Definition 4.1

The motivation for this terminology comes from

### Lemma 4.2

*f*satisfies an algebraic relation

*f*is integral over \(\mathbb {C}[t]\)) if and only if

### Proof

A crucial observation for the process to obtain an integral basis for \(M^\infty (N)\) from an order-complete basis is stated in the following.

### Proposition 4.3

*q*(

*x*) and \(p_j(x)\) in \(\mathbb {C}[x]\), such that

### Proof

*S*is an additive submonoid of \((\mathbb {Z}_{\ge 0},+)\). Moreover, \(\mathbb {Z}_{\ge 0}\backslash S\) has only finitely many elements; let

*k*be the maximal element in this set. Then there exist \(c_j\in \mathbb {C}\), not all zero, such that

*q*-expansions of both sides (which are functions in \(M^\infty (N)\)) gives \(k+1\) equations in \(k+2\) variables \(c_j\). Hence, the dimension of the \(\mathbb {C}\)-vector space

*G*, which is generated by all the \(g_j\), \(j\ge 0\), is bounded by \(k+1\). Using \(g_j:=t^j f-h_j\) with \(h_j\in M\), (4.2) rewrites into the form:

*M*; hence, this gives the desired relation for

*f*with \(q(t)=c_0 + c_1 t + \cdots +c_{k+1} t^{k+1}\). \(\square \)

### Corollary 4.4

### Proof

*g*and noting that \(c_i\ne 0\) proves the first part of the statement on \(h_\alpha \). To prove (4.4), consider

*k*be the index for which \({{\,\mathrm{pord}\,}}b_k\) becomes maximal with \(c_k\ne 0\). Recalling \({{\,\mathrm{pord}\,}}b_j\equiv j \, ({\mathrm {mod}}\, n)\), \(j=1,\ldots , n\), proves \({{\,\mathrm{pord}\,}}b_k\ge k + n\). Because otherwise \({{\,\mathrm{pord}\,}}b_k = k\) which owing to the choice of

*k*would imply \({{\,\mathrm{pord}\,}}b_j = j\) for all \(j=1,\ldots , n\), and the given order-complete basis would be integral. This proves (4.4). \(\square \)

Corollary 4.4 motivates the following.

### Definition 4.5

We summarize in the form of

### Proposition 4.6

- (i)By a finite sequence of pole-order-reduction steps the order-complete basis \((1,b_1,\ldots ,b_{n-1})\) can be transformed into an integral basis \((1,\beta _1,\ldots ,\beta _{n-1})\), such that$$\begin{aligned} \langle 1,\beta _1,\ldots ,\beta _{n-1} \rangle _{\mathbb {C}[t]} = M^\infty (N). \end{aligned}$$
- (ii)If \((1,\beta '_1,\ldots ,\beta '_{n-1})\) is any another integral basis, that is,then$$\begin{aligned} \langle 1,\beta '_1,\ldots ,\beta '_{n-1} \rangle _{\mathbb {C}[t]} = M^\infty (N), \end{aligned}$$$$\begin{aligned} \{{{\,\mathrm{pord}\,}}\beta _1,\ldots , {{\,\mathrm{pord}\,}}\beta _{n-1}\}= \{{{\,\mathrm{pord}\,}}\beta '_1,\ldots , {{\,\mathrm{pord}\,}}\beta '_{n-1}\}. \end{aligned}$$(4.5)

### Proof

The proof of part (i) is an immediate consequence of Corollary 4.4. Namely, owing to (4.4), each step reduces the pole order of one of the basis elements by *n*. This guarantees termination in finitely many steps. To prove (ii), without loss of generality, we can assume that \({{\,\mathrm{pord}\,}}\beta _j \equiv {{\,\mathrm{pord}\,}}\beta '_j\equiv j \pmod {n}\) for all *j*. Suppose \({{\,\mathrm{pord}\,}}\beta _j\ne {{\,\mathrm{pord}\,}}\beta '_j\) for some \(j\in \{1,\ldots , n-1\}\), i.e., \({{\,\mathrm{pord}\,}}\beta '_j= {{\,\mathrm{pord}\,}}\beta _j + k n\) with \(k\ge 1\). However, this implies that \(\beta _j \not \in \langle 1,\beta '_1,\ldots ,\beta '_{n-1} \rangle _{\mathbb {C}[t]}\), because then, no element in this module can have the same pole order as \(\beta _j\), a contradiction. \(\square \)

## 5 Order-Reduction Polynomials

It was shown in the previous section that by applying a procedure using finitely many steps, any order-complete basis of a subalgebra of \(M^\infty (N)\) can be extended to an integral basis of \(M^\infty (N)\). Moreover, by (4.5), the pole orders of the integral basis functions are uniquely determined. It turns out that under particular circumstances, one can keep track of the number of order-reduction steps, which then gives a proof of the Weierstraß gap Theorem 12.2. To do this bookkeeping, one can use “order-reduction” polynomials. To our knowledge, for the first time such polynomials have been used by Dedekind and Weber [3], see [2] for Stillwell’s translation into English.

*t*is a holomorphic function on \(\mathbb {H}\). Moreover, the induced function \(t^*\), which is meromorphic on the compact Riemann surface \(X_0(N)\), has a pole only at \([\infty ]_N\).

### Remark 5.1

Depending on the context, we will freely move between considering *t* as a function on \(\mathbb {H}\), resp. \({\hat{\mathbb {H}}}\), and its induced version \(t^*:X_0(N)\rightarrow {\hat{\mathbb {C}}}\).

*n*pairwise distinct points \(x_j=[\tau _j]_N\in X_0(N)\) with \(\tau _j\in \mathbb {H}\), such that

*V*of \(v_0\) and neighborhoods \(U_j\) of \(x_j\), such that

*V*. Carrying out the same construction on neighborhoods

*V*for all \(v_0\in \mathbb {C}\backslash {\mathrm {BranchPts}}(t^*)\), and gluing the resulting functions \(D_t(1,b_1,\ldots ,b_{n-1}): V\rightarrow \mathbb {C}\) together, gives a global holomorphic function:

### Lemma 5.2

The meromorphic function \(D_t(1,b_1,\ldots ,b_{n-1})(v)\) constructed above is a polynomial function in *v*.

### Definition 5.3

### Example 5.4

### Example 5.5

*t*and the \(b_j\) as in Example 5.4, one obtains

### Remark 5.6

How such polynomials are computed is explained in Sect. 6.

*n*pairwise distinct preimages of \(v_0\) as follows:

### Lemma 5.7

^{3}

## 6 How to Compute Order-Reduction Polynomials

Next, we explain how to compute the order-reduction polynomials in (5.4) and (5.5).

To this end, it will be convenient to introduce the following notation:

### Definition 6.1

*q*-expansion at infinity for some \(f\in M^\infty (N)\), we define

^{4}

*V*of \(v_0\):

*q*-expansion of

*t*is 1. Now, if

*W*be such that \(U(W(q))=W(U(q))=q\), and define

### Lemma 6.2

### Proof

*v*close to \(v_0\), these values are pairwise different for \(j=1,\ldots , n\), because

*v*. Consequently, we can compute it by taking suitable truncated versions of the expansions (6.5).

## 7 Discriminant Polynomials

Important special cases of order-reduction polynomials are produced by order-complete module bases of \(\mathbb {C}[t,f]\) of the form as in Proposition 3.5.

### Definition 7.1

### Lemma 7.2

## 8 Reduction Steps and Order-Reduction Polynomials

In Sect. 4, we described how order-complete bases can be transformed into integral bases of \(M^\infty (N)\) by a finite sequence of pole-order-reduction steps. In this section, we establish a link between pole-order-reduction steps and order-reduction polynomials.

### Proposition 8.1

### Proof

After filling the right side of (8.1) into the determinant definition (5.3) of \(D_t(1, b_1,\ldots ,b_{k-1}, h_\alpha , b_{k+1},\ldots , b_{n-1})(v)\) and noticing that \(t^*(T_j(v))=v\), \(j=1,\ldots ,n\), the proof is a straightforward consequence of determinant calculus. \(\square \)

### Example 8.2

## 9 Local Puiseux Expansions

### Remark 9.1

*elliptic*point, i.e., if

^{5}

*V*is an open subset of \(V_0\) not containing \(v_0\), then for each \(j=1,\ldots ,\ell \), there exist pairwise disjoint open subsets \(U_{j,k}\subseteq U_j\), \(k=1,\ldots ,k_j\), such that

*j*,

*k*), \(j=1,\ldots ,k_j\) and \(k\in \{1,\ldots , k_j\}\), a uniquely determined \(\tau =\tau (j,k)\in U_{j,k}\), such that

*v*, one has

### Lemma 9.2

*V*are chosen as above. Then, there exist series expansions with complex coefficients \(c_{j,p}\), such that for all \(v \in V:\)

### Proof

### Proposition 9.3

### Proof

*V*is an open subset of a neighborhood \(V_0\) of \(v_0\), such that

*V*does not contain \(v_0\). However, invoking the identity theorem from complex analysis, the statement extends to all \(v\in \mathbb {C}\). \(\square \)

Properties (9.12) and (9.13) are sufficiently important to deserve a

### Definition 9.4

*separation property*). Let \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\), let \(f\in M^\infty (N)\) be such that \(\gcd (n,{{\,\mathrm{pord}\,}}f)=1\). For \(v_0\in \mathbb {C}\), suppose that

*f*has the separation property for \((t,v_0)\) if

*f*satisfies (9.12) and (9.13).

### Remark 9.5

In Sect. 14, we describe how to construct such an *f* having the separation property.

An immediate consequence of Proposition 9.3 is

### Corollary 9.6

*f*have the separation property for \((t, \beta )\) with \(\beta \in \mathbb {C}\). Then

Another consequence of our analysis above is

### Proposition 9.7

*f*has the separation property for \((t,v_0)\), then

^{6}

### Proof

*B*(

*x*) above would be the zero polynomial, we are done. Otherwise, invoking condition (9.12) implies that

### Corollary 9.8

*f*has the separation property for \((t,v_0)\), then

### Proof

If one of the \(a_j\) would be non-zero, Proposition 9.7 would imply a pole of \(F^*\) at some \([\tau _j]_N\ne [\infty ]_N\), \(\tau _j\in \mathbb {H}\cup \mathbb {Q}\). \(\square \)

## 10 Order Reduction and Discriminant Polynomials

*f*gives rise to an order-complete basis \((1,f,\ldots ,f^{n-1})\) of the \(\mathbb {C}[t]\)-module:

*v*from a neighborhood

*V*of \(v_0\in V\). With the setting as in (5.3), one has

*v*:

## 11 Order-Reduction Polynomials: Further Results

In this section, we continue the considerations made in the previous section. Again, \(t\in M^\infty (N)\) with \(n:={{\,\mathrm{pord}\,}}t \ge 1\), and \((1,b_1,\ldots ,b_{n-1})\) with \(b_j\in M^\infty (N)\) is assumed to be an integral basis for \(M^\infty (N)\) over \(\mathbb {C}[t]\).

### Lemma 11.1

### Proof

### Lemma 11.2

### Lemma 11.3

*f*has the separation property for \((t, \beta )\) for some \(\beta \in \mathbb {C}\). Then

### Proof

*f*has the separation property for \((t,\beta )\), one has by Corollary 9.8:

*Q*(

*x*), such that

### Proposition 11.4

### Proof

*f*having the separation property for \((t,\beta )\).

^{7}For the “\(\Rightarrow \)” direction of the statement, suppose \(D_t(1,b_1,\ldots ,b_{n-1})(\beta )=0\). Then (11.4) implies \(D_t(f)(\beta )=0\) which, owing to Corollary 9.6, is true if and only if \(\beta \in {\mathrm {BranchPts}}(t)\). For the other direction, we use the reverse direction of this “if and only if” relation: \(\beta \in {\mathrm {BranchPts}}(t^*)\) implies \(x-\beta \mid D_t(f)(x)\). Next, we apply Lemma 11.3 to the equation (10.5) and obtain

### Proposition 11.5

### Proof

*f*to have the separation property for \((t,\beta )\). Then (9.14) implies the existence of a polynomial \(p(x)\in \mathbb {C}[x]\), such that

*f*on a (compact) Riemann surface

*X*. For \(x_0=[\tau _0]_N\in X_0(N)\), one has (e.g., [11, Lemma 4.7] and [5, Sect. 2.4]) with respect to our charts \(\phi _{\tau _0}(\tau )\) centered at 0:

^{8}

### Corollary 11.6

*f*on a compact Riemann surface

*X*:

^{9}for a non-constant holomorphic map \(F:X\rightarrow Y\) between compact Riemann surfaces:

### Corollary 11.7

## 12 Proof of the Weierstraß Gap Theorem

In this section, we prove the gap theorem for modular functions in \(M^\infty (N)\).

### Definition 12.1

*Gaps in modular function algebras*) Let

*M*be a subalgebra of \(M^\infty (N)\), the modular functions for \(\Gamma _0(N)\) which are holomorphic in \(\mathbb {H}\) and with a pole at \(\infty \). A positive integer

*n*is called a gap in

*M*if there is no \(f\in M\) with \({{\,\mathrm{pord}\,}}f =n\). We also define the gap number \(g_M\) as the total number of gaps in

*M*, that is,

In this section, we prove the gap theorem in the following version.

### Theorem 12.2

*g*gaps \(n_j\) with

^{10}

*n*for \(j\in \{1,\ldots ,n-1\}\) makes clear that one cannot find any function of pole order

*j*, \(l_j\) pole orders are missing; summing

*j*from 1 to \(n-1\) gives the total number of missing pole orders of functions in \(\mathbb {C}[t,f]\):

### Lemma 12.3

### Proof

If \(n=1\) or \(\ell =1\) then \(\mathbb {C}[t,f]=M^\infty (N)\); i.e., there is no gap. The case both *n* and \(\ell \) greater or equal to 2 was treated above. \(\square \)

### Proof of Theorem 12.2.

- (i)
- (ii)

*t*and

*f*are chosen from \(M^\infty (N)\), such that \(n:={{\,\mathrm{pord}\,}}t\ge 2\), \(l:={{\,\mathrm{pord}\,}}f \ge 2\), and \(\gcd (l,n)=1.\) Such a pair (

*t*,

*f*) can be easily constructed, see, for instance, Example 2.3.

*r*reduction steps, we arrive at the integral basis \((1,b_1,\ldots ,b_{n-1})\) of \(M^\infty (N)\). Defining

*g*:

*g*gaps. If \(g=0\) there are no gaps; i.e., in this case, after relabelling indices,

To prove the remaining part of the gap theorem, namely, the bound (12.1) for the gaps \(\{n_1=1,n_2,\ldots ,n_g\}\) where \(g\ge 1\), we will use a general combinatorial argument. Notice that \(n_1=1\), because otherwise there would be no gap, which, as we proved, is only possible if \(g=0\).

*t*and

*f*from \(M^\infty (N)\) as above, by applying pole-order-reduction steps, we arrived, after relabelling indices, at an integral basis \((1,b_1,\ldots , b_{n-1})\) for \(M^\infty (N)\) where \({{\,\mathrm{pord}\,}}b_j \equiv j \pmod n\), \(j=1,\ldots ,n-1\). Defining

*S*with respect to \(m+1\). Namely, it is easy to see that there exist positive integers \(s_1,\ldots , s_m \in \mathbb {Z}_{>0}\), such that \(s_j\equiv j \pmod {m+1}\) for all \(j\in \{1,\ldots ,m\}\) and

*S*by \(\gamma (S)\). Hence, in the given context, \(g=\gamma (S)\). Recall that \(m+1\) is chosen to be the smallest non-gap of \(M^\infty (N)\). Therefore, we choose a monoid representation with respect to \(m+1\). Concretely, in this case, there are \(k_j\in \mathbb {Z}_{>0}\), such that

## 13 A Gap Property of Monoids

*S*.

^{11}Let \(\gamma (S)\) be the total number of gaps of

*S*. Relating to our proof setting in Sect. 12, we choose this representation of

*S*under the assumption that \(m+1\) is the smallest non-gap of

*S*. By the definition of

*S*, there are

*positive*integers \(k_j\), such that

### Lemma 13.1

In other words, the largest possible gap is bounded by \(2 \gamma (S)-1\). Before proving this statement, we prove two elementary observations.

### Lemma 13.2

*i*and \(\ell \) in \(\mathbb {Z}_{>0}\) are such that \(i + \ell = j\) for \(j\in \{1,\ldots , m\}\), then

### Proof

### Lemma 13.3

If *i* and \(\ell \) in \(\mathbb {Z}_{>0}\) are such that \(i + \ell = j+m+1\) for \(j\in \{1,\ldots , m\}\), then \(k_i + k_\ell +1\ge k_j\).

### Proof

### Proof of Lemma 13.1

## 14 Functions with Separation Property

^{12}

In Definition 9.4, we defined the separation property of *f* for \((t,v_0)\) with \(v_0=\alpha \) as in (14.1). At various places, we required *f* to have this property, for instance, in Proposition 11.5. In this section, we prove the existence of such *f*. In addition, here, we have to use the charts as in (9.1), (9.2), and (9.3).

### Lemma 14.1

### Proof

*j*runs form 1 to \(\ell \):

^{13}

*n*unknowns \(a_0,\ldots , a_{n-1}\). This implies that there exists a solution to the system with the \(a_j\) not all 0 . This produces a contradiction: since \(g\in M^\infty (N)\), there are polynomials \(p_j(x)\in \mathbb {C}[x]\), such that

### Lemma 14.2

### Proof

*m*. If \(m=2\), then by the previous lemma, there exists an \(i\in \{1,\ldots ,n-1\}\), such that \(b_i(\tau _{i_1})\ne b_i(\tau _{i_2})\), and we choose \(f:=b_i\). Suppose \(m\ge 2\). Because of index relabeling, we can choose \(S_m:=\{\tau _1,\ldots ,\tau _m\}\), and the induction hypothesis gives an \(F\in M^\infty (N)\) as a \(\mathbb {C}\)-linear combination of \(b_j\), such that the values \(F(\tau _1),\ldots ,F(\tau _{m})\) are pairwise distinct. If \(F(\tau _{m+1})\ne F(\tau _i)\) for all \(i=1,\ldots ,m\), the induction step is done. Otherwise, \(F(\tau _{m+1})=F(\tau _r)\) for some \(r\in \{1,\ldots ,m\}\). By the previous lemma, there is some \(k\in \{1,\ldots ,n-1\}\), such that \(b_k(\tau _{m+1})\ne b_k(\tau _r)\), and we can choose a non-zero \(c\in \mathbb {C}\), such that

*s*, we are done with \(f:=F_r\). Otherwise, by the previous lemma, there is some \(l\in \{1,\ldots ,n-1\}\), such that \(b_l(\tau _{m+1})\ne b_l(\tau _s)\), and we can choose a non-zero \(d\in \mathbb {C}\), such that

*f*additionally has to satisfy the conditions (9.13) which rewritten as order conditions are

### Remark 14.3

### Lemma 14.4

### Proof

*n*unknowns \(a_0,\ldots , a_{n-1}\).

^{14}This implies that there exists a solution to the system with the \(a_j\) not all 0, which produces a contradiction as in the proof of Lemma 14.2. \(\square \)

### Lemma 14.5

### Proof

*j*is chosen according to Lemma 14.4, namely, such that \({{\,\mathrm{ord}\,}}_{\phi _{\tau _i}(\tau )}(b_j(\tau )-b_j(\tau _i))=1\). We will show: given \(F=a_0 t + a_1 b_1 +\cdots + a_{n-1} b_{n-1}\) with \(a_j\in \mathbb {C}\), such that

*k*.

*F*such that (14.8) holds is by Lemma 14.2. If, in addition,

*f*the condition (14.6) holds and also that

*F*of required form, such that (14.8) and (14.9) hold. If, in addition,

*f*has the properties (14.6) and (14.10). The extra requirement (14.11) is needed to guarantee the first

*k*instances of the latter condition. This completes the proof of the induction step and also the proof of Lemma 14.5. \(\square \)

Summarizing, using an integral basis \((1,b_1,\ldots , b_{n-1})\) for \(M^\infty (N)\) over \(\mathbb {C}[t]\), we constructed an *f* which proves

### Corollary 14.6

For every \(\alpha \in {\mathbb {C}}\) , there is an \(f\in M^{\infty }(N)\), such that *f* has the separation property for \((t,\alpha )\).

## 15 Appendix: Modular Functions–Basic Notions

To make this article as much self-contained as possible, in this section, we recall most of the facts we need about modular functions.

*N*is a fixed positive integer. Note that \(\Gamma _0(1)={\mathrm {SL}}_2({\mathbb {Z}})\). These subgroups have a finite index in \({\mathrm {SL}}_2({\mathbb {Z}})\):

*meromorphic modular function for*\(\Gamma _0(N)\), if (i) for all \(\left( {\begin{matrix} a&{}b\\ c&{}d \end{matrix}} \right) \in \Gamma _0(N)\):

*M*(

*N*), we denote the set of meromorphic modular functions for \(\Gamma _0(N)\).

*q*with finite principal part, that is

*f*to \({\mathbb {H}}\cup \{\infty \}\) by defining \(f(\infty ):=\infty \), if \(M>0\), and \(f(\infty ):= f_0\), otherwise. Subsequently, a Laurent expansion of

*f*as in (15.2) will be also called

*q*

*-expansion of*

*f*

*at infinity*.

^{15}

*q*

*-expansion of*

*f*

*at*

*a*/

*c*. Understanding that \(a/0=\infty \), this also covers the definition of

*q*-expansions at \(\infty \). Concerning the uniqueness of such expansions, let \(\gamma '\in {\mathrm {SL}}_2({\mathbb {Z}})\) be such that \(\gamma ' \infty = \gamma \infty = a/c\), then the

*q*-expansion of \(f(\gamma ' \tau )\) differs from that of \(f(\gamma \tau )\) only by a root-of-unity factor in the coefficients, namely, we have then \(\gamma ' = \gamma \left( {\begin{matrix} \pm 1&{}m\\ 0&{}\pm 1 \end{matrix}} \right) \) for some \(m \in \mathbb {Z}\), which implies

*f*from \(\mathbb {H}\) to \({\hat{\mathbb {H}}}:=\mathbb {H}\cup \{\infty \}\cup \mathbb {Q}\) by defining \(f(a/c):=\lim _{{\mathrm {Im}}(\tau )\rightarrow \infty } f(\gamma \tau )\), where \(\gamma \in {\mathrm {SL}}_2({\mathbb {Z}})\) is chosen, such that \(\gamma \infty = a/c\). Another consequence is that the

*q*-expansions of

*f*at \(\infty \) are uniquely determined owing to

*N*is clear from the context, one also writes \([\tau ]\) instead of \([\tau ]_N\). The set of all such orbits is denoted by

*cusps*of \(X_0(N)\). One has, for example,

### Lemma 15.1

- (1)
\(X_0(\ell ) \text { has two cusps}: [\infty ]_\ell \, \text {and }\ [0]_\ell \);

- (2)
\(X_0(\ell ^2) \ \text { has }\ell +1\text { cusps}: [\infty ]_{\ell ^2},[0]_{\ell ^2}, \text {and }\ [k/\ell ]_{\ell ^2},\, \, k=1,\ldots , \ell -1\).

### Proof

This fact can be found in many sources; a detailed description of how to construct a set of representatives for the cusps of \(\Gamma _0(N)\), for instance, is given in [16, Lemma 5.3]. \(\square \)

*f*.

As described in detail in [5], \(X_0(N)\) can be equipped with the structure of a compact Riemann surface. This analytic structure turns the induced functions \(f^*\) into meromorphic functions on \(X_0(N)\). The following classical lemma [11, Theorem 1.37], a Riemann surface version of Liouville’s theorem, is crucial for zero recognition of modular functions.

### Lemma 15.2

Let *X* be a compact Riemann surface. Suppose that \(g:X \rightarrow \mathbb {C}\) is a holomorphic function on all of *X*. Then, *g* is a constant function.

*j*is the modular invariant (the Klein

*j*function). The subset

*finite*number of cusps:

*q*-expansions of

*f*at \(a_j/c_j\); i.e., of

*q*-expansions of \(f(\gamma _j \tau )\) as in (15.3) with \(\gamma _j\in {\mathrm {SL}}_2({\mathbb {Z}})\) such that \(\gamma _j \infty = a_j/c_j\). We call these expansions also local

*q*-expansions of \(f^*\) at the cusps \([a_j/c_j]_N\); \(w_N(c_j)\) is called the width of the cusp \([a_j/c_j]_N\). It is straightforward to show that it is independent of the choice of the representative \(a_j/c_j\) of the cusp \([a_j/c_j]_N\), and that \(w_N(c_j)=N/\gcd (c_j^2,N)\) for relatively prime \(a_j\) and \(c_j\). Note that \([\infty ]_N= [1/0]_N\).

### Definition 15.3

*Order and*\(\phi \)

*-order*). Let \(f=\sum _{n=m}^\infty a_n q^n\) with \(m\in \mathbb {Z}\), such that \(a_m\ne 0\). Then we define the order of

*f*as

*f*as

*q*-expansion of \(f^*\) at \([a/c]_N\), that is,

## 16 Appendix: Meromorphic Functions on Riemann Surfaces–Basic Notions

To make this article as much self-contained as possible, in this second appendix section, we recall most of the facts that we need about meromorphic functions on Riemann surfaces. For the terminology, we basically follow [7]; other classic texts are [6] and [11].

Lemma 15.2 states the fundamental fact that any analytic function on a compact Riemann surface is constant. In Example 2.1, we have seen that \(z_5^*\) has its only zero of order 1 at \([\infty ]_5\) and its only pole at \([0]_5\) with multiplicity 1, i.e., \(z_5^*\) has order \(-1\) at \([\infty ]_5\).^{16} This is also in accordance with Lemma 16.1, a corollary of another fundamental fact which says that meromorphic functions on compact Riemann surfaces have exactly as many zeroes as poles (counting multiplicities); see, for instance, [11, Proposition 4.12]:

### Lemma 16.1

*g*be a non-constant meromorphic function on a compact Riemann surface

*X*. Then

Here, \({{\,\mathrm{ord}\,}}_{x_0} g\) is defined as follows. Suppose \(g(x)= \sum _{n=m}^\infty c_n (\varphi (x)-\varphi (x_0))^n\), \(c_m\ne 0\), is the local Laurent expansion of *g* at \(x_0\) using the local coordinate chart \(\varphi :U_0\rightarrow \mathbb {C}\) which homeomorphically maps a neighborhood \(U_0\) of \(x_0\in X\) to an open set \(V_0\subseteq \mathbb {C}\). Then, \({{\,\mathrm{ord}\,}}_{x_0} g :=m\).

Let \({\mathcal {M}}(S)\) denote the field of meromorphic functions \(f:S\rightarrow {\hat{\mathbb {C}}}\) on a Riemann surface *S*.^{17} Let \(f\in {\mathcal {M}}(S)\) be non-constant: then for every neighborhood *U* of \(x\in S\), there exist neighborhoods \(U_x\subseteq U\) of *x* and *V* of *f*(*x*), such that the set \(f^{-1}(v)\cap U_x\) contains exactly *k* elements for every \(v\in V\backslash \{f(x)\}\). This number *k* is called the multiplicity of *f* at *x*; notation: \(k={{\,\mathrm{mult}\,}}_x(f)\).^{18} If *S* is compact, \(f\in {\mathcal {M}}(S)\) is surjective and each \(v \in {\hat{\mathbb {C}}}\) has the same number of preimages, say *n*, counting multiplicities; i.e., \(n=\sum _{x\in f^{-1}(v)}{{\,\mathrm{mult}\,}}_x(f)\), see, e.g., [7, Theorem 4.24]. This number *n* is called the degree of *f*; notation: \(n={\mathrm {Deg}}(f)\). One of the consequences is that non-constant functions on compact Riemann surfaces have as many (finitely many) zeros as poles counting multiplicities; this is Lemma 16.1.

\({\mathrm {RamiPts}}(f):= \{x\in S: {{\,\mathrm{mult}\,}}_x(f)\ge 2\}\) denotes the set of ramification points of *f*; \({\mathrm {BranchPts}}(f):= f({\mathrm {RamiPts}}(f))\subseteq {\hat{\mathbb {C}}}\) denotes the set of branch points of *f*. Ramification points, and also branch points, of a function *f* form sets having no accumulation point. Hence, for functions on compact Riemann surfaces, these sets have finitely many elements.

## 17 Conclusion

In this article, we present the first proof of the Weierstraß gap theorem (for modular functions) without using the Riemann–Roch theorem. The main ingredient in our proof is the concept of order-reduction polynomials which corresponds to the discriminant of a field extension of \(\mathbb {Q}\) in the setting of algebraic number theory, see, for instance, [10, III, §3]. In the field case, the structure of this discriminant is related to the ramification index [10, III, §2, Proposition 8, and III, §3, Proposition 14]. Analogously, in Proposition 11.5, we give a factorization of the order-reduction polynomial which in direct fashion relates to the branch points of the modular function *t*. This relation allows us to connect the degree of this polynomial to the genus of \(X_0(N)\). This observation is crucial for our proof of the Weierstraß gap theorem.

In addition, our approach gives new algebraic and algorithmic insight based on module presentations of modular function algebras, in particular, the usage of integral bases. For example, our proof also gives a method to compute the order-reduction polynomial by using the Puiseux series expansions at infinity. Another new feature concerns the gap bound: the main task of our proof is to show that there are exactly *g* gaps for any modular function algebra. The proof that the corresponding pole orders are bounded by \(2g-1\), with the help of an elementary combinatorial argument turns out to be an immediate consequence of our approach. Another by-product of our framework is a natural explanation of the genus \(g=0\) case as a consequence of the reduction to an integral basis.

Summarizing, our setting generalizes ideas from algebraic number theory, but still stays close to “first principles.” Hence, we feel that our approach has potential for further extensions and applications. For example, we are planning to exploit the algorithmic content of our approach for computer algebra applications, for instance, for the effective computation of suitable module bases for modular function algebras.

## Footnotes

- 1.
A \(\mathbb {C}\)-algebra is a commutative ring with 1 which is also a vector space over \(\mathbb {C}\).

- 2.
Notice that \({{\,\mathrm{pord}\,}}1/z_{11}=5\).

- 3.
The existence of such a \(v_0\) is owing to Corollary 4.4.

- 4.
Notice that, in particular, \(\tau \ne \infty \).

- 5.
Note that, in particular, \(\tau _j\ne \infty \).

- 6.
Note that \(\tau _j\in \mathbb {H}\cup \mathbb {Q}\) are such that \([\tau _j]_N\ne [\infty ]_N\). Hence, (9.16) implies that \(F\not \in M^\infty (N)\).

- 7.
How to construct such

*f*is described in Sect. 14. - 8.
I.e., \(\phi _{\tau _0}(\tau _0)=0\).

- 9.
Actually the special case we need, \(Y={\hat{\mathbb {C}}}\), was given by Riemann; e.g., [2].

- 10.
This means, in this case, one has \(M^\infty (N)=\mathbb {C}[h]\).

- 11.
The largest gap is called the Frobenius number of

*S*. - 12.
Recall that \(t^*([\tau ]_n)=t(\tau )\).

- 13.
Notice that the charts \(\phi _{\tau _j}(\tau )\) are centered at 0; i.e., \(\phi _{\tau _j}(\tau _j)=0\). Consequently, for fixed

*j*, the numerator in (14.2) has to be of the form: \(\phi _{\tau _j}(\tau )^{k_j}(c_0+c_1\phi _{\tau _j}(\tau )+ \cdots ).\) - 14.
Notice that for this argument to work, we invoke \(k_i>1\).

- 15.
This expansion and also those for \(f(\gamma \tau )\) are required to converge for all \(\tau \in \mathbb {H}\) with \({\mathrm {Im}}(\tau )\) sufficiently large.

- 16.
Notice that we also say that \(z_5^*\) has pole order 1 at \([\infty ]_5\).

- 17.
In this context, \({\hat{\mathbb {C}}}:=\mathbb {C}\cup \{\infty \}\) is understood to be a compact Riemann surface isomorphic to the Riemann sphere.

- 18.
If

*x*is a pole of \(f:\)\({{\,\mathrm{mult}\,}}_x f =-{{\,\mathrm{ord}\,}}_x f\); otherwise, \({{\,\mathrm{mult}\,}}_x f = {{\,\mathrm{ord}\,}}_x (f - f(x))\).

## Notes

### Acknowledgements

Open access funding provided by Johannes Kepler University Linz. Both authors were supported by grant SFB F50-06 of the Austrian Science Fund (FWF). In November 2018, while working on parts of this paper, the first named author enjoyed the overwhelming hospitality of Bill Chen and his team at the Center for Applied Mathematics, Tianjin University. Finally, we thank Krishnaswami Alladi and an anonymous referee for constructive remarks which helped to improve the presentation of this paper.

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