On the modulo p zeros of modular forms congruent to theta series

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Introduction
For k ∈ Z ≥0 and a congruence subgroup Γ ⊂ SL 2 (Z), denote by M k (Γ) the Q-vector space of modular forms with rational q-expansion of weight k for Γ.If Γ = SL 2 (Z) we simply write M k for M k (SL 2 (Z)).Basic examples of modular forms include the Eisenstein series of weight k, given by the q-expansion for even k ≥ 4, where B k is the k-th Bernoulli number and τ ∈ H = {τ ∈ C | Im τ > 0}.Another example is the modular discriminant It is well known that the space M k is finite-dimensional: writing k ≥ 4 uniquely as an explicit basis for M k is given by Furthermore, define the modular j-invariant as it is a weakly holomorphic modular form (poles at the cusps are allowed) of weight 0. For any f ∈ M k , using the notation in (1), consider the quotient this is a weakly holomorphic modular form of weight 0. It follows from ( 2) that there exists a polynomial P [f ](j) ∈ Q[j] of degree ≤ n k such that Explicitly, these polynomials can be written as where F ⊂ H denotes the standard fundamental domain and where i and ρ = e 2πi/3 are the elliptic points of F. Thus, the zeros of the polynomial P [f ] contain the zeros of f as input; naturally this polynomial plays an important role in the study of zeros of modular forms.For example, one can consider the polynomial P [f ] for f = E k .Since the orders at the elliptic points are given by ord ρ (E k ) = a k and ord i (E k ) = b k , see [15], it follows that (j − j(τ )).
In 1970, it was shown by F.K.C. Rankin and H.P.F.Swinnerton-Dyer [15] that the non-elliptic zeros of E k (τ ) in F are all simple and located on the arc Since the j-image of the arc is j(A) = (0, 1728), it follows that the zeros of P [E k ](j) are all simple and located in the real interval (0, 1728).Moreover, P [E k ](j) ∈ Q[j] implies that the j-invariants of the zeros are all algebraic numbers.Another feature of this polynomial is the location of the zeros over finite fields.For primes p ≥ 5, the q-series coefficients of E p−1 are all p-integral, so that the coefficients of the polynomials P [E p−1 ](j) are also all p-integral.Hence these polynomials can all be reduced modulo p.We have the following surprising result for them.
In fact, it was shown by M. Deuring [6] that j(E) lies in F p 2 for supersingular elliptic curves E over F p .Therefore, we have the following theorem.
The number of linear factors of this polynomial is related [3] to the class number of the field Q( √ −p).More arithmetic properties for these polynomials can be found for example in [8].Natural extensions of these polynomials to rational function fields have been studied in [5].
The polynomials P [E p−1 ](j) mod p can also be cast as truncated hypergeometric functions.We first introduce some notation.For x ∈ C and n ∈ Z ≤0 , define the Pochhammer symbol as the series defines an absolutely convergent series in the disk |z| < 1.Finally, define the polynomials U 0 n (j) and U 1 n (j) for n ≥ 0 as the unique polynomials of degree n satisfying These truncations of hypergeometric functions satisfy the following congruences.
Theorem 1.3 (Kaneko-Zagier [10,Proposition 5]).If k = p − 1, we have the congruence The main goal of this paper is to prove analogues of Theorems 1.1, 1.2 and 1.3 for a different modular setup that we outline in the next section.

Theta modular forms
Given a formal power series f (q) ∈ C[[q]] and an even integer k ≥ 4, there is a unique modular form where n k is defined in (1).Example 1.4 below shows that there is indeed a natural construction of this form.We first introduce some notation.For a prime p and f, g ∈ Q[[q]] with p-integral coefficients write f ≡ g + O(q m ) mod p for some m ≥ 0 if the first m Fourier coefficients of f and g agree modulo p.Furthermore, write f ≡ g mod p if all Fourier coefficients agree modulo p.
Example 1.4.Consider the formal power series ].The resulting modular forms C k 1 are called extremal modular forms and are related to the theory of extremal lattices, see for example [9].In 2007, W. Duke and P. Jenkins [7] showed that the non-elliptic zeros of C k 1 are all simple and located on the arc A inside F, see ) mod p when k = p − 1.We will see later in Lemma 2.1 that this implies the congruence We now introduce theta modular forms.For a positive definite lattice L, define its theta series as This is a holomorphic function on H.We call C k θ L (τ ) the theta modular form of weight k corresponding to the lattice L. In this paper we will consider the (one-dimensional) lattice Z and the (hexagonal 3).Their theta series are known as the Jacobi theta series and For these theta series it is known that θ 2 Z ∈ M 1 (Γ 1 (4)) and θ H ∈ M 1 (Γ 1 (3)), see for example [4].
Example 1.5.For k = 52, the theta modular form corresponding to the lattice Z is In this paper we find analogues of Theorems 1.1, 1.

Results
In this section we state our results about factorisations of the polynomials P [C k θ Z ](j) and P [C k θ H ](j) over finite fields.Since C k θ Z (τ ) and C k θ H (τ ) have integer Fourier coefficients, the polynomials P [C k θ Z ](j) and P [C k θ H ](j) have integer coefficients as well.This means that reduction modulo primes is always well-defined.
Define the polynomials W 0 n (j) and W 1 n (j) for n ≥ 0 as the unique polynomials of degree n satisfying For these polynomials we have the following congruences.
Theorem 1.6.Let p ≥ 7 be a prime and k = p+1 2 .Then where b k and n k are defined in (1).
Note how the parameters in Equations ( 8) and ( 9) are approximately halved compared to ( 5) and (6).Using this hypergeometric expression for P [C k θ Z ](j) mod p, we will show that this polynomial splits over the ground field F p .
Theorem 1.7.Let p ≥ 7 be a prime and k = p+1 2 .Then P [C k θ Z ](j) splits over F p into linear factors.
Thus, Theorems 1.6 and 1.7 are analogues of Theorems 1.3 and 1.2 respectively.The modular forms C k θ Z for k = p+1 2 are congruent, up to a constant, to the weight k modular forms T h(ϕ H 1 2k ) defined by T. Miezaki in [12].Here ϕ H 1 2k are certain invariant polynomials related to the genus 1 average weight enumerator of binary, self-dual and doubly even codes of length 2k, see [14].
Explicitly, the zero set of P [C k θ Z ](j) mod p is given by This zero set has an interpretation via elliptic curves.Let p ≥ 7 be a prime and E an elliptic curve over F p given by the equation E : where a, b ∈ F p , with 4a holds.The product here is over elliptic curves defined over F p , up to F p -isomorphism, with full rational 2-torsion and no rational points of order 4.
Define the polynomials V 0 n (j) and V 1 n (j) for n ≥ 0 as the unique polynomials of degree n satisfying We have the following congruences for them.
Theorem 1.10.Let p be a prime congruent to 5 or 11 modulo 12 and k = p + 1.Then we have the congruence Furthermore, the polynomials P [C k θ H ](j) have a specific factorisation modulo p.
Theorem 1.11.Let p be a prime congruent to 5 or 11 modulo 12 and k = p + 1.The polynomials P [C k θ H ](j) split over F p 2 .Moreover, these polynomials factor over F p as a product of quadratic factors only if the degree n k is even and as (j + 1728) times a product of quadratic factors if the degree n k is odd.
Explicitly, its zero set is where 2 1/3 is the unique cube root of 2 in F p .
There is no clear analogy with Theorem 1.8 for the polynomials P [C k θ H ](j).However, by an explicit calculation we get the following characterisation of the zero set (13).
Proposition 1.12.The zero set (13) coincides with the set of j-invariants of the elliptic curves whenever it is non-singular, with b ∈ F p 2 satisfying b p+1 ≡ −2 mod p.
The We check the congruence properties in Theorems 1.11 and 1.10.We see that

Proofs
We start with the following basic observation.
Lemma 2.1.Let k ≥ 4 and p ≥ 5 a prime.Suppose f ∈ M k has p-integral Fourier coefficients and f ≡ O(q n k +1 ) mod p (that is, the first n k Fourier coefficients of f are divisible by p).Then f ≡ 0 mod p.
Proof.Suppose f satisfies the conditions of the lemma.Note that the F p -vector space is a basis for this space, thus f ≡ 0 mod p.
Given two modular forms f, g ∈ M k with p-integral Fourier coefficients, Lemma 2.1 implies that they agree modulo p if their first n k + 1 Fourier coefficients coincide modulo p.

Proofs for C k θ Z
We start with a hypergeometric identity for the function θ Z (τ ).Lemma 2.2.In a neighborhood of τ = i∞, we have Proof.The theta series θ Z (τ ) 2 is a modular form of weight 1 for the congruence subgroup Γ 1 (4).Therefore, by [18, Proposition 21], it satisfies a second order linear differential equation as a function in 1728/j.As see [18, Eq. ( 74)], and the 2 F 1 satisfies a second order linear differential equation, it is easy to check that the left-hand side and the right-hand side of ( 16) satisfy the same differential equation and initial values.
Proof of Theorem 1.6.Let p ≥ 7 be a prime and k = p+1 2 .Consider the modular form of weight k.In order to show that P [C k θ Z ](j) and W b k n k (j) agree modulo p as polynomials in j, it suffices to show the congruence for power series in q.We first assume k ≡ 0, 4 mod 12, i.e. b k = 0. Using Lemma 2.2 we find As 3n k + a k − 1/8 = p/8, we see that From As before we have 3n k + a k + 11/8 = p/8, so that Again, Lemma 2.1 implies We will now work towards the proof of Theorem 1.7.The goal is to make a choice of a Hauptmodul t 2 (τ ) for the group Γ(2) and write j(τ ) as a rational function in t 2 (τ ).This induces a variable transformation for the polynomial P [C k θ Z ](j) that simplifies the treatment of the zeros.Consider the modular lambda function is the Dedekind eta function.Choose λ(τ ) as Hauptmodul t 2 (τ ) for the group Γ(2) and write j(τ ) as a rational function in t 2 (τ ): Using Lemmas 2.5 and 2.7 below, we will write the zeros of P [C k θ Z ] mod p in terms of t 2 .We first start with some technical statements.For a prime p congruent to 3 modulo 4, define the truncated hypergeometric function i.e. the hypergeometric function truncated at λ where Proof.Cases k ≡ 0, 4 mod 12.In this case we have b k = 0. Lemma 2.3 implies Applying the hypergeometric identity ; λ , see [17, Eq. ( 28)], we find out that and the left-hand side is a polynomial of degree (p + 1)/4 in λ, we conclude that Cases k ≡ 6, 10 mod 12.In this case we have b k = 1.The identity gives, together with Lemma 2.4, Again comparing the first (p + 1)/4 coefficients on both sides we find out that This finishes the proof of the lemma.
The goal is to show that the polynomial G p splits over F p ; by Lemma 2.5 this will imply that j=0 c j λ j .We need to show It is easy to see that c 0 ≡ 1 ≡ c p+1 4 mod p.The congruence for the remaining coefficients follows by induction on j and from the congruence Lemma 2.7.For a prime p congruent to 3 mod 4, the polynomial G p factors as Proof.For a polynomial P (x) ∈ F p [x] of degree d > 0 and v ≥ 0, define the v-th power sums of P as S v (P ) = Let R p (λ) be the polynomial on the right-hand side of (21).The power sums can be written in terms of Legendre symbols as where Euler's criterion is used to obtain the expression in the last line.Since Using Lemma 2.6 and Newton's identities, we relate the coefficients of G p to the power sums of G p as follwos: in the notation from the proof of Lemma 2.6.Using induction on v and the identity for all integers 0 ≤ v ≤ p+1 4 .As both polynomials have the same leading coefficient in F p , we conclude that G p (λ) ≡ R p (λ) mod p.
Proof of Theorem 1.7.It follows from Lemmas 2.5 and 2.7 that the zero set of the polynomials Therefore, P [C k θ Z ](j) splits over F p .

Elliptic curves with a prescribed rational 4-torsion group
The goal of this section is to classify all elliptic curves over F p for p ≡ 3 mod 4 with a prescribed rational 4-torsion group.We will relate this to the zeros of P [C k θ Z ](j) mod p, where k = p+1 2 .Every E/F p with rational 2-torsion and p ≡ 3 mod 4 is isomorphic (over F p ) to a Legendre elliptic curve E λ /F p given by the equation where λ ∈ F p \ {0, 1}.This follows from the explicit F p -isomorphism given in [16, Proposition 1.7a], see also [2].For these elliptic curves, we can classify all possible F p -rational 4-torsion groups.The next lemma is comparable with [2, Proposition 2.1].
Lemma 2.8.For λ ∈ F p \ {0, 1} and p ≡ 3 mod 4, Proof.For the proof consider the division polynomial ψ 4 (X, Y ) ∈ F p [X, Y ], see [16, p. 105], of the elliptic curve E λ : Y 2 = X(X − 1)(X − λ).The zeros of this polynomial correspond precisely to points in E λ (F p ) [4].A computation shows p , we see that ψ 4 (X, Y ) has no zeros in F p apart from Y = 0.The remaining case uses a similar computation.For example, if λ, λ − 1 ∈ F * 2 p , we see that A combination of Lemmas 2.8 and 2.7 gives the following result.
Lemma 2.9.For primes p ≡ 3 mod 4, we have the congruence As a consequence of Lemma 2.9, these are exactly the j-invariants, different from 0, 1728 , of the elliptic curves over

Proofs for C k θ H (τ )
We start with a hypergeometric identity for the function θ H (τ ).
Lemma 2.10.In a neighborhood of τ = i∞, we have the identity Proof.Since θ H (τ ) is a modular form of weight 1 for the congruence subgroup Γ 1 (3), by [18, Proposition 21], it satisfies a second order linear differential equation as a function in 1728/j.The proof now follows the lines of the proof of Lemma 2.2.
Proof of Theorem 1.10.This proof is similar to the proof of Theorem 1.6.For weights k ≡ 0, 6 mod 12 consider the modular form ) mod p and, therefore, g k ≡ C k θ H mod p by Lemma 2.1.
Similar to the case k ≡ 0 mod 12 we find out that As in the case of C k θ Z (τ ), we next choose a Hauptmodul for the group Γ 1 (3).Here we pick as a power series in F p [[y]].Here the O-term is dropped since the left-hand side is a polynomial of degree (at most) (p + 1)/3 in y.It remains to notice that (−4) (p+1)/12 ≡ 2 1/3 mod p.
We next show that P [C k θ H ](j) mod p has at most one zero in F p .Suppose β = 2 and 1.3 for the modular forms C k θ Z and C k θ H . Namely, we will write the polynomials P [C k θ Z ] and P [C k θ H ] as truncated hypergeometric functions modulo primes, similar to what is done for the Eisenstein series in Theorem 1.3, see Theorems 1.6 and 1.10 below.Furthermore, we will see that the factorisations of the polynomials P [C k θ Z ] and P [C k θ H ] over finite fields have a structure reminiscent to that of P [E k ], as recorded in Theorems 1.1 and 1.2.Over finite fields, P [C k θ Z ] factors as a product of linear factors, see Theorem 1.7, while P [C k θ H ] factors as a product of quadratic factors only (and one linear factor if the degree of the polynomial is odd), see Theorem 1.11, whereas P [E k ] factors as the product of both linear and quadratic factors.
mod p, for the choice of b k and n k as in (1).

3 3 4 4 (2a−1) 3 a(a+4) 3 ∈ F p 2
is a zero of P [C k θ H ](j). From the relation a p = −2/a we see thatβ p = 3 3 4 4 (2a p − 1)3 a p (a p + 4) Since β ∈ F p if and only if β p = β, it is clear that β = −1728.Furthermore, by comparing the degree of P [C k θ H ](j) with the cardinality of (26), we find that β = −1728 is a simple zero of P [C k θ H ](j) mod p.Therefore, P [C k θ H ](j) factors as a product of quadratic factors if n k is even and as (j + 1728) times quadratic factors if n k is odd.
, just as in the case of E k .Thus, we see that the polynomials P [C k 1](j) and P [E k ](j) have similar factorisation behaviour in the ring R[j].But this is also the case in the ring F p [j]: indeed, one can study congruence properties of these modular forms.For primes p ≥ 5 the congruence of the Bernoulli numbers [16,7b 2 ≡ 0 mod p. Denote by E(F p ) the group of F p -rational points on E and byE(F p )[n] the n-torsion subgroup of E(F p ), where n ∈ N. It is well known that E(F p )[n] is isomorphic to a subgroup of Z/nZ × Z/nZ[16, Corollary 6.4]. W have the following result.
Lemma 2.1 we conclude that h k ≡ C k θ Z mod p.