Traces of Hecke operators in level 1 and Gaussian hypergeometric functions

We provide formulas for traces of p-th Hecke operators in level 1 in terms of values of finite field 2F1-hypergeometric functions, extending previous work of the author to all odd primes p, instead of only those p=1 (mod 12). We first give a general level 1 trace formula in terms of the trace of Frobenius on a family of elliptic curves, and then we draw on recent work of Lennon to produce level 1 trace formulas in terms of hypergeometric functions for all primes p>3.


Introduction
In recent years, relationships between traces of Hecke operators and counting points on families of varieties have been explored. For example, Ahlgren and Ono [2] described traces of p th Hecke operators in weight 4 and level 8 in terms of the number of F p -points on a Calabi-Yau threefold, while in [1], Ahlgren related traces of Hecke operators in weight 6 and level 4 to counting F p -points on the Legendre family of elliptic curves. The level 2 formula (for all weights) was made explicit in terms of the number of F p -points on a family of elliptic curves by Frechette, Ono, and Papanikolas in [5]. In [7], the author considered the level 1 case and provided a formula in terms of the number of F p -points on a one parameter family of elliptic curves for primes p ≡ 1 (mod 12). Most recently, Lennon [12] considered the levels 3 and 9 scenarios. Earlier work of Ihara [10] and Birch [4] gave reason to believe such formulas were possible.
Interestingly, these trace formulas also have a link to finite field hypergeometric functions introduced by Greene in the 1980s [8]. Various authors [3,5,12] have used relations between values of Greene's hypergeometric functions and counting F ppoints on varieties to produce trace formulas in terms of hypergeometric functions. In [7,Thm. 1.2], the author proved an explicit relationship between counting F ppoints on a one-parameter family of elliptic curves and the values of a particular 2 F 1 function over F p , which led to a level 1 trace formula in terms of hypergeometric functions. However, this formula was only proved for primes p ≡ 1 (mod 12). Recently, Lennon [11, Thms. 1.1 and 2.1] has removed this restriction on the congruence class of p to produce formulas that relate #E(F q ) to values of a 2 F 1 function over F q for any q = p e where q ≡ 1 (mod 12).
In this paper, we provide a level 1 trace formula that holds for all p > 3. Then, we use Lennon's result to produce formulas for traces of Hecke operators in level 1 in terms of finite field hypergeometric functions.

Statement of Main Results
Let p > 3 be prime and let k ≥ 2 be an even integer.
Then letting x + y = s and xy = p gives rise to polynomials G k (s, p) = F k (x, y). These polynomials can be written alternatively as Throughout, results will depend on the congruence class of p mod 12. As such, we set up some notation for various congruence classes of p to be used throughout the remainder of the paper. Whenever p ≡ 1 (mod 4), we let a, b ∈ Z be such that p = a 2 + b 2 and a + bi ≡ 1 (2 + 2i) in Z[i]. In that case, we define Similarly, whenever p ≡ 1 (mod 3), we let c, d ∈ Z be such that where ω = e 2πi/3 . This this case, we define We consider a one-parameter family of elliptic curves having j-invariant 1728 t . Specifically, for t ∈ F p \{0, 1}, we let Let a(t, p) denote the trace of the Frobenius endomorphism on E t . In particular, for t = 0, 1, we have a(t, p) = p + 1 − #E t (F p ). Let Γ = SL 2 (Z) and let M k and S k , respectively, denote the spaces of modular forms and cusp forms of weight k for Γ. Further, let Tr k (Γ, p) denote the trace of the Hecke operator T k (p) on S k . Our first main result completely classifies the traces of cusp forms in level 1: Theorem 2.1. Let p > 3 be prime. Then for even k ≥ 4, Next, we move to results which link these traces of Hecke operators in level 1 with hypergeometric functions over finite fields. We begin with some preliminaries. Let p be a prime and let q = p e . Let F × q denote the group of all multiplicative characters on F × q . We extend χ ∈ F × q to all of F q by setting χ(0) = 0. We let ε denote the trivial character. For A, B ∈ F × p , let J(A, B) denote the usual Jacobi symbol and define Greene defined hypergeometric functions over F q in the following way: Defn. 3.10). If n is a positive integer, x ∈ F q , and A 0 , A 1 , . . . , A n , In [7, Thm. 1.2], the author proved a formula giving an explicit relationship between a(t, p) and a 2 F 1 hypergeometric function over F p , but required that p ≡ 1 (mod 12). In this case, the result of Theorem 2.1 can be rewritten to be in terms of a hypergeometric function over F p . However, [7] did not address the other classes of primes mod 12. Notice that either p ≡ 1 (mod 12) or, if not, then p 2 ≡ 1 (mod 12). With this in mind, for the remainder of the paper we define q = p e(p) , where We consider the same family of elliptic curves E t , as defined in (4), but now over F q , and with a(t, q) = q + 1 − #E t (F q ). Thanks to Lennon's results [11] and an inverse pair given in [13], we can now describe the traces of Hecke operators in level 1 in terms of a 2 F 1 function over F q for the other classes of primes mod 12: Theorem 2.3. Let p > 3 be prime such that p = 5, 7, 11 (mod 12) with q = p e(p) and T a generator of F × q . Let k ≥ 4 be even and m = k where λ(k, p) is as in Theorem 2.1.
Our final result is a generalization of [7, Thm. 1.4], giving a recursive formula for traces of Hecke operators in level 1 in terms of hypergeometric functions, now for all primes p > 3: Theorem 2.4. Let p > 3 be prime, and q = p e(p) . Let k ≥ 4 be even, and and λ(k, p) is as in Theorem 2.1.

Proof of Theorem 2.1
To prove Theorem 2.1, we begin with Hijikata's version of the Eichler-Selberg trace formula [9]. The statement of this theorem requires some notation.
and 0 otherwise, and where we classify integers s with s 2 − 4p < 0 by some positive integer ℓ and square-free integer m via To link Theorem 3.1 to a(t, p), we need to consider all isomorphism classes of elliptic curves over F p . If E is any elliptic curve defined over F p , let a(E) = p + 1 − #E(F p ). Additionally, for a perfect field K, we define We first address the cases j(E) = 1728 and j(E) = 0. Lemma 3.2. Let p be an odd prime. Whenever p ≡ 1 (mod 4), define a, b ∈ Z be such that p = a 2 + b 2 and a + bi ≡ 1 (2 + 2i) in Z[i]. Then, for n ≥ 2 even, Proof. The case p ≡ 1 (mod 4) was proved by the author in [ Proof. The case p ≡ 1 (mod 3) was proved by the author in [6, Lemma IV.3.5]. If p ≡ 2 (mod 3) and [E] Fp ∈ Ell Fp , then #E(F p ) = p + 1, so a(E) = 0.
The proof of Theorem 2.1 proceeds along the same line as the proof of the p ≡ 1 (mod 12) case proved by the author in [7]. In particular, we begin with the following extension of [7, Lemma 5.3].
Lemma 3.4. Let p > 3 be prime. Then for n ≥ 2 even, Proof. The proof for primes p ≡ 1 (mod 12) is provided in [7,Lemma 5.3]. It can be adapted to hold for all p > 3 once one verifies that the following two identities remain true: a(E) n First consider (7). The proof given in [7] holds for all primes p ≡ 1 (mod 4). In light of Lemma 3.2, we must verify that if p ≡ 3 (mod 4), then We verify this by proving that no s, f exist to contribute to the sums. For, suppose s, f ∈ Z such that 0 < s < 2 √ p and s 2 − 4p f 2 = −4. Then 4|s 2 , so s must be even.
Substituting s = 2r and rearranging gives r 2 + f 2 = p, which is not possible since p ≡ 3 (mod 4). This verifies (7) for the remaining primes. We handle (8) in a similar way. The proof in [7] verifies the equation for p ≡ 1 (mod 3). Keeping in mind Lemma 3.3, we must prove that if p ≡ 2 (mod 3), then Suppose then that we have s, f ∈ Z such that 0 < s < 2 √ p and Then 4p = 3f 2 + s 2 and hence p ≡ 2 ≡ s 2 (mod 3). However, since 2 3 = −1, this is impossible. This verifies (8) for the remaining primes, and completes the proof of the lemma.
The following proposition generalizes [7,Prop. 5.4] by removing the restriction on the congruence class of p (mod 12).
Proposition 3.5. Let p > 3 be prime and n ≥ 2 be even. Then Regardless of the congruence class of p (mod 12), the first sum in the last line above can still be written in terms of class numbers by combining Hasse's theorem with a theorem of Schoof [14,Thm. 4.6]. This results in by Lemma 3.4. One now applies Lemmas 3.2 and 3.3 in each appropriate congruence class to obtain the result.
With the these tools in place, we now complete the proof of Theorem 2.1.
Proof of Theorem 2.1. The proof proceeds in a similar fashion to the author's proof of the p ≡ 1 (mod 12) case in [7,Thm. 1.3], with a few modifications. We still begin with an application of Theorem 3.1 and then substitute the definition of G k (s, p). This gives Now, notice that the k = 2 case of Theorem 3.1 gives Substituting and applying Proposition 3.5 with n = k − 2j − 2 gives To complete the proof, we distribute the copies of (−p) k 2 −1 to the three summations in a specific way. Notice that First, since G 2 = 1, we see that (11) − (−p) a(t, p), p).
A straightforward calculation for each of the congruence classes of p (mod 12) verifies

Trace formulas in terms of hypergeometric functions
We now prove Theorems 2.3 and 2.4. As mentioned before, one essential tool is a theorem of Lennon, which writes the trace of Frobenius of any elliptic curve in Weierstrass form in terms of a finite field hypergeometric function: Theorem 4.1. [11, Thm. 2.1] Let q = p e , where p > 3 is prime and q ≡ 1 (mod 12). Let E : y 2 = x 3 + ax + b be an elliptic curve over F q in Weierstrass form with j(E) = 0, 1728. Then the trace of the Frobenius map on E can be expressed as We now specify this theorem to our family of curves.
Corollary 4.2. Let p > 3 be prime and q = p e(p) , where e(p) is defined as in (6). Then To prove this corollary, we require a transformation law proved by Greene: Proof of Cor. 4.2. After putting E t into Weierstrass form, we have a = b = −27

4(1−t)
in Theorem 4.1. Then a 3 27 = −3 6 4 3 (1−t) 3 and − 27b 2 4a 3 = 1 − t. Combining these simplifications with Greene's theorem above gives (12) a(t, q) = −qT Now, using multiplicativity and the fact that T has order q − 1, we have is its own inverse and q ≡ 1 (mod 12). The proof is completed by making this substitution for T   We require two more tools to prove our trace theorems. First, note that whenever e(p) = 2 (i.e. q = p 2 ), Theorem 4.1 relates a(t, p 2 ) to a hypergeometric function over F p 2 . Even though our trace formula Theorem 2.1 is in terms of a(t, p), we can still gain new information, since (13) a(t, p) 2 = a(t, p 2 ) + 2p.
The last tool is an inverse pair given in [13]. As in the statement of Theorems 2.3 and 2.4, we let m = k 2 − 1 and also define H m (x) := m i=0 m+i m−i x i . Then, as in [7], notice Consider the inverse pair [13, p. 67] given by Applying this to the definition of H m , we see By combining (14) with the choice x = −s 2 p , we have We may now prove Theorems 2.3 and 2.4.
Proof of Theorem 2.3. Recall that in the statement of this theorem, p ≡ 5, 7, 11 (mod 12), so q = p 2 . By (14) and (13), we have We isolate G 2m+2 (s, p) and take s = a(t, p). Substituting into (17)  The final statement of the theorem combines these two cases together, completing our proof.