Supersingular zeros of divisor polynomials of elliptic curves of prime conductor
 Matija Kazalicki^{1}Email author and
 Daniel Kohen^{2, 3}
DOI: 10.1186/s4068701700998
© The Author(s) 2017
Received: 6 December 2016
Accepted: 21 February 2017
Published: 8 May 2017
Abstract
For a prime number p, we study the zeros modulo p of divisor polynomials of rational elliptic curves E of conductor p. Ono (CBMS regional conference series in mathematics, 2003, vol 102, p. 118) made the observation that these zeros are often jinvariants of supersingular elliptic curves over \({\overline{{\mathbb {F}}_{p}}}\). We show that these supersingular zeros are in bijection with zeros modulo p of an associated quaternionic modular form \(v_E\). This allows us to prove that if the root number of E is \(1\) then all supersingular jinvariants of elliptic curves defined over \({\mathbb {F}}_{p}\) are zeros of the corresponding divisor polynomial. If the root number is 1, we study the discrepancy between rank 0 and higher rank elliptic curves, as in the latter case the amount of supersingular zeros in \({\mathbb {F}}_p\) seems to be larger. In order to partially explain this phenomenon, we conjecture that when E has positive rank the values of the coefficients of \(v_E\) corresponding to supersingular elliptic curves defined over \({\mathbb {F}}_p\) are even. We prove this conjecture in the case when the discriminant of E is positive, and obtain several other results that are of independent interest.
Keywords
Divisor polynomial Supersingular elliptic curves Brandt module1 Background
It is worth noting that the root number of \(E_{83}\) is \(1\). The behavior of the roots of the divisor polynomial is explained by the following theorem.
Theorem 1
Let \(E/\mathbb {Q}\) be an elliptic curve of prime conductor p with root number \(1\), and let \(F(F_E,x)\) be the corresponding divisor polynomial. If \(j\in {\mathbb {F}}_{p}\) is a supersingular jinvariant mod p, then \(F(F_E,j)\equiv 0 \pmod {p}\).
Figure 1 shows the graph of the function \(\frac{N_p(E)}{s_p}\) where E ranges over all elliptic curves of root number 1 and conductor p where \(p<10{,}000\). The elliptic curves of rank zero (158 of them) are colored in blue, while the elliptic curves of rank two (59 of them) are colored in red.
It would be interesting to understand these data. In particular,
 1.
Why are there “so many” \({\mathbb {F}}_{p}\)supersingular zeros?
 2.
How can we explain the difference between rank 0 and rank 2 curves?
 3.
What about the outlying rank 0 curves (e.g., of conductor \(p=4283\) and \(p=5303\)) with the “large” number of zeros?
Remark
It seems that there is no obvious connection between the number of \(\mathbb {F}_{p^2}\)supersingular zeros of the divisor polynomial \(F(F_E,x)\) and the rank of elliptic curve E.
The key idea to study these questions is to show (following [13]) how to associate with \(F_E\) a modular form \(v_E\) on the quaternion algebra B over \(\mathbb {Q}\) ramified at p and \(\infty \). Such modular form is a function on the (finite) set of isomorphism classes of supersingular elliptic curves over \({\overline{{\mathbb {F}}_{p}}}\). In order to explain this precisely, we combine the expositions from [3, 4].
Theorem 2
This theorem allows us to give a more explicit description of the supersingular zeros of the divisor polynomial. Furthermore, it enables us to obtain computational data in a much more efficient manner. The proof of Theorems 1 and 2 will be the main goal of Sect. 2. In order to prove them, we will use both Serre’s and Katz’s theory of modular forms modulo p and the modular forms introduced in [13].
Now, let \(D_E\) be the congruence number of \(f_E\), i.e., the largest integer such that there exists a weight two cusp for on \(\Gamma _0(p)\), with integral coefficients, which is orthogonal to \(f_E\) with respect to the Petersson inner product and congruent to \(f_E\) modulo \(D_E\). The congruence number is closely related to \(\deg \phi _{f_E}\), the modular degree of \(f_E\), which is the degree of the minimal parametrization \(\phi _{f_E}:X_0(p)\rightarrow E'\) of the strong Weil elliptic curve \(E'/\mathbb {Q}\) associated with \(f_E\) (\(E'\) is isogenous to E but they may not be equal). In general, \(\deg \phi _{f_E}D_E\), and if the conductor of E is prime, we have that \(\deg \phi _{f_E}=D_E\) (see [1]).
We have the following theorem due to Mestre [7, Theorem 3].
Theorem 3
Conjecture 1
If E is an elliptic curve of prime conductor p, root number 1, and \(rank(E)>0\), then \(v_E(e_i)\) is an even number for all \(e_i\) with \(j(E_i)\in {\mathbb {F}}_{p}\)
While this is true for all 59 rank 2 curves we observed, it holds for 35 out of 158 rank 0 curves. This explains in a way a difference in the number of \({\mathbb {F}}_{p}\)supersingular zeros between rank 0 and rank 2 curves (Question 2), since, heuristically, it seems more likely for a number to be zero if we know it is even (especially in light of Theorem 3 which suggests that the numbers \(v_E(e_i)\) are small).
Thirtytwo out of 35 elliptic curves of rank 0 for which the conclusion of Conjecture 1 holds (the remaining three curves have conductors \(p=571, 6451\) and 8747) are distinguished from the other rank 0 curves by the fact that their set of real points \(E(\mathbb {R})\) is not connected (i.e., E has positive discriminant). In general, we have the following theorem, which will be the subject of Sect. 3.
Theorem 4
 1.
E has positive discriminant
 2.
E has no rational point of order 2,
Note that this gives a partial answer to Question 3 since, for example, all outlying elliptic curves of rank 0 for which \(\frac{N_p}{s_p}>0.5\) have positive discriminant and no rational point of order 2.
Note that among 59 rank 2 curves, for 25 of them \(E(\mathbb {R})\) is not connected (and have no rational point of order 2). For the rest of the rank 2 elliptic curves, we do not have an explanation of why they satisfy the conjecture.
Lastly, in the final section we will show how the Gross–Waldspurger formula might answer Question 2. More precisely, we will show that the quaternion modular form \(v_E\) associated with an elliptic curve E of rank 2 must be orthogonal to divisors arising from optimal embeddings of certain imaginary quadratic fields into maximal orders of the quaternion algebra B, leading to a larger amount of supersingular zeros.
2 Proof of the main theorems
2.1 Katz’s modular forms
We will recall the definition of modular forms given by Katz [5].
Definition 5
 1.
\(g({\tilde{E}}/R,\omega )\) depends only on the Risomorphism class of \(({\tilde{E}}/R,\omega )\).
 2.For any \(\lambda \in R^{\times }\),$$\begin{aligned} g({\tilde{E}}/R,\lambda \omega )=\lambda ^{k} g({\tilde{E}}/R,\omega ) . \end{aligned}$$
 3.
\(g({\tilde{E}}/R,\omega )\) commutes with base change by morphisms of \(R_{0}\)algebras.
Remark
The reader should notice that the notations used here are not the same as the ones used by Katz.
In the rest of the article, we will only consider the case when \(R_{0}={\overline{{\mathbb {F}}_{p}}}\), for \(p \ge 5\) a prime number.
In [10, 11], Serre considers the space of modular forms modulo p of weight k and level 1 as the space consisting of all elements of \({\overline{{\mathbb {F}}_{p}}}[[q]]\) that are the reduction modulo p of the qexpansions of elements in \(M_{k}\) that have pinteger coefficients. The following proposition shows that under mild assumptions, this definition agrees with the previous definition.
Proposition 6
Example

\(E_{4}({\tilde{E}}/{\overline{{\mathbb {F}}_{p}}},\omega _{can}):= c_{4}\) defines an element in \(M({\overline{{\mathbb {F}}_{p}}},4,1)\) whose qexpansion is the same as the reduction modulo p of the classical Eisenstein series \(E_{4}\).

\(E_{6}({\tilde{E}}/{\overline{{\mathbb {F}}_{p}}},\omega _{can}):= c_{6}\) defines an element in \(M({\overline{{\mathbb {F}}_{p}}},6,1)\) whose qexpansion is the same as the reduction modulo p of the classical Eisenstein series \(E_{6}\).

\(\Delta ({\tilde{E}}/{\overline{{\mathbb {F}}_{p}}},\omega _{can}):= \frac{c_{4}^3c_{6}^2}{1728}=\Delta ({\tilde{E}})\) defines an element in \(M({\overline{{\mathbb {F}}_{p}}},12,1)\) whose qexpansion is the same as the the reduction modulo p of the classical cusp form \(\Delta \).

\(j({\tilde{E}}/{\overline{{\mathbb {F}}_{p}}},\omega _{can}):= \frac{c_{4}^3}{\Delta }=j({\tilde{E}})\) defines an element in \(\mathcal {M}({\overline{{\mathbb {F}}_{p}}},0,1)\) whose qexpansion is the same as the the reduction modulo p of the classical jinvariant.
Proposition 7

\(\Delta ({\tilde{E}},\omega )\) never vanishes.

\( E_{4}({\tilde{E}},\omega )\) vanishes if and only if \(j({\tilde{E}})=0\).

\(E_{6}({\tilde{E}},\omega )\) vanishes if and only if \(j({\tilde{E}})=1728\).

\(j(({\tilde{E}},\omega ))=j({\tilde{E}})\), i.e., it only depends on the isomorphism class of \({\tilde{E}}\).
Proof
If we evaluate \(\Delta ({\tilde{E}}, \omega _{can})\), we recover the discriminant of \({\tilde{E}}\). This is nonzero as, by definition, an elliptic curve is nonsingular. The remaining statements are analogous. \(\square \)
Now we have the ingredients to prove the following proposition that relates the zeros of the divisor polynomial of E with the zeros of the modular form \(F_E\) modulo p.
Proposition 8
Proof
2.2 The spaces \(S({\overline{{\mathbb {F}}_{p}}},k,1)\)
Following [13], we introduce a definition:
Definition 9
\({S}({\overline{{\mathbb {F}}_{p}}},k,1)\) is the space of rules g that assign to every pair \(({\tilde{E}}/{\overline{{\mathbb {F}}_{p}}}, \omega )\), where \({\tilde{E}}\) is a supersingular elliptic curve and \(\omega \) is a nowhere vanishing differential on \({\tilde{E}}\), an element \(g({\tilde{E}}/{\overline{{\mathbb {F}}_{p}}},\omega ) \in {\overline{{\mathbb {F}}_{p}}}\) that satisfies the same properties as in Definition 5.
Definition 10
Proposition 11
We have a natural inclusion \(M({\overline{{\mathbb {F}}_{p}}},k,1) \subset {S}({\overline{{\mathbb {F}}_{p}}},k,1)\). If \(g \in M({\overline{{\mathbb {F}}_{p}}},k,1)\) is an eigenform for the Hecke operators \(T_{\ell }\) (\(\ell \ne p\)) with eigenvalues \(\lambda _{\ell } \in {\overline{{\mathbb {F}}_{p}}}\), then the image of g in \( {S}({\overline{{\mathbb {F}}_{p}}},k,1)\) is an eigenform for the Hecke operators with the same eigenvalues \(\lambda _{\ell }\).
Proof
This is clear from the definitions. \(\square \)
We have the following proposition that allows us to shift from weight \(p+1\) to weight 0.
Proposition 12
If we consider the isobaric polynomials A, B such that \(A(E_4,E_6)=E_{p1}\) and \(B(E_4,E_6)=E_{p+1}\), the reductions \({\tilde{A}}\), \({\tilde{B}}\) have no common factor ([10, Corollary 1 of Theorem 5]). Since \(E_{p1}\) vanishes at supersingular elliptic curves, we obtain that \(E_{p+1}\) does not vanish at supersingular elliptic curves over \({\overline{{\mathbb {F}}_{p}}}\).
Proposition 13
Finally, we state a proposition that will be useful later.
Proposition 14
The element \(\overline{F_{E}} \in {S}({\overline{{\mathbb {F}}_{p}}},0,1)[1]\) has the same eigenvalues for \(T_{\ell }\) (\(\ell \ne p\)) as \(F_{E}\). In addition, it has the same eigenvalues modulo p as \(f_{E}\).
2.3 Modular forms on quaternion algebras
This pairing identifies \({\mathcal {X}}^{*}=Hom(\mathcal {X},\mathbb {Z})\) with the subgroup of \(\mathcal {X}\otimes \mathbb {Q}\) with basis \(e^{*}_{i}=\frac{e_{i}}{w_{i}}\).
Proposition 15
Proposition 16
\(\overline{F_{E}}^{*}=\sum \overline{F_{E}}( e_i) e^{*}_{i}= \sum \overline{F_{E}}( e_i)(1/w_{i}) e_{i}\) and \(v_{E}=\sum {v_{E}}( e_i) e_{i}\) have the same eigenvalues modulo p for the Hecke operators \(t_{\ell }\) (\(\ell \ne p\)).
Proof
By Proposition 14, \(\overline{F_{E}}\) has the same eigenvalues as \(F_{E}\) for \(T_\ell \) (\(\ell \ne p\)), but with the action twisted, that is, \(T_{\ell } \overline{F_{E}}= \ell \lambda _{\ell } \overline{F_{E}}\). Since the pairing defining the duality is Hecke linear and by the above remarks, we must have \(t_{\ell } \overline{F_{E}}=\lambda _{\ell } \overline{F_{E}}\), which has the same eigenvalues modulo p as \(v_{E}\). \(\square \)
Corollary 17
\(\overline{F_{E}}( e_i) \equiv 0 \bmod {p} \iff {v_{E}}(e_i) \equiv 0 \bmod {p}\).
Proof
The forms \(\overline{F_{E}}^{*}=\sum \overline{F_{E}}( e_i)(1/w_{i}) e_{i}\) and \(v_{E}=\sum {v_{E}}( e_i) e_{i}\) have the same eigenvalues for \(T_{\ell }\) (\(\ell \ne p\)) by Proposition 16. By the work of Emerton [3, Theorem 0.5 and Theorem 1.14], we have the multiplicity one property for \(\mathcal {X}\) modulo p, since p is a prime different from 2.
Therefore, up to a nonzero scaling, the coefficients of these two quaternion modular forms agree modulo p. Finally, noting that the \(w_{i}\) are not divisible by p, the result follows.
\(\square \)
Now we are in position to prove Theorem 2.
Proof of Theorem 2
Let \(S_p \subset \{1,\ldots , n\}\) be a subset of indices such that \(i \in S_p\) if and only if \(j(E_i)\in {\mathbb {F}}_{p}\) (hence \(\#S_p=s_p\)). For \(i \in \{1,\ldots , n\}\) let \({\bar{i}}\) be the unique element of \(\{1,\ldots , n\}\) such that \(E_i^p \cong E_{{\bar{i}}}\). Note that, \({\bar{i}}=i\) if and only if \(i \in S_p\).
Proposition 18
Now we finish the section with the proof of Theorem 1.
Proof of Theorem 1
Let \(E_{i}\) be a supersingular elliptic curve with \(j(E_{i}) \in {\mathbb {F}}_{p}\). The operator \(t_{p}\) acts as \(W_{p}\) on \(M_2\), and since the elliptic curve has root number \(1\), we get that \(t_{p}\) acts as \(1\). By Proposition 18, we have that \(t_{p}e_i=e_i\), hence \({v_{E}}( e_i)=0\), and the result follows from Theorem 2. \(\square \)
3 Proof of Theorem 4
3.1 Some basic properties of Brandt matrices
Alternatively, \(M_{ij} \cong Hom_{\overline{{\mathbb {F}}_{p}}}{(E_i, E_j)}\) and \(B_{ij}(m)\) is equal to the number of subgroup schemes C of order m in \(E_i\) such that \(E_i/C \simeq E_j\) [4, Proposition 2.3].
Following the discussion before 18, we can state the following results.
Proposition 19
We have the equality \(v_E(e_j)=\lambda _{p} v_E(e_{{\bar{j}}})\). In particular, \(v_E(e_j)\) and \(v_E(e_{{\bar{j}}})\) have the same parity.
Proof
The first assertion follows from the fact that \(\sum _{i} v_E(e_i)e_{i}\) is an eigenvector for the action of \(t_p\) and Proposition 18. The last assertion follows from the fact that \(\lambda _p= \pm 1\).
\(\square \)
Proposition 20
Proof
Proposition 21
Proof
 (a)
\(i=j\) (i.e., \(M_{ij}=R_i\))
Direct calculation shows that the vectors 1 and \(\phi _i\) span the eigenspace with eigenvalue 1. The eigenspace with eigenvalue \(1\) is the orthogonal complement of \(\phi _i\) in the trace zero subspace \(B^{0}\) of B (since for \(f \in B^{0}\) we have \(f \perp \phi _i \iff \mathrm{Nr}(f+\phi _i)=\mathrm{Nr}(f)+\mathrm{Nr}(\phi _i) \iff f \hat{\phi _i}+{\hat{f}} \phi _i=0 \iff f\phi _i = \phi _i f \iff \Theta (f)=f\)).
 (b)
\(i \ne j\)
Let \(\phi _{ji}:= \phi _j \phi _i\). The matrix representations of \(\Theta \) in the invariant subspaces generated by \(\{ 1, \phi _{ji}\}\) and \(\{ \phi _i, \phi _j \}\) are equal to \(\left( {\begin{matrix}0&{}p\\ 1/p&{}0 \end{matrix}} \right) \) and \(\left( {\begin{matrix}0&{}1\\ 1&{}0 \end{matrix}} \right) \); hence, \(\Theta \) has two eigenspaces of dimension 2 with eigenvalues \(1\) and 1.
Assume that b is an eigenvector of \(\Theta \). Then \(b \in V_+\) or \(b\in V_\). In any case, since \(l=q(b)\), it follows that l is representable by a binary quadratic form of discriminant \(p\) or \(4p\) which is not possible since \(\left( \frac{p}{l}\right) =\left( \frac{4p}{l}\right) = 1\). \(\square \)
3.2 Fourier coefficients of \(f_E(\tau )\) mod 2
Proposition 22
Let \(E/\mathbb {Q}\) be an elliptic curve of prime conductor p such that E has positive discriminant and E has no rational point of order 2. There is a positive proportion of odd primes \(\ell \) such that \(\left( \frac{p}{\ell }\right) = 1\) and \(\lambda _{\ell }\equiv 1 \pmod {2}\), where \(f_E(\tau )=\sum \lambda _{n} q^n\) is the qexpansion of \(f_E(\tau )\).
Proof
If \(\ell \equiv 3 \pmod {4}\) then \(\left( \frac{p}{\ell }\right) = 1\) implies that \(\ell \) splits in F. If, in addition, \(\ell \) does not split completely in K, then the order of \(Frob_\ell \) is 3 and \(\lambda _{\ell }\) is odd. There is a positive proportion of such primes \(\ell \) since by Chebotarev density theorem (applied to the field \(L=\mathbb {Q}(\sqrt{1})K\)) there is a positive proportion of primes \(\ell \) which are inert in \(\mathbb {Q}(\sqrt{1})\), split in F and do not split completely in K. \(\square \)
3.3 Proof of Theorem 4
Proof
We are going to give a different proof of Theorem 4 under the additional assumption that E is supersingular at 2. The idea is to use the results of Le Hung and Li [6] on level raising modulo 2 together with the multiplicity one mod 2 results from [3] to obtain mod 2 congruences between modular forms of the same level p, but with different signs of the Atkin–Lehner involution. We hope that by extending these ideas to level \(2^r p\) one will be able to understand Conjecture 1 better.
Theorem 23
Let E be a rational elliptic curve of conductor p, without rational 2torsion and with positive discriminant. Suppose further that E is supersingular at 2 . Then, there exist a newform \(g \in S_{2}(\Gamma _{0}(p))\) and a prime \(\lambda \) above two in the field of coefficients of g such that \(f \equiv g \bmod {\lambda }\) and such that \(W_{p}\) acts as \(1\) on g.
Proof
We will verify the assumptions of Le Hung and Li [6, Theorem 2.9], starting with our elliptic curve E of prime conductor and in the scenario where we choose no primes as level raising primes (so we are looking for a congruence between level p newforms). As we explained before, the hypotheses imply that \({\rho }_{2}: G_{\mathbb {Q}} \rightarrow Gl_{2}(\mathbb {F}_{2})\) is surjective and the only quadratic subfield of \(\mathbb {Q}(E[2])\) is given by \(\mathbb {Q}(\sqrt{p})\). Therefore, the conductor of \({\rho }_{2}\) is p and it is not induced from \(\mathbb {Q}(i)\). Moreover, \({\rho }_{2}\) restricted to \(G_{\mathbb {Q}_{2}}\) is not trivial if E is supersingular at 2. Thus, we are in position to use the theorem and find a g as in the statement, because, since \(\Delta (E)>0\), we can prescribe the sign of the Atkin–Lehner involution at p. \(\square \)
Now we are in condition to give another proof of Theorem 4, under the additional assumption that E is supersingular at 2. Since g has eigenvalue \(1\) for the Atkin–Lehner operator, we have that \(v_g(e_i)=0\) for every \(i \in S_{p}\) by Proposition 18. As we did earlier, Theorem 0.5 and Theorem 1.14 in [3] imply, since E is supersingular at 2, that we have multiplicity one mod 2 in the \(f_E\)isotypical component in \(\mathcal {X}\); therefore, \(v_E(e_{i})\) is even for \(i \in S_{p}\) as we wanted to show.
4 Further remarks
Proposition 24
Acknowledgements
We would like to thank the ICTP and the ICERM for provding the oportunity of working on this project. We would like to thank A. Pacetti for his comments on an early version of this draft. MK acknowledges support from the QuantiXLie Center of Excellence. DK was partially supported by a CONICET doctoral fellowship.
Declarations
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Authors’ Affiliations
References
 Agashe, A., Ribet, K.A., Stein, W.A.: The modular degree, congruence primes, and multiplicity one. In: Number Theory, Analysis and Geometry, pp. 19–49. Springer, New York (2012)
 Edixhoven, B.: Serre’s conjecture. In: Modular Forms and Fermat’s Last Theorem, pp. 209–242. Springer, New York (1997)
 Emerton, M.: Supersingular elliptic curves, theta series and weight two modular forms. J. Am. Math. Soc. 15(3), 671–714 (2002)MathSciNetView ArticleMATHGoogle Scholar
 Gross, B.: Heights and the special values of Lseries. Number theory. In: CMS Conference Proceedings, vol. 7, pp. 115–187 (1987)
 Katz, N. M.: \(p\)adic properties of modular schemes and modular forms. In: Modular Functions of One Variable, III (Proceedings of the International Summer School, University of Antwerp 1972. Lecture Notes in Mathematics, vol. 350, pp. 60–190 (1973)
 Le Hung, B., Li, C.: Level raising mod \(2\) and arbitrary \(2\)Selmer ranks. Compos. Math. 152(8), 1576–1608 (2016)MathSciNetView ArticleGoogle Scholar
 Mestre, J. F.: La méthode des graphes. Exemples et applications. In: Proceedings of the International Conference on Class Numbers and Fundamental Units of Algebraic Number Fields, pp. 217–242. Nagoya University, Katata (1986)
 Ono, K.: The web of modularity: arithmetic of the coefficients of modular forms and qseries. In: CBMS Regional Conference Series in Mathematics, vol. 102 p. 118 (2003)
 Robert, G.: Congruences entre séries d’Eisenstein, dans le cas supersingulier. Invent. Math. 61(2), 103–158 (1980)MathSciNetView ArticleMATHGoogle Scholar
 Serre, J.P.: Congruences et formes modulaires [d’après H. P. F. SwinnertonDyer]. In: Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416. Lecture Notes in Mathematics, vol. 317, pp. 319–338 (1973)
 Serre, JP.: Formes modulaires et fonctions zêta \(p\)adiques. In: Modular Functions of One Variable. In: III Proceedings of the International Summer School, University of Antwerp 1972. Lecture Notes in Mathematics, vol. 350, 191–268 (1973)
 Serre, J.P.: Sur les représentations modulaires de degré 2 de \(Gal (Q^{} / Q )\). Duke Math. J. 54(1), 179–230 (1987)MathSciNetView ArticleGoogle Scholar
 Serre, J.P.: Two letters on quaternions and modular forms (mod p). Isr. J. Math. 95, 281–299 (1996)MathSciNetView ArticleMATHGoogle Scholar
 Watkins, M.: Explicit lower bounds on the modular degree of an elliptic curve. Mathematics Faculty Scholarship. Paper 104 (2004)
 Zagier, D.: Modular parametrizations of elliptic curves. Can. Math. Bull. 28, 372–384 (1985)MathSciNetView ArticleMATHGoogle Scholar