2 Background and conventions

This section establishes some relevant background on Galois representations and L-functions of elliptic curves and Dirichlet characters, as well as some notational conventions that might be deemed less standard in the literature.

For a primitive $ n $ th root of unity $ \zeta _n $ , the ring of integers of the cyclotomic field $ \mathbb {Q}(\zeta _n) $ will be denoted $ \mathbb {Z}[\zeta _n] $ , and denote its norm map by $ \operatorname {Nm}_n : \mathbb {Q}(\zeta _n) \to \mathbb {Q} $ . The ring of integers of its maximal totally real subfield $ \mathbb {Q}(\zeta _n)^+ $ will be denoted $ \mathbb {Z}[\zeta _n]^+ $ , and denote its norm map by $ \operatorname {Nm}_n^+ : \mathbb {Q}(\zeta _n)^+ \to \mathbb {Q} $ . The isomorphism in class field theory from the unit group $ (\mathbb {Z} / n)^\times $ of $ \mathbb {Z} $ modulo $ n $ to the cyclotomic Galois group $ \operatorname {Gal}(\mathbb {Q}(\zeta _n) / \mathbb {Q}) $ will be given by $ a \mapsto (\zeta _n \mapsto \zeta _n^a) $ , which identifies Dirichlet characters of modulus $ n $ with Artin representations that factor through $ \mathbb {Q}(\zeta _n) $ .

Denote the two-dimensional special, general, and projective linear group functors by $ \operatorname {SL}_2 $ , $ \operatorname {GL}_2 $ , and $ \operatorname {PGL}_2 $ . For a matrix $ M $ in such a matrix group, its trace will be denoted $ \operatorname {tr}(M) $ and its determinant will be denoted $ \det (M) $ . For a prime $ \ell $ , the conjugacy classes of $ \operatorname {SL}_2(\mathbb {Z} / \ell ) $ can be grouped by their traces as follows [Reference Bonnafé3, Table 1.1 and Exercise 1.4].

• • There are three conjugacy classes of trace $ 2 $ represented by $ (\begin {smallmatrix} 1 & z \\ 0 & 1 \end {smallmatrix}) $ for $ z \in \mathbb {F}_\ell $ , one for each of the three Legendre symbols $ \left (\tfrac {z}{\ell }\right ) \in \{0, \pm 1\} $ , each of which has cardinality equal to $ ((\ell ^2 - 1) / 2)^{\left |\left (\frac {z}{\ell }\right )\right |} $ and has elements of order equal to $ \ell ^{\left |\left (\frac {z}{\ell }\right )\right |} $ .

• • There are three conjugacy classes of trace $ \ell - 2 $ represented by $ (\begin {smallmatrix} -1 & z \\ 0 & -1 \end {smallmatrix}) $ for $ z \in \mathbb {F}_\ell $ , one for each of the three Legendre symbols $ \left (\tfrac {z}{\ell }\right ) \in \{0, \pm 1\} $ , each of which has cardinality equal to $ ((\ell ^2 - 1) / 2)^{\left |\left (\frac {z}{\ell }\right )\right |} $ and has elements of order equal to $ 2\ell ^{\left |\left (\frac {z}{\ell }\right )\right |} $ .

• • There are $ (\ell - 3) / 2 $ conjugacy classes of trace $ x + x^{-1} $ represented by $ (\begin {smallmatrix} x & 0 \\ 0 & x^{-1} \end {smallmatrix}) $ for $ x \in \mathbb {F}_\ell ^\times \setminus \{\pm 1\} $ , one for each unordered pair $ \{x^{\pm 1}\} $ , each of which has cardinality equal to $ \ell (\ell + 1) $ and has elements of order equal to the order of $ x $ .

• • There are $ (\ell - 1) / 2 $ conjugacy classes of trace $ \xi + \xi ^\ell $ represented by

  $$ \begin{align*}\begin{pmatrix} \tfrac{1}{2}(\xi + \xi^\ell) & \tfrac{\zeta}{2}(\xi - \xi^\ell) \\ \tfrac{1}{2\zeta}(\xi - \xi^\ell) & \tfrac{1}{2}(\xi + \xi^\ell) \end{pmatrix}, \qquad \xi \in (\mathbb{F}_{\ell^2}^\times / \mathbb{F}_\ell^\times) \setminus \{\pm1\},\end{align*} $$
  where $ \zeta $ is a fixed element of $ \mathbb {F}_{\ell ^2}^\times $ satisfying $ \zeta + \zeta ^\ell = 0 $ , one for each pair $ \{\xi ^{\pm 1}\} $ , each of which has cardinality $ \ell (\ell - 1) $ and elements of order equal to the order of $ \xi $ .

This will be useful for Theorem 4.4 and Proposition 6.1.

Throughout, an elliptic curve will always refer to an elliptic curve $ E $ over $ \mathbb {Q} $ of conductor $ N $ , and any explicit example of an elliptic curve will be given by its Cremona label [Reference Cremona11, Table 1]. For a prime $ \ell $ , the $ \ell $ -adic Galois representation associated with the $ \ell $ -adic Tate module of $ E $ is denoted $ \rho _{E, \ell } $ , and its $ \ell $ -adic Galois image $ \operatorname {im}(\rho _{E, \ell }) $ will be given by its Rouse–Sutherland–Zureick-Brown label as a subgroup of $ \operatorname {GL}_2(\mathbb {Z}_\ell ) $ up to conjugacy [Reference Rouse, Sutherland and Zureick-Brown25, Section 2.4]. For any $ n \in \mathbb {N} $ , the projection of $ \rho _{E, \ell } $ onto $ \operatorname {GL}_2(\mathbb {Z} / \ell ^n) $ is denoted $ \overline {\rho _{E, \ell ^n}} $ , and its mod- $ \ell ^n $ Galois image $ \operatorname {im}(\overline {\rho _{E, \ell ^n}}) $ will be given by its Sutherland label as a subgroup of $ \operatorname {GL}_2(\mathbb {Z} / \ell ^n) $ up to conjugacy [Reference Sutherland31, Section 6.4]. Note that if $ \operatorname {Fr}_v $ is an arithmetic Frobenius at a prime $ v \ne \ell $ , then its trace is given by

Let $ \omega _E $ denote a global invariant differential on a minimal Weierstrass equation of $ E $ . Let $ X_0(N) $ denote the modular curve associated with the Hecke congruence subgroup $ \Gamma _0(N) $ of $ \operatorname {SL}_2(\mathbb {Z}) $ , and let $ S_2(N) $ denote the space of weight two cusp forms of level $ \Gamma _0(N) $ . By the modularity theorem, there is a surjective morphism $ \phi _E : X_0(N) \twoheadrightarrow E $ of minimal degree and an eigenform $ f_E \in S_2(N) $ with Fourier coefficients $ a_v(E) $ for each prime $ v \nmid N $ . These constructions define two differentials on $ X_0(N) $ , namely $ 2\pi if_E(z)\operatorname {d}\! z $ and the pullback $ \phi _E^*\omega _E $ of $ \omega _E $ by $ f_E $ , which are related by

$$ \begin{align*}\phi_E^*\omega_E = \pm c_0(E)2\pi if_E(z)\operatorname{d}\! z,\end{align*} $$

where $ c_0(E) $ is a positive integer called the Manin constant [Reference Edixhoven13, Proposition 2].

It is conjectured that $ c_0(E) = 1 $ when $ E $ is $ \Gamma _0(N) $ -optimal in its isogeny class, which was recently proven for semistable $ E $ [Reference Česnavičius10, Theorem 1.2], but it is possible that $ c_0(E) \ne 1 $ in general. Nevertheless, every modular parameterization by $ X_0(N) $ factors through a parameterization by the modular curve $ X_1(N) $ associated with the congruence subgroup $ \Gamma _1(N) $ of $ \operatorname {SL}_2(\mathbb {Z}) $ [Reference Stevens30, Theorem 1.9]. An analogous construction using $ X_1(N) $ yields the Manin constant $ c_1(E) $ , with the following important conjecture.

For a complex Galois representation $ \rho $ , its local Euler factor at a prime $ v $ is given by

where $ \rho ^{I_v} $ is the subrepresentation of $ \rho $ invariant under the inertia subgroup $ I_v $ at $ v $ . The L-function $ L(E, s) $ of $ E $ is then defined to be the infinite Euler product of $ L_v(\rho _{E, \ell }^\vee , v^{-s})^{-1} $ over all primes $ v $ , where $ \rho _{E, \ell }^\vee $ is the dual of the complex Galois representation associated with $ \rho _{E, \ell } \otimes _{\mathbb {Z}_\ell } \mathbb {Q}_\ell $ for some prime $ \ell \ne v $ . The modularity theorem says that $ L(E, s) $ is the Hecke L-function of $ f_E $ , so that its order of vanishing at $ s = 1 $ , and hence its leading term $ L^*(E, 1) $ , are both well-defined.

The Birch–Swinnerton-Dyer conjecture predicts this order of vanishing and its leading term in terms of arithmetic invariants as follows. Let $ \operatorname {tor}(E) $ and $ \operatorname {rk}(E) $ denote the torsion subgroup and the rank of the Mordell–Weil group $ E(\mathbb {Q}) $ respectively. Let $ \Omega (E) $ denote the real period given by $ \int _{E(\mathbb {R})} \omega _E $ , with orientation chosen such that $ \Omega (E)> 0 $ . Let $ \operatorname {Tam}(E) $ denote the Tamagawa number, given as the product of local Tamagawa numbers $ \operatorname {Tam}_v(E) $ over all primes $ v $ . Let $ \operatorname {Reg}(E) $ denote the elliptic regulator defined in terms of the Néron–Tate pairing $ \langle P, Q\rangle = \tfrac {1}{2}h_E(P + Q) - \tfrac {1}{2}h_E(P) - \tfrac {1}{2}h_E(Q) $ , where $ h_E $ is the canonical height on $ E $ . Finally, let denote the Tate–Shafarevich group, which is implicitly assumed to be finite in this article.

Conjecture 2.2 (Birch–Swinnerton-Dyer)

Let $ E $ be an elliptic curve. Then the order of vanishing of $ L(E, s) $ at $ s = 1 $ is equal to $ \operatorname {rk}(E) $ , and its leading term satisfies

Here, the left hand side is the modified L-value of $ E $ , which will be denoted $ \mathscr {L}(E) $ , and the right hand side is the Birch–Swinnerton-Dyer quotient of $ E $ , which will be denoted $ \operatorname {BSD}(E) $ . For the base change $ E / K $ of $ E $ to an extension $ K $ of $ \mathbb {Q} $ , there are analogous quantities $ \mathscr {L}(E / K) $ and $ \operatorname {BSD}(E / K) $ [Reference Dokchitser, Evans and Wiersema12, Section 1.5]. If $ \operatorname {ord}_\ell : \mathbb {Q} \to \mathbb {Z} \cup \{\infty \} $ denotes the $ \ell $ -adic valuation for some prime $ \ell $ , the conjecture that $ \operatorname {ord}_\ell (\mathscr {L}(E)) = \operatorname {ord}_\ell (\operatorname {BSD}(E)) $ is called the $ \ell $ -part of the Birch–Swinnerton-Dyer conjecture.

Throughout, a character will always refer to a nontrivial even primitive Dirichlet character $ \chi $ of order $ q> 1 $ and prime conductor $ p \nmid N $ , which automatically means that $ \chi (-1) = 1 $ and $ p \equiv 1 \mod q $ . The L-function $ L(E, \chi , s) $ of $ E $ twisted by $ \chi $ is defined to be the Euler product of $ L_v(\rho _{E, \ell }^\vee \otimes \overline {\chi }, v^{-s})^{-1} $ over all primes $ v $ , so that in particular $ \mathscr {L}(E, 1) = \mathscr {L}(E) $ . The modularity theorem says that $ L(E, \chi , s) $ is the Hecke L-function of $ f_E $ twisted by $ \chi $ [Reference Shimura28, Theorem 3.66], so that its order of vanishing at $ s = 1 $ , and hence its leading term $ L^*(E, \chi , 1) $ , are again well-defined. When $ L(E, \chi , 1) \ne 0 $ , Kato showed that $ \operatorname {rk}(E) = \operatorname {rk}(E / K) $ and is finite [Reference Kato16, Corollary 14.3]. The analogous modified twisted L-value is given by

where $ \tau (\chi ) $ is the Gauss sum of $ \chi $ .

3 Modular symbols

This section recalls some classical facts on modular symbols. Most arguments here are well-known since the time of Manin [Reference Manin22], with some recent results by Wiersema–Wuthrich [Reference Wiersema and Wuthrich33], but they are provided here for reference. Nevertheless, the main tool is the congruence in Corollary 3.7. Note that similar congruences were explored by Fearnley–Kisilevsky–Kuwata [Reference Fearnley, Kisilevsky and Kuwata14, Theorem 3.5], and are essentially equivalent to the equivariant Tamagawa number conjecture as shown by Bley [Reference Bley2, Section 2].

Let $ N \in \mathbb {N} $ . The congruence subgroup $ \Gamma _0(N) $ of $ \operatorname {SL}_2(\mathbb {Z}) $ acts on the extended upper half plane $ \mathscr {H} $ of $ \mathbb {C} $ by fractional linear transformations, and a smooth path between two points in the same $ \Gamma _0(N) $ -orbit projects onto a closed path in the quotient $ X_0(N) = \mathscr {H} / \Gamma _0(N) $ , which defines an integral homology class $ \gamma \in H_1(X_0(N), \mathbb {Z}) $ . This is independent of the smooth path chosen because $ \mathscr {H} $ is simply connected, and any integral homology class $ \gamma \in H_1(X_0(N), \mathbb {Z}) $ arises in such a way. On the other hand, any cusp form $ f \in S_2(N) $ induces a differential $ 2\pi if(z)\operatorname {d}\! z $ on $ X_0(N) $ , and integrating this over the closed path $ \gamma $ gives a complex number $ \int _\gamma 2\pi if(z)\operatorname {d}\! z $ called a modular symbol. A general definition for paths with arbitrary endpoints is given by Manin [Reference Manin22, Section 1.2], but for the purposes of this article, it suffices to consider the modular symbol associated with the path from $ 0 $ to cusps $ c \in \mathbb {Q} \cup \{\infty \} $ . When $ c $ is rational with denominator coprime to $ N $ , the image of any smooth path between $ 0 $ and $ c $ is closed [Reference Manin22, Proposition 2.2], so that it makes sense to write the modular symbol

The key example for $ f $ will be the normalized cuspidal eigenform $ f_E \in S_2(N) $ associated with an elliptic curve $ E $ of conductor $ N $ . In this case, it turns out that $ \mathscr {L}(E) $ , as well as $ \mathscr {L}(E, \chi ) $ for any character $ \chi $ of conductor coprime to $ N $ , can be written as sums of for some $ c \in \mathbb {Q} $ . Furthermore, the terms in these sums can be paired up in a way that guarantees integrality, using the following trick.

Lemma 3.1 Let $ c \in \mathbb {Q} $ with denominator coprime to some $ N \in \mathbb {N} $ . If $ f \in S_2(N) $ , then

$$ \begin{align*}\mu_f(c) + \mu_f(1 - c) = 2\Re(\mu_f(c)).\end{align*} $$

In particular, if $ E $ is an elliptic curve, then $ \mu _E(c) + \mu _E(1 - c) $ is an integer multiple of $ c_0(E)^{-1}\Omega (E) $ .

For this exact reason, the modular symbols $ \mu _E(c) $ can be normalized to be integers. More precisely, for an elliptic curve $ E $ of conductor $ N $ with normalized cuspidal eigenform $ f_E \in S_2(N) $ , define the normalized modular symbol

which is now an integer. The integrality of $ \mathscr {L}(E) $ is now a formal consequence of the action of Hecke operators on the space of modular symbols.

Proposition 3.3 Let $ E $ be an elliptic curve of conductor $ N $ . Let $ v $ be an odd prime such that $ v \nmid N $ . Then

$$ \begin{align*}c_0(E)\mathscr{L}(E)\#E(\mathbb{F}_v) = \sum_{a = 1}^{\lfloor\tfrac{v - 1}{2}\rfloor} \mu_E^+(a / v).\end{align*} $$

In particular, both sides lie in $ \mathbb {Z} $ .

The same argument can be adapted for $ \mathscr {L}(E, \chi ) $ using Birch’s formula.

Proposition 3.5 Let $ E $ be an elliptic curve of conductor $ N $ . Let $ \chi $ be a character of order $ q $ and odd prime conductor $ p \nmid N $ . Then

$$ \begin{align*}c_0(E)\mathscr{L}(E, \chi) = \sum_{a = 1}^{\lfloor\tfrac{p - 1}{2}\rfloor} \overline{\chi(a)}\mu_E^+(a / p).\end{align*} $$

In particular, both sides lie in $ \mathbb {Z}[\zeta _q] $ . Furthermore, if $ c_1(E) = 1 $ , then $ \mathscr {L}(E, \chi ) \in \mathbb {Z}[\zeta _q] $ .

Now observe that the right hand sides of Propositions 3.3 and 3.5 are highly similar. More precisely, since $ \overline {\chi (a)} \equiv 1\ \mod (1 - \zeta _q) $ except when $ \ell \mid a $ , the right hand sides are congruent modulo $ (1 - \zeta _q) $ . This is summarized in the following result, which will be the main tool behind much of the rest of the article.

Corollary 3.7 Let $ E $ be an elliptic curve of conductor $ N $ . Let $ \chi $ be a character of order $ q $ and odd prime conductor $ p \nmid N $ . Then

$$ \begin{align*}c_0(E)\mathscr{L}(E, \chi) \equiv -c_0(E)\mathscr{L}(E)\#E(\mathbb{F}_p) \quad\mod (1 - \zeta_q).\end{align*} $$

Furthermore, if $ q \nmid c_0(E) $ , then

$$ \begin{align*}\mathscr{L}(E, \chi) \equiv -\mathscr{L}(E)\#E(\mathbb{F}_p) \quad\mod (1 - \zeta_q),\end{align*} $$

where the denominators of both sides are inverted modulo $ (1 - \zeta _q) $ .

Remark 3.9 Modified twisted L-values $ \mathscr {L}(E, \chi ) $ are Galois equivariant as predicted by Deligne’s period conjecture [Reference Bouganis and Dokchitser4, Theorem 2.7], in the sense that $ \mathscr {L}(E, \sigma \circ \chi ) = \sigma (\mathscr {L}(E, \chi )) $ for any $ \sigma \in \operatorname {Gal}(\mathbb {Q}(\zeta _q) / \mathbb {Q}) $ . With this property, $ \mathscr {L}(E) $ can be expressed in terms of the sum of $ \mathscr {L}(E, \chi ) $ for all characters $ \chi $ of a given conductor and order. For instance, when $ \chi $ is a cubic character of conductor $ p $ ,

$$ \begin{align*}1 + \chi(a) + \overline{\chi(a)} = \begin{cases} 1 & \text{if} \ a \ \text{is not a unit in} \ \mathbb{F}_p, \\ 3 & \text{if} \ a \ \text{is the cube of a unit in} \ \mathbb{F}_p, \\ 0 & \text{otherwise}, \end{cases} \end{align*} $$

so that the identities in Propositions 3.3 and 3.5 combine to yield

$$ \begin{align*}c_0(E)\mathscr{L}(E, \chi) + c_0(E)\mathscr{L}(E, \overline{\chi}) + c_0(E)\mathscr{L}(E)\#E(\mathbb{F}_p) = 3\sum_a \mu_E^+(a / p),\end{align*} $$

where the sum runs over the cubic residues $ a $ in $ \mathbb {F}_p $ such that $ 1 \le a \le \lfloor \tfrac {p - 1}{2}\rfloor $ . By Galois equivariance, the first two terms combine to $ 2c_0(E)\Re (\mathscr {L}(E, \chi )) $ , so that this expresses $ \Re (\mathscr {L}(E, \chi )) $ in terms of $ \mathscr {L}(E) $ up to a few error terms consisting of modular symbols. By reducing modulo $ 3 $ , this recovers the congruence in Corollary 3.7, but also shows that the congruence would not a priori hold modulo $ 9 $ , unless the modular symbols $ \mu _E^+(a / p) $ for each cubic residue $ a $ in $ \mathbb {F}_p $ sum to a multiple of $ 3 $ .

4 Denominators of L-values

This section proves a few results on the $ \ell $ -adic valuations of denominators of modified L-values, where $ \ell $ is an odd prime, which may be of independent interest. Since $ c_0(E)\mathscr {L}(E)\#E(\mathbb {F}_v) $ is integral, the $ \ell $ -adic valuation of the rational number $ c_0(E)\mathscr {L}(E) $ can be bounded from below by the $ \ell $ -adic valuation of $ \#E(\mathbb {F}_v) $ , which is in turn controlled by $ \operatorname {tor}(E) $ in the denominator of $ \operatorname {BSD}(E) $ . When $ \ell \ne 3 $ , assuming the $ \ell $ -part of the Birch–Swinnerton-Dyer conjecture, such a lower bound follows from Lorenzini’s result that $ \operatorname {ord}_\ell (\#\operatorname {tor}(E)) \le \operatorname {ord}_\ell (\operatorname {Tam}(E)) $ with finitely many exceptions [Reference Lorenzini20, Proposition 1.1], but the case $ \ell = 3 $ requires more work.

Proof By the assumption that $ E $ has a point of order $ 3 $ , $ E $ is isomorphic either to the elliptic curve given by $ y^2 + cy = x^3 $ for some cube-free $ c \in \mathbb {N} $ , which has complex multiplication by $ \mathbb {Z}[\zeta _3] $ , or to the elliptic curve $ E_{1, \pm b / a} $ given by

$$ \begin{align*}y^2 + xy \pm \dfrac{b}{a}y = x^3,\end{align*} $$

for some coprime $ a, b \in \mathbb {N} $ [Reference Barrios and Roy1, Proposition 2.4]. If $ 3 \nmid \operatorname {ord}_v(a) $ for some prime $ v $ , then $ 3 \mid \operatorname {Tam}_v(E_{1, \pm b / a}) $ [Reference Barrios and Roy1, Theorem 3.5], which contradicts the assumption that ${ 3 \nmid \operatorname {Tam}(E) }$ , so that $ a = d^3 $ for some $ d \in \mathbb {N} $ coprime to $ b $ . The change of variables $ (x, y) \mapsto (x / d^2, y / d^3) $ yields an isomorphism from $ E_{1, \pm b / a} $ to the elliptic curve $ E_{d, \pm b} $ given by

$$ \begin{align*}y^2 + dxy \pm by = x^3,\end{align*} $$

which is now a minimal model and has discriminant $ \Delta = \pm b^3(d^3 - 27b) $ . Now let $ v $ be a prime such that $ v \mid b $ , so that $ v \mid \Delta $ and $ \operatorname {ord}_v(d^3 - 27b) = 0 $ by coprimality. By step $ 2 $ of Tate’s algorithm, since $ T^2 + dT $ splits in $ \mathbb {F}_v $ , the elliptic curve $ E_{d, \pm b} $ has Kodaira symbol $ \mathbf {I}_{\operatorname {ord}_v(\Delta )} $ with split mutiplicative reduction at $ v $ , so that

$$ \begin{align*}\operatorname{Tam}_v(E) = \operatorname{Tam}_v(E_{d, \pm b}) = \operatorname{ord}_v(\Delta) = 3\operatorname{ord}_v(b),\end{align*} $$

which contradicts the assumption that $ 3 \nmid \operatorname {Tam}(E) $ . This forces $ b = 1 $ , but the j-invariant of $ E_{d, \pm b} = E_{d, \pm 1} $ computes to be

$$ \begin{align*}\dfrac{d^3(d^3 \mp 24)^3}{\pm d^3 - 27} = 27\dfrac{\left(\tfrac{27}{\pm d^3 - 27} + 1\right)\left(\tfrac{27}{\pm d^3 - 27} + 9\right)^3}{\left(\tfrac{27}{\pm d^3 - 27}\right)^3},\end{align*} $$

which implies that $ \operatorname {im}(\overline {\rho _{E, 3}}) $ is the Borel subgroup 3B.1.1 [Reference Zywina35, Theorem 1.2].

Assuming just one direction of the $ 3 $ -part of the Birch–Swinnerton-Dyer conjecture, a clean divisibility result for $ \operatorname {BSD}(E) $ can be derived from the integrality of $ c_0(E)\mathscr {L}(E)\#E(\mathbb {F}_v) $ via a case-by-case analysis on $ \operatorname {im}(\rho _{E, 3}) $ .

Proof The final statement follows from the first statement, so it suffices to prove the latter. Assume that $ 3 \nmid c_0(E) $ . By Proposition 3.3 and the assumptions,

so it suffices to find an odd prime $ v \nmid N $ such that $ \#E(\mathbb {F}_v) \equiv 3 \ \mod 9 $ . By Chebotarev’s density theorem, this reduces to finding a matrix $ M \in \operatorname {im}(\overline {\rho _{E, 9}}) $ such that $ 1 + \det (M) - \operatorname {tr}(M) = 3 $ . By inspecting the table in Section A.2, such matrices exist for all $ \operatorname {im}(\rho _{E, 3}) $ except for the two $ 3 $ -adic Galois images 9.72.0.1 and 9.72.0.5, so these two cases have to be handled separately. If $ \operatorname {im}(\rho _{E, 3}) $ is 9.72.0.1, then $ \operatorname {im}(\overline {\rho _{E, 3}}) $ is 3Cs.1.1 and not 3B.1.1, so Lemma 4.1 implies that $ 3 \mid \operatorname {Tam}(E) $ . Otherwise $ \operatorname {im}(\rho _{E, 3}) $ is 9.72.0.5, then $ \operatorname {im}(\overline {\rho _{E, 9}}) $ fixes a subspace of the group of $ 9 $ -torsion points of $ E $ , so that $ E(\mathbb {Q}) \cong \mathbb {Z} / 9 $ , which contradicts the assumption that $ E(\mathbb {Q}) \cong \mathbb {Z} / 3 $ .

A lower bound on the $ \ell $ -adic valuation of $ c_0(E)\mathscr {L}(E) $ then follows for any odd prime $ \ell $ , assuming the $ \ell $ -part of the Birch–Swinnerton-Dyer conjecture, but when $ E $ has no rational $ \ell $ -isogeny the bound is unconditional by simple group theory.

Proof For the first statement, by Proposition 3.3, it suffices to find an odd prime $ v \nmid N $ such that $ \ell \nmid \#E(\mathbb {F}_v) $ . By Chebotarev’s density theorem, this reduces to finding a matrix $ M \in \operatorname {im}(\overline {\rho _{E, \ell }}) $ such that $ \operatorname {tr}(M) \ne 1 + \det (M) $ . Suppose otherwise that $ \operatorname {tr}(M) = 1 + \det (M) $ for all matrices $ M \in \operatorname {im}(\overline {\rho _{E, \ell }}) $ , so that in particular $ \operatorname {tr}(M) = 2 $ for all matrices $ M \in \operatorname {im}(\overline {\rho _{E, \ell }}) \cap \operatorname {SL}_2(\mathbb {Z} / \ell ) $ . In this case, by inspecting the orders of elements in each conjugacy class of $ \operatorname {SL}_2(\mathbb {Z} / \ell ) $ as in Section 2, it is clear that $ \operatorname {im}(\overline {\rho _{E, \ell }}) \cap \operatorname {SL}_2(\mathbb {Z} / \ell ) $ is necessarily an $ \ell $ -group, so that in particular $ \ell \mid \#\operatorname {im}(\overline {\rho _{E, \ell }}) $ . Then either $ \operatorname {im}(\overline {\rho _{E, \ell }}) $ is contained in a Borel subgroup of $ \operatorname {GL}_2(\mathbb {Z} / \ell ) $ or $ \operatorname {im}(\overline {\rho _{E, \ell }}) $ contains $ \operatorname {SL}_2(\mathbb {Z} / \ell ) $ [Reference Serre26, Proposition 15]. The former contradicts the assumption that $ E $ has no rational $ \ell $ -isogeny, and the latter is impossible by comparing orders.

For the second statement, the assumption that $ \operatorname {ord}_\ell (\mathscr {L}(E)) = \operatorname {ord}_\ell (\operatorname {BSD}(E)) $ reduces the statement to proving that $ \operatorname {ord}_\ell (c_0(E)\operatorname {BSD}(E)) \ge -1 $ . By Mazur’s torsion theorem, since $ \ell $ is odd, it suffices to consider $ \operatorname {tor}(E) $ being one of the eight subgroups

$$ \begin{align*}\mathbb{Z} / 3, \ \mathbb{Z} / 5, \ \mathbb{Z} / 6, \ \mathbb{Z} / 7, \ \mathbb{Z} / 9, \ \mathbb{Z} / 10, \ \mathbb{Z} / 12, \ \mathbb{Z} / 2 \oplus \mathbb{Z} / 6,\end{align*} $$

If $ E(\mathbb {Q}) \not \cong \mathbb {Z} / 3 $ , then a case-by-case analysis of Lorenzini’s classification yields $ \operatorname {ord}_\ell (\operatorname {Tam}(E)) \ge \operatorname {ord}_\ell (\#\operatorname {tor}(E)) $ except for the elliptic curve 11a3 with $ \ell = 5 $ , and the elliptic curves 14a4 and 14a6 with $ \ell = 3 $ [Reference Lorenzini20, Proposition 1.1], but these exceptions all have $ \operatorname {ord}_\ell (c_0(E)) = 1 $ and $ \operatorname {ord}_\ell (\operatorname {BSD}(E)) = -2 $ , so that in particular $ \operatorname {ord}_\ell (c_0(E))\operatorname {BSD}(E)) \ge -1 $ . If $ E(\mathbb {Q}) \cong \mathbb {Z} / 3 $ , then Proposition 4.2 implies that

as required.

The following is another easy result on the $ \ell $ -adic valuation of $ \mathscr {L}(E)\#E(\mathbb {F}_v) $ . The factors arising from the denominator of the rational number $ \mathscr {L}(E)\#E(\mathbb {F}_v) $ could a priori cancel the factors appearing in $ c_0(E) $ , but the congruence of L-values says that this should not happen, assuming Stevens’s conjecture that $ c_1(E) = 1 $ .

5 Units of twisted L-values

Under the standard arithmetic conjectures, Dokchitser–Evans–Wiersema computed the norm of $ \mathscr {L}(E, \chi ) $ in terms of $ \operatorname {BSD}(E) $ and $ \operatorname {BSD}(E / K) $ , where $ K $ is the degree $ q $ subfield of $ \mathbb {Q}(\zeta _p) $ cut out by the kernel of $ \chi $ [Reference Dokchitser, Evans and Wiersema12, Theorem 38]. Some of their main results can be summarized in the notation of this article as follows.

In ideal-theoretic language, Proposition 5.1.1 predicts that the ideal $ I $ of $ \mathbb {Z}[\zeta _q] $ generated by $ \mathscr {L}(E, \chi ) $ has norm equal to the nonzero positive rational number $ \operatorname {BSD}(E / K) / \operatorname {BSD}(E) $ , and Proposition 5.1.2 says that this rational number is the square of a positive rational number $ B $ equal to the norm of the ideal of $ \mathbb {Z}[\zeta _q]^+ $ generated $ \mathscr {L}(E, \chi )\zeta $ . Thus there are only finitely many possibilities for the prime ideal factorization of $ I $ , and the fact that $ I $ is invariant under complex conjugation narrows down the possibilities further. The precise ideal factorization can then be recovered from the $ \operatorname {Gal}(K / \mathbb {Q}) $ -module structure of [Reference Burns and Castillo5, Remark 7.4], such as in the case of $ \operatorname {Gal}(K / \mathbb {Q}) \cong \mathbb {Z} / 5 $ explored by Maistret–Shukla [Reference Maistret and Shukla21, Theorem 1.4].

Assuming that $ I $ has been computed as an ideal of $ \mathbb {Z}[\zeta _q] $ , any generator of $ I $ is only equal to the actual value of $ \mathscr {L}(E, \chi ) $ up to a unit $ u \in \mathbb {Z}[\zeta _q] $ . Proposition 5.1.2 refines this prediction slightly by adding a condition on the norm of $ \mathscr {L}(E, \chi )\zeta $ , which determines the actual value of $ \mathscr {L}(E, \chi ) $ up to a unit $ u \in \mathbb {Z}[\zeta _q]^+ $ . In the special case of $ q = 3 $ , this is still ambiguous up to a sign, since the units of $ \mathbb {Z}[\zeta _3]^+ = \mathbb {Z} $ are $ \pm 1 $ . Corollary 3.7 comes into the picture by pinning down the sign in terms of $ \#E(\mathbb {F}_p) $ .

Corollary 5.2 Let $ E $ be an elliptic curve of conductor $ N $ such that $ L(E, 1) \ne 0 $ . Let $ \chi $ be a cubic character of odd prime conductor $ p \nmid N $ such that $ 3 \nmid c_0(E)\operatorname {BSD}(E)\#E(\mathbb {F}_p) $ . Assume further that $ c_1(E) = 1 $ , and that $ \mathscr {L}(E) = \operatorname {BSD}(E) $ and $ \mathscr {L}(E / K) = \operatorname {BSD}(E / K) $ . Then

$$ \begin{align*}\mathscr{L}(E, \chi) = u\overline{\chi(N)}B,\end{align*} $$

where the positive rational number $ B \in \mathbb {Q}^\times $ is the positive square root of the positive rational square $ \operatorname {BSD}(E / K) / \operatorname {BSD}(E) \in (\mathbb {Q}^\times )^2 $ , and the sign $ u = \pm 1 $ is such that

$$ \begin{align*}u \equiv -\#E(\mathbb{F}_p)\operatorname{BSD}(E)B^{-1} \quad\mod 3.\end{align*} $$

This follows immediately from Corollary 3.7 and Proposition 5.1. Corollary 5.2 clarifies much of the phenomena observed by Dokchitser–Evans–Wiersema [Reference Dokchitser, Evans and Wiersema12, Example 45], where they gave many pairs of examples of arithmetically similar elliptic curves $ E_1 $ and $ E_2 $ with $ \mathscr {L}(E_1, \chi ) \ne \mathscr {L}(E_2, \chi ) $ for a few cubic characters $ \chi $ , in the sense that $ \mathscr {L}(E_1, \chi ) \ne \mathscr {L}(E_2, \chi ) $ precisely because $ \#E_1(\mathbb {F}_p) \not \equiv \#E_2(\mathbb {F}_p) \ \mod 3 $ .

Example 5.3 Let $ E_1 $ and $ E_2 $ be the elliptic curves 1356d1 and 1356f1, and let $ \chi $ be the cubic character of conductor $ 7 $ given by $ \chi (3) = \zeta _3^2 $ . Then $ c_0(E_i) = \operatorname {BSD}(E_i) = \operatorname {BSD}(E_i / K) = 1 $ for $ i \in \{1, 2\} $ , so Proposition 5.1 implies that $ \mathscr {L}(E_i, \chi ) = \pm \overline {\chi (1356)} = \pm \zeta _3^2 $ , but it was a priori unclear why $ \mathscr {L}(E_1, \chi ) = \zeta _3^2 $ and $ \mathscr {L}(E_2, \chi ) = -\zeta _3^2 $ . Corollary 5.2 explains this by requiring that this sign agrees with $ -\#E_i(\mathbb {F}_7) $ modulo $ 3 $ , and in this case $ \#E_1(\mathbb {F}_7) = 11 $ and $ \#E_2(\mathbb {F}_7) = 7 $ , which are distinct modulo $ 3 $ . They provided other examples satisfying $ c_0(E) = \operatorname {BSD}(E) = \operatorname {BSD}(E / K) = 1 $ with different $ \mathscr {L}(E, \chi ) $ for a few different cubic characters $ \chi $ , and they can all be explained similarly. The values of $ \mathscr {L}(E, \chi ) $ for the above character are tabulated as follows.

When $ q> 3 $ but $ \operatorname {BSD}(E) = \operatorname {BSD}(E / K) $ , Proposition 5.1 says that $ \mathscr {L}(E, \chi ) $ is a unit in $ \mathbb {Z}[\zeta _q] $ , and Corollary 3.7 places a congruence on this unit in terms of $ \#E(\mathbb {F}_p) $ .

Again, this follows immediately from Corollary 3.7 and Proposition 5.1. Corollary 5.4 explains the remaining phenomena observed by Dokchitser–Evans–Wiersema [Reference Dokchitser, Evans and Wiersema12, Example 44], where they gave many pairs of examples of arithmetically trivial elliptic curves $ E_1 $ and $ E_2 $ with $ \mathscr {L}(E_1, \chi ) \ne \mathscr {L}(E_2, \chi ) $ for quintic characters $ \chi $ , in the sense that $ \mathscr {L}(E_1, \chi ) \ne \mathscr {L}(E_2, \chi ) $ precisely because $ \#E_1(\mathbb {F}_p) \not \equiv \#E_2(\mathbb {F}_p) \ \mod 5 $ .

Example 5.5 Let $ E_1 $ and $ E_2 $ be the elliptic curves 307a1 and 307c1, and let $ \chi $ be the quintic character of conductor $ 11 $ given by $ \chi (2) = \zeta _5 $ . Then $ c_0(E_i) = \operatorname {BSD}(E_i) = \operatorname {BSD}(E_i / K) = 1 $ for $ i \in \{1, 2\} $ , so Proposition 5.1 implies that $ \mathscr {L}(E_i, \chi ) $ is a unit, but it was a priori unclear why $ \mathscr {L}(E_1, \chi ) = 1 $ and $ \mathscr {L}(E_2, \chi ) = \zeta _5u^2 $ , where . Corollary 5.4 explains this by requiring that $ \mathscr {L}(E_i, \chi ) \equiv -\#E_i(\mathbb {F}_{11}) \ \mod (1 - \zeta _5) $ , and in this case $ \#E_1(\mathbb {F}_{11}) = 9 $ and $ \#E_2(\mathbb {F}_{11}) = 16 $ , which are distinct modulo $ 5 $ . They provided other examples satisfying $ c_0(E) = \operatorname {BSD}(E) = \operatorname {BSD}(E / K) = 1 $ with different $ \mathscr {L}(E, \chi ) $ for this character, and they can all be explained similarly as follows.

When $ q> 3 $ and $ \operatorname {BSD}(E) \ne \operatorname {BSD}(E / K) $ , it is awkward to rephrase Proposition 5.1 to apply Corollary 3.7, but it can be illustrated with an example [Reference Dokchitser, Evans and Wiersema12, Example 46].

6 Residual densities of twisted L-values

For a fixed elliptic curve $ E $ of conductor $ N $ , a natural problem is to determine the asymptotic distribution of $ \mathscr {L}(E, \chi ) $ , as $ \chi $ varies over characters of some fixed prime order $ q $ but of arbitrarily high odd prime conductor $ p \nmid N $ . However, for each such $ p $ , there are $ q - 1 $ characters $ \chi $ of conductor $ p $ and order $ q $ , giving rise to $ q - 1 $ conjugates of $ \mathscr {L}(E, \chi ) $ , so that a uniform choice of $ \chi $ for each $ p $ has to be made for any meaningful analysis. One solution is to observe that the residue class of $ \mathscr {L}(E, \chi ) $ modulo $ (1 - \zeta _q) $ is independent of the choice of $ \chi $ for each $ p $ , so that a simpler problem would be to determine the asymptotic distribution of these residue classes instead. As in the introduction, let $ X_{E, q}^{< n} $ be the set of characters of odd order $ q $ and odd prime conductor $ p < n $ not dividing $ N $ . Define the residual densities $ \delta _{E, q} $ of $ \mathscr {L}(E, \chi ) $ to be the natural densities of $ \mathscr {L}(E, \chi ) $ modulo $ (1 - \zeta _q) $ . In other words, this is the value

if such a limit exists. Note that as these residue classes only depend on $ p $ rather than $ \chi $ , the set $ X_{E, q}^{< n} $ can be replaced with the set of equivalence classes of characters in $ X_{E, q}^{< n} $ , where two characters are equivalent if they have the same conductor. When $ q \nmid c_0(E) $ , this can be computed for each $ \lambda \in \mathbb {F}_q $ using Corollary 3.7, with the only subtlety being the possible cancellations between $ \mathscr {L}(E) $ and $ \#E(\mathbb {F}_p) $ . In the generic scenario when $ \operatorname {im}(\overline {\rho _{E, q}}) $ is maximal, there is a clean description in terms of Legendre symbols.

Proposition 6.1 Let $ E $ be an elliptic curve such that $ L(E, 1) \ne 0 $ . Let $ q $ be an odd prime such that $ q \nmid c_0(E) $ .

1. (1) If $ \operatorname {ord}_q(\mathscr {L}(E))> 0 $ , then $ \delta _{E, q}(0) = 1 $ and $ \delta _{E, q}(\lambda ) = 0 $ for any $ \lambda \in \mathbb {F}_q^\times $ .

2. (2) If $ \operatorname {ord}_q(\mathscr {L}(E)) \le 0 $ , then set and

   

   Then for any $ \lambda \in \mathbb {F}_q $ ,

   $$ \begin{align*}\delta_{E, q}(\lambda) = \dfrac{\#\left\{M \in G_{E, q^m} \ \middle| \ 1 + \det(M) - \operatorname{tr}(M) \equiv -\lambda\mathscr{L}(E)^{-1} \quad\mod q^m\right\}}{\#G_{E, q^m}}.\end{align*} $$

In particular, if $ \operatorname {ord}_q(\mathscr {L}(E)) \le 0 $ and $ \overline {\rho _{E, q}} $ is surjective, then compute the Legendre symbols

Then for any $ \lambda \in \mathbb {F}_q $ ,

$$ \begin{align*}\delta_{E, q}(\lambda) = \begin{cases} \tfrac{1}{q - 1} & \text{if} \ \epsilon_{E, q}(\lambda) = 1, \\[5pt] \tfrac{q}{q^2 - 1} & \text{if} \ \epsilon_{E, q}(\lambda) = 0, \\[5pt] \tfrac{1}{q + 1} & \text{if} \ \epsilon_{E, q}(\lambda) = -1. \end{cases} \end{align*} $$

Assuming the $ q $ -part of the Birch–Swinnerton-Dyer conjecture, Theorem 4.4.3 says $ \operatorname {ord}_q(\operatorname {BSD}(E)) \ge -1 $ , so that nontrivial values of $ \delta _{E, q} $ are only visible when $ \operatorname {ord}_q(\operatorname {BSD}(E)) \in \{0, -1\} $ . Once this is determined, computing $ \delta _{E, q} $ then reduces to identifying $ \operatorname {im}(\overline {\rho _{E, q}}) $ or $ \operatorname {im}(\overline {\rho _{E, q^2}}) $ , and then weighing the proportion of matrices with a certain determinant and trace. To illustrate this in action, the next result describes the possible ordered triples $ (\delta _{E, 3}(0), \delta _{E, 3}(1), \delta _{E, 3}(2)) $ of residual densities, which is only made possible thanks to the classification of $ 3 $ -adic Galois images by Rouse–Sutherland–Zureick-Brown [Reference Rouse, Sutherland and Zureick-Brown25, Corollaries 1.3.1 and 12.3.3]. As in the introduction, these ordered triples will also be denoted $ \delta _{E, 3} $ for ease of notation.

Theorem 6.4 Let $ E $ be an elliptic curve such that $ L(E, 1) \ne 0 $ and that $ 3 \nmid c_0(E) $ . Assume further that $ \operatorname {ord}_3(\mathscr {L}(E)) = \operatorname {ord}_3(\operatorname {BSD}(E)) $ . Then precisely one of the following holds.

1. (1) If $ \operatorname {ord}_3(\operatorname {BSD}(E))> 0 $ , then $ \delta _{E, 3} = (1, 0, 0) $ .

2. (2) If $ \operatorname {ord}_3(\operatorname {BSD}(E)) = 0 $ and $ 3 \mid \#\operatorname {tor}(E) $ , then $ \delta _{E, 3} = (1, 0, 0) $ .

3. (3) If $ \operatorname {ord}_3(\operatorname {BSD}(E)) = 0 $ and $ 3 \nmid \#\operatorname {tor}(E) $ , then $ \delta _{E, 3} $ is given by the table in Section A.1.

4. (4) If $ \operatorname {ord}_3(\operatorname {BSD}(E)) = -1 $ , then $ \delta _{E, 3} $ is given by the table in Section A.2.

In particular, $ \delta _{E, 3} $ only depends on $ \operatorname {BSD}(E) $ and $ \operatorname {im}(\overline {\rho _{E, 9}}) $ , and can only be one of

$$ \begin{align*}(1, 0, 0), \ \left(\tfrac{3}{8}, \tfrac{3}{8}, \tfrac{1}{4}\right), \ \left(\tfrac{3}{8}, \tfrac{1}{4}, \tfrac{3}{8}\right), \ \left(\tfrac{1}{2}, \tfrac{1}{2}, 0\right), \ \left(\tfrac{1}{2}, 0, \tfrac{1}{2}\right), \ \left(\tfrac{1}{8}, \tfrac{3}{4}, \tfrac{1}{8}\right),\end{align*} $$

$$ \begin{align*}\left(\tfrac{1}{8}, \tfrac{1}{8}, \tfrac{3}{4}\right), \ \left(\tfrac{1}{4}, \tfrac{1}{2}, \tfrac{1}{4}\right), \ \left(\tfrac{1}{4}, \tfrac{1}{4}, \tfrac{1}{2}\right), \ \left(\tfrac{5}{9}, \tfrac{2}{9}, \tfrac{2}{9}\right), \ \left(\tfrac{1}{3}, \tfrac{2}{3}, 0\right), \ \left(\tfrac{1}{3}, 0, \tfrac{2}{3}\right).\end{align*} $$

Proof The fact that there are only four possibilities is immediate from Theorem 4.4. By Proposition 6.1, the first statement follows immediately under the assumption that $ \operatorname {ord}_3(\mathscr {L}(E)) = \operatorname {ord}_3(\operatorname {BSD}(E)) $ , while the second statement follows from $ 3 \mid 1 + \det (M) - \operatorname {tr}(M) $ for all matrices $ M \in \operatorname {im}(\overline {\rho _{E, 3}}) $ whenever $ 3 \mid \#\operatorname {tor}(E) $ .

For the third statement, it suffices to consider $ G_{E, 3} = \operatorname {im}(\overline {\rho _{E, 3}}) \cap \operatorname {SL}_2(\mathbb {Z} / 3) $ , and there are only $ 5 $ possibilities for $ \operatorname {im}(\overline {\rho _{E, 3}}) $ when $ \overline {\rho _{E, 3}} $ is not surjective, as tabulated in the table in Section A.1. If $ \overline {\rho _{E, 3}} $ is surjective, then $ \delta _{E, 3} $ is already computed in the final statement in Proposition 6.1, while the other $ 5 $ cases are similar but easier computations. For instance, if $ \operatorname {im}(\overline {\rho _{E, 3}}) $ is 3B.1.2, then $ G_{E, 3} $ is conjugate to the subgroup of unipotent upper triangular matrices in $ \operatorname {SL}_2(\mathbb {Z} / 3) $ . There are $ 6 $ matrices in this subgroup, all of which have trace $ 0 $ , so that $ \delta _{E, 3} = (1, 0, 0) $ . Note that when $ \delta _{E, 3}(1) \ne \delta _{E, 3}(2) $ , the nonzero residue of $ \operatorname {BSD}(E) $ modulo $ 3 $ would swap the values of $ \delta _{E, 3}(1) $ and $ \delta _{E, 3}(2) $ . For instance, if $ \overline {\rho _{E, 3}} $ is surjective, then $ G_{E, 3} = \operatorname {SL}_2(\mathbb {Z} / 3) $ , and

$$ \begin{align*}\delta_{E, 3} = \begin{cases} \left(\tfrac{3}{8}, \tfrac{1}{4}, \tfrac{3}{8}\right) & \text{if} \ \operatorname{BSD}(E) \equiv 1 \quad\mod 3, \\ \left(\tfrac{3}{8}, \tfrac{3}{8}, \tfrac{1}{4}\right) & \text{if} \ \operatorname{BSD}(E) \equiv 2 \quad\mod 3. \end{cases} \end{align*} $$

For the fourth statement, it suffices to consider $ G_{E, 9} $ , and by the classification this is simply the projection onto $ \operatorname {GL}_2(\mathbb {Z} / 9) $ of $ 21 $ different possible $ \operatorname {im}(\rho _{E, 3}) $ , as tabulated in the table in Section A.2. For instance, if $ \operatorname {im}(\rho _{E, 3}) $ is 3.8.0.1, then $ G_{E, 9} $ is the preimage of the subgroup of $ \operatorname {SL}_2(\mathbb {Z} / 3) $ generated by $ (\begin {smallmatrix} 1 & 2 \\ 0 & 1 \end {smallmatrix}) $ and $ (\begin {smallmatrix} 1 & 2 \\ 0 & 2 \end {smallmatrix}) $ under the canonical projection $ \operatorname {GL}_2(\mathbb {Z} / 9) \twoheadrightarrow \operatorname {GL}_2(\mathbb {Z} / 3) $ . This preimage in $ \operatorname {GL}_2(\mathbb {Z} / 9) $ consists of $ 243 $ matrices, of which $ 135 $ have trace $ 0 $ and $ 54 $ have trace $ 1 $ and $ 2 $ each, so that

$$ \begin{align*}\delta_{E, 3} = \left(\tfrac{135}{243}, \tfrac{54}{243}, \tfrac{54}{243}\right) = \left(\tfrac{5}{9}, \tfrac{2}{9}, \tfrac{2}{9}\right).\end{align*} $$

The other $ 20 $ cases are similar but easier computations, noting again that the nonzero residue of $ 3\operatorname {BSD}(E) $ modulo $ 3 $ would swap the values of $ \delta _{E, 3}(1) $ and $ \delta _{E, 3}(2) $ when $ \delta _{E, 3}(1) \ne \delta _{E, 3}(2) $ . For instance, if $ \operatorname {im}(\rho _{E, 3}) $ is 27.648.18.1, then

$$ \begin{align*}\delta_{E, 3} = \begin{cases} \left(\tfrac{1}{3}, 0, \tfrac{2}{3}\right) & \text{if} \ 3\operatorname{BSD}(E) \equiv 1 \quad\mod 3, \\ \left(\tfrac{1}{3}, \tfrac{2}{3}, 0\right) & \text{if} \ 3\operatorname{BSD}(E) \equiv 2 \quad\mod 3. \end{cases} \end{align*} $$

Finally, the final statement follows immediately from the first four.

7 Twisted L-values of Kisilevsky–Nam

The computation of residual densities was originally motivated by patterns in the statistical data by Kisilevsky–Nam [Reference Kisilevsky and Nam18, Section 7], where they numerically computed millions of modified twisted L-values by fixing the elliptic curve and varying the character. However, they considered an alternative normalization, given by

in contrast to the normalization factor $ \zeta $ given in Proposition 5.1. Under the implicit assumption that $ \mathscr {L}(E, \chi ) \in \mathbb {Z}[\zeta _q] $ , they showed that $ \mathscr {L}^+(E, \chi ) \in \mathbb {Z}[\zeta _q]^+ $ [Reference Kisilevsky and Nam18, Proposition 2.1], so that $ \operatorname {Nm}_q^+(\mathscr {L}^+(E, \chi )) $ is an integer. Fixing six elliptic curves $ E $ and five small integers $ q $ , they varied the character $ \chi $ of order $ q $ over millions of conductors $ p $ with an arbitrary choice of $ \chi $ for each $ p $ , empirically determined the greatest common divisor $ \gcd _{E, q} $ of all the integers $ \operatorname {Nm}_q^+(\mathscr {L}^+(E, \chi )) $ , and considered the integer

As their normalization differs from that in Proposition 5.1 [Reference Kisilevsky and Nam18, Remark 1], the resulting residual densities are skewed. More precisely, define the set $ X_{E, q}^{< n} $ as before, and analogously define the skewed residual densities $ \widetilde {\delta }_{E, q} $ of $ \widetilde {\mathscr {L}}^+(E, \chi ) $ to be the natural densities of $ \widetilde {\mathscr {L}}^+(E, \chi ) $ modulo $ q $ . In other words, this is the value

if such a limit exists. In the simplest case where $ q = 3 \nmid \gcd _{E, 3} $ ,

$$ \begin{align*} \widetilde{{\mathscr{L}}}^+(E, \chi) \equiv\begin{cases}{\mathscr{L}}(E, \chi)\gcd_{E, 3} & \text{if} \ \chi(N) = 1, \\2{\mathscr{L}}(E, \chi)\gcd_{E, 3} & \text{if} \ \chi(N) \ne 1,\end{cases} \end{align*} $$

since there is no norm. This becomes amenable to a computation similar to that of Proposition 6.1 provided that the condition $ \chi (N) = 1 $ can be controlled. For many elliptic curves, including half of those considered by Kisilevsky–Nam, $ \chi (N) $ depends completely on $ \#E(\mathbb {F}_p) $ due to a shared action of Frobenius in $ \operatorname {GL}_2(\mathbb {Z} / 3) $ .

Proof Let $ L $ be the extension of $ K $ where all points in $ E[3] $ are defined. By the assumption that $ E $ has no rational $ 3 $ -isogeny and the classification of $ \operatorname {im}(\overline {\rho _{E, 3}}) $ , if $ \overline {\rho _{E, 3}} $ were not surjective, then $ \operatorname {Gal}(L / \mathbb {Q}) $ is either 3Nn or 3Ns. Neither of this could occur, since by the assumption that $ F \subseteq K $ , there are subfield inclusions

$$ \begin{align*}\mathbb{Q} \subseteq \mathbb{Q}(\zeta_3) \subseteq F \subseteq K \subseteq L.\end{align*} $$

In particular, $ \operatorname {Gal}(L / \mathbb {Q}) $ surjects onto $ \operatorname {Gal}(F / \mathbb {Q}) \cong \mathcal {S}_3 $ , which forces $ \operatorname {Gal}(L / \mathbb {Q}) \cong \operatorname {GL}_2(\mathbb {Z} / 3) $ . On the other hand, $ \operatorname {Gal}(K / \mathbb {Q}) $ permutes the roots of the degree $ 4 $ polynomial $ \psi _{E, 3} $ , which forces it to be $ \operatorname {PGL}_2(\mathbb {Z} / 3) \cong \mathcal {S}_4 $ . Its subgroup $ \operatorname {Gal}(K / \mathbb {Q}(\zeta _3)) \cong \mathcal {A}_4 $ surjects onto $ \operatorname {Gal}(F / \mathbb {Q}(\zeta _3)) \cong \mathbb {Z} / 3 $ , with kernel the unique subgroup $ \operatorname {Gal}(K / F) \cong (\mathbb {Z} / 2)^2 $ of index $ 4 $ consisting precisely of all elements of $ \mathcal {A}_4 $ of order $ 1 $ or $ 2 $ .

Now $ \operatorname {Fr}_p \in \operatorname {Gal}(K / \mathbb {Q}) $ acts on the residue field of a prime $ \pi $ of $ F $ above $ p $ by

$$ \begin{align*}\operatorname{Fr}_p(\zeta_3) \equiv \zeta_3^p \quad\mod \pi, \qquad \operatorname{Fr}_p(\sqrt[3]{N}) \equiv \sqrt[3]{N}^p \quad\mod \pi.\end{align*} $$

Clearly $ \operatorname {Fr}_p $ fixes $ \zeta _3 $ , so that $ \operatorname {Fr}_p \in \operatorname {Gal}(K / \mathbb {Q}(\zeta _3)) $ . If $ p $ does not split completely in $ K $ , the condition $ \operatorname {Fr}_p \in \operatorname {Gal}(K / F) $ turns out to be equivalent to $ \#E(\mathbb {F}_p) \equiv 2 \ \mod 3 $ and to $ \chi (N) = 1 $ . To see this, on one hand, this means that $ \operatorname {Fr}_p $ fixes $ \sqrt [3]{N} $ , or equivalently that $ \sqrt [3]{N}^{p - 1} \equiv 1 \ \mod p $ , which is precisely the condition that $ \chi (N) = 1 $ . On the other hand, this also means that $ \operatorname {Fr}_p^2 = 1 $ in $ \operatorname {Gal}(K / \mathbb {Q}(\zeta _3)) $ , which is equivalent to $ \operatorname {Fr}_p $ having order exactly $ 2 $ in $ \operatorname {Gal}(K / \mathbb {Q}(\zeta _3)) $ . By the Cayley–Hamilton theorem, these are precisely the trace $ 0 $ matrices in $ \operatorname {PGL}_2(\mathbb {Z} / 3) $ , or equivalently the trace $ 0 $ matrices in $ \operatorname {GL}_2(\mathbb {Z} / 3) $ , which proves the equivalence with $ a_p(E) = 0 $ . Finally, if $ p $ splits completely in $ K $ , then $ \operatorname {Fr}_p = 1 $ in $ \operatorname {Gal}(K / \mathbb {Q}(\zeta _3)) $ , and hence $ \chi (N) = 1 $ , but these never have trace $ 0 $ in $ \operatorname {PGL}_2(\mathbb {Z} / 3) $ or in $ \operatorname {GL}_2(\mathbb {Z} / 3) $ , so that $ \#E(\mathbb {F}_p) \not \equiv 2 \ \mod 3 $ .

For these elliptic curves, the residual density of $ \widetilde {\mathscr {L}}^+(E, \chi ) $ is now easy to compute.

Proposition 7.7 Let $ E $ be an elliptic curve of conductor $ N $ with no rational $ 3 $ -isogeny such that $ 3 \nmid c_0(E)\gcd _{E, 3} $ and that the splitting field $ F $ of $ X^3 - N $ lies in the splitting field $ K $ of the $ 3 $ -division polynomial $ \psi _{E, 3} $ . Let $ \chi $ be a cubic character of odd prime conductor $ p \nmid N $ . Then

$$ \begin{align*}\widetilde{\mathscr{L}}^+(E, \chi) \equiv \begin{cases} 0 \quad\mod 3 & \text{if} \ \#E(\mathbb{F}_p) \equiv 0 \quad\mod 3, \\ 2 \quad\mod 3 & \text{if} \ \#E(\mathbb{F}_p) \equiv 1 \quad\mod 3 \ \text{and} \ p \ \text{splits completely in} \ K, \\ 1 \quad\mod 3 & \text{otherwise}. \end{cases} \end{align*} $$

In particular,

$$ \begin{align*}\widetilde{\delta}_{E, 3}(0) = \tfrac{9}{24}, \qquad \widetilde{\delta}_{E, 3}(1) = \tfrac{15}{24}, \qquad \widetilde{\delta}_{E, 3}(2) = \tfrac{1}{24}.\end{align*} $$

Proof By Corollary 3.7 and the assumption that $ 3 \nmid c_0(E)\gcd _{E, 3} $ ,

$$ \begin{align*}\widetilde{\mathscr{L}}^+(E, \chi) \equiv \begin{cases} 2\#E(\mathbb{F}_p)\mathscr{L}(E)\gcd_{E, 3} & \text{if} \ \chi(N) = 1, \\ \#E(\mathbb{F}_p)\mathscr{L}(E)\gcd_{E, 3} & \text{if} \ \chi(N) \ne 1. \end{cases} \end{align*} $$

Clearly $ \widetilde {\mathscr {L}}^+(E, \chi ) \equiv 0 \ \mod 3 $ when $ \#E(\mathbb {F}_p) \equiv 0 \ \mod 3 $ . By Lemma 7.3, $ \chi (N) = 1 $ occurs either when $ \#E(\mathbb {F}_p) \equiv 1 \ \mod 3 $ but $ p $ splits completely in $ K $ , or when $ \#E(\mathbb {F}_p) \equiv 2 \ \mod 3 $ but $ p $ does not split completely in $ K $ , the only remaining case being when $ \#E(\mathbb {F}_p) \equiv 1 \ \mod 3 $ and $ \chi (N) \ne 1 $ . The first statement then follows by substituting the residues of $ \#E(\mathbb {F}_p) $ modulo $ 3 $ , and noting that $ \gcd _{E, 3} $ cancels out the factors in $ \mathscr {L}(E) $ by definition. Now the description of the groups in Lemma 7.3 implies that $ \#E(\mathbb {F}_p) \equiv \lambda \ \mod 3 $ occurs with the proportion of matrices $ M \in \operatorname {SL}_2(\mathbb {Z} / 3) $ with $ \operatorname {tr}(M) = 2 - \lambda $ , by Chebotarev’s density theorem. If $ p $ splits completely in $ K $ , then $ \operatorname {Fr}_p = 1 $ in $ \operatorname {PGL}_2(\mathbb {Z} / 3) $ , so that $ \operatorname {Fr}_p = \pm 1 $ in $ \operatorname {GL}_2(\mathbb {Z} / 3) $ and in $ \operatorname {SL}_2(\mathbb {Z} / 3) $ , but the condition $ \#E(\mathbb {F}_p) \equiv 1 \ \mod 3 $ forces $ \operatorname {Fr}_p = -1 $ , which has trace $ 1 $ . The final statement then follows by counting matrices in $ \operatorname {SL}_2(\mathbb {Z} / 3) $ with given trace.

Proposition 7.7 completely explains the numerical data by Kisilevsky–Nam for the elliptic curve 11a1 where $ \gcd _{E, 3} = 5 $ and the elliptic curves 15a1 and 17a1 where $ \gcd _{E, 3} = 4 $ , all of which satisfy the assumptions of Proposition 7.7. Unfortunately, the same argument cannot explain the density patterns when $ 3 \mid \gcd _{E, 3} $ , such as for the remaining three elliptic curves 14a1, 19a1, and 37b1 considered by Kisilevsky–Nam, since Corollary 3.7 is a priori not valid modulo $ 9 $ , as noted in Remark 3.9.