For an elliptic curve E/QE/\mathbb{Q} and an Artin representation ρ\rho of Q\mathbb{Q}, the Hasse–Weil LL-function of EE twisted by ρ\rho has a conjectural rank formula (order of vanishing of L(E,ρ,s)L(E,\rho,s) at s=1s=1 equal to the multiplicity of ρ\rho in E(K)⊗CE(K)\otimes\mathbb{C}), but no formula of Birch–Swinnerton-Dyer type is known for the leading term. Dokchitser, Evans and Wiersema axiomatised the required behaviour: an invariant BSD(E,ρ)∈C×\mathrm{BSD}(E,\rho)\in\mathbb{C}^\times satisfying axioms (C1)–(C6) (compatibility with the classical BSD quotient over number fields, multiplicativity, Galois equivariance, and integrality), and proved that such an invariant exists under the full Birch–Swinnerton-Dyer conjecture together with analytic continuation, functional equations, Deligne's period conjecture, the Manin constant conjecture and the Riemann hypothesis for all twists. The open problem is twofold: construct BSD(E,ρ)\mathrm{BSD}(E,\rho) by intrinsic arithmetic means, not by normalising LL-values, and justify the axioms from purely arithmetic hypotheses such as finiteness of the Tate–Shafarevich group and the structure of Selmer groups, thereby explaining the conjectural multiplicativity, Galois equivariance and integrality through a Selmer-theoretic mechanism. Both requests remain open.

Background

Let E/QE/\mathbb{Q} be an elliptic curve and L(E,s)L(E,s) its Hasse–Weil LL-function. The Birch–Swinnerton-Dyer (BSD) conjecture predicts, for every finite extension F/QF/\mathbb{Q}, both the order rr of vanishing of L(E/F,s)L(E/F,s) at s=1s=1 (the algebraic rank r=rank E(F)r=\mathrm{rank}\,E(F)) and an explicit formula for the leading term: the modified leading term L(E/F)\mathscr{L}(E/F), obtained by dividing by (s−1)r(s-1)^r and rescaling by the archimedean periods Ω±(E)r1+r2\Omega_\pm(E)^{r_1+r_2} and the primes dividing the discriminant of FF, should equal BSD(E/F)=RegE/F ∣IIIE/F∣ CE/F/∣E(F)tors∣2\mathrm{BSD}(E/F)=\mathrm{Reg}_{E/F}\,|\mathrm{III}_{E/F}|\,C_{E/F}/|E(F)_{\mathrm{tors}}|^2. Here RegE/F\mathrm{Reg}_{E/F} is the regulator of a basis of the free part of the Mordell–Weil group E(F)E(F) (the finitely generated abelian group of FF-rational points of EE), IIIE/F\mathrm{III}_{E/F} is the Tate–Shafarevich group (the group measuring the failure of the local-global principle for principal homogeneous spaces under EE), and CE/FC_{E/F} is the product of local Tamagawa numbers (measures of the component groups of the Néron model at finite places). An Artin representation ρ\rho of Q\mathbb{Q} is a continuous finite-image complex representation of the absolute Galois group Gal(Q‾/Q)\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}); Dirichlet characters are the abelian examples. Just as the Dedekind zeta function of a number field factors into Artin LL-functions, L(E/F,s)L(E/F,s) factors as a product of twisted LL-functions L(E,ρ,s)L(E,\rho,s). The rank part of BSD then takes the equivariant form ords=1L(E,ρ,s)=⟨ρ,E(K)⊗C⟩\mathrm{ord}_{s=1}L(E,\rho,s)=\langle\rho,E(K)\otimes\mathbb{C}\rangle (inner product of characters, for ρ\rho factoring through Gal(K/Q)\mathrm{Gal}(K/\mathbb{Q})). For the leading terms, however, no equivariant formula is known: writing L(E,ρ)\mathscr{L}(E,\rho) for the modified leading term of L(E,ρ,s)L(E,\rho,s) at s=1s=1, defined in the source (Definition 12) by

L(E,ρ)=lim⁡s→1L(E,ρ,s)(s−1)r⋅fρΩ+(E)d+(ρ) ∣Ω−(E)∣d−(ρ) wρ,\mathscr{L}(E,\rho)=\lim_{s\to 1}\frac{L(E,\rho,s)}{(s-1)^{r}}\cdot\frac{\sqrt{\mathfrak{f}_\rho}}{\Omega_+(E)^{d^+(\rho)}\,|\Omega_-(E)|^{d^-(\rho)}\,w_\rho},

where r=ords=1L(E,ρ,s)r=\mathrm{ord}_{s=1}L(E,\rho,s), fρ\mathfrak{f}_\rho is the Artin conductor of ρ\rho, d±(ρ)d^\pm(\rho) are the dimensions of the ±1\pm1-eigenspaces of complex conjugation acting on ρ\rho, and wρw_\rho is the Artin root number, Dokchitser–Evans–Wiersema showed there is a barrier to any naive expression, because arithmetically indistinguishable curves can have different twisted LL-values (their Example 3: the curves 307a1 and 307c1, labelled by Cremona's standard catalogue for elliptic curves over Q\mathbb{Q}, have identical conductor (the integer measuring bad reduction), discriminant, trivial Mordell–Weil groups, trivial III\mathrm{III} and Tamagawa numbers over both Q\mathbb{Q} and Q(ζ11)+\mathbb{Q}(\zeta_{11})^+, yet L(E,χ)=1≠L(E′,χ)\mathscr{L}(E,\chi)=1\neq\mathscr{L}(E',\chi) for a Dirichlet character χ\chi of order 5 and conductor 11). Their approach is instead axiomatic: Conjecture 4 postulates an invariant BSD(E,ρ)∈C×\mathrm{BSD}(E,\rho)\in\mathbb{C}^\times for every elliptic curve E/QE/\mathbb{Q} and Artin representation ρ\rho, with L(E,ρ)=BSD(E,ρ)\mathscr{L}(E,\rho)=\mathrm{BSD}(E,\rho), satisfying (C1) compatibility with the classical BSD quotient BSD(E/F)\mathrm{BSD}(E/F) under induction, (C2) multiplicativity BSD(E,ρ⊕τ)=BSD(E,ρ)BSD(E,τ)\mathrm{BSD}(E,\rho\oplus\tau)=\mathrm{BSD}(E,\rho)\mathrm{BSD}(E,\tau), (C3) behaviour under duality with the root number wE,ρw_{E,\rho} (the sign in the conjectural functional equation), (C4) reality and sign for self-dual ρ\rho, and, when ⟨ρ,E(K)⊗C⟩=0\langle\rho,E(K)\otimes\mathbb{C}\rangle=0 (rank zero), (C5) BSD(E,ρ)∈Q(ρ)×\mathrm{BSD}(E,\rho)\in\mathbb{Q}(\rho)^\times with Galois equivariance (Q(ρ)\mathbb{Q}(\rho) being the field generated by the values of ρ\rho), and (C6) integrality in Z[ζd]\mathbb{Z}[\zeta_d] when ρ\rho is a non-trivial primitive Dirichlet character of order dd and the conductors are coprime or EE is semistable (all bad reductions of multiplicative type) with no non-trivial isogenies (non-constant homomorphisms of elliptic curves). Theorem 5 establishes that such an invariant exists — namely L(E,ρ)\mathscr{L}(E,\rho) — assuming the analytic continuation and functional equations of all twists, the BSD conjecture itself, Deligne's period conjecture (a general conjecture describing the algebraicity of critical LL-values relative to motivic periods), Stevens's Manin constant conjecture (that the Manin constant attached to a modular parametrisation of EE is 1), and the Riemann hypothesis for L(E,ρ,s)L(E,\rho,s). Axiomatically, BSD(E,ρ)\mathrm{BSD}(E,\rho) is uniquely determined for Q\mathbb{Q}-valued characters by (C1)–(C4) through Brauer-type inductions. The conjecture has striking conditional arithmetic consequences that appear out of reach of classical Selmer-group techniques: Theorem 6 forces cross-prime interdependencies between pp-primary and qq-primary parts of III\mathrm{III} over cyclic extensions, Theorem 7 forces non-trivial pp-Selmer groups (the Selmer group is a Galois-cohomological group controlling E(F)/pE(F)/p and pp-torsion in III\mathrm{III}, here with p∤[F:Q]p\nmid[F:\mathbb{Q}]) from purely local data, and Theorem 8 gives positive-rank predictions for dihedral twists beyond the currently known cases of the parity conjecture (the conjecture that the parity of the algebraic rank equals the parity of the root number). The authors stress that Conjecture 4 ought to be purely a statement about Selmer groups and Tate–Shafarevich groups, but that no such structure is currently understood to produce it; their justification passes entirely through LL-function conjectures. The question sits inside the broader equivariant/refined BSD programme, where the equivariant Tamagawa number conjecture (ETNC, the equivariant refinement of the Bloch–Kato Tamagawa number conjecture predicting the leading terms of Artin LL-functions in terms of regulators and Galois-cohomological invariants) provides a conjectural higher-rank framework. The source also flags further directions that it explicitly declines to pursue there (Remark 14 and the closing remark of §4): extending the algebraicity, integrality and norm predictions beyond the nonvanishing (rank-zero) setting, possibly via the motivic LL-value conjectures (Beilinson, Bloch–Kato, equivariant Tamagawa number conjecture), and pinning down more precisely the ideal generated by L(E,ρ)\mathscr{L}(E,\rho) in the ring of integers of Q(ρ)\mathbb{Q}(\rho), plausibly from the Galois module structure of III\mathrm{III}.

Problem Statement

For every elliptic curve E/QE/\mathbb{Q} and every Artin representation ρ\rho of Q\mathbb{Q}, construct an invariant BSD(E,ρ)∈C×\mathrm{BSD}(E,\rho)\in\mathbb{C}^\times by intrinsic arithmetic means — not defined by normalising the LL-function L(E,ρ,s)L(E,\rho,s) — together with a conjectural identity L(E,ρ)=BSD(E,ρ)\mathscr{L}(E,\rho)=\mathrm{BSD}(E,\rho) for the modified leading term at s=1s=1, thereby answering Problem 2 of Dokchitser–Evans–Wiersema; and justify that this invariant satisfies the axioms (C1)–(C6) of their Conjecture 4 — compatibility with BSD(E/F)\mathrm{BSD}(E/F) under induction, multiplicativity under direct sums, the duality relation with root numbers, reality and sign for self-dual ρ\rho, and in rank zero the rationality, Galois equivariance and (for non-trivial Dirichlet characters of order dd under their stated hypotheses) Z[ζd]\mathbb{Z}[\zeta_d]-integrality — from purely arithmetic hypotheses such as finiteness of the Tate–Shafarevich group and the Galois module or Selmer group structure of EE, without assuming the Birch–Swinnerton-Dyer conjecture, thereby answering their Problem 9.

References

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