BSD and the p-converse (Guest Post: Ashay Burungale)

(Here is the first of a series of guest blog posts by number theorists on the recent OpenAI results, this one by Ashay Burungale.)

Let \(E/\mathbf{Q}\) be an elliptic curve. Its rational points form a finitely generated abelian group. The Birch–Swinnerton-Dyer conjecture, formulated in the 1960s, predicts that its rank equals the order of vanishing of the Hasse–Weil \(L(E,s)\) at \(s=1\). It also predicts finiteness of the Tate–Shafarevich group \(\text{Ш}(E/\mathbf{Q})\) and expresses the leading coefficient in terms of \(\#\text{Ш}(E/\mathbf{Q})\). Gross–Zagier’s work in the early 1980s and Kolyvagin’s work later that decade, together with modularity, establish the rank equality and finiteness when the analytic rank is zero or one.

The announcements

For a prime \(p\), recall the Kummer exact sequence

Put \(s_p(E)=\mathop{\mathrm{corank}}\nolimits_{\mathbf{Z}_p}\mathop{\mathrm{Sel}}\nolimits_{p^\infty}(E/\mathbf{Q})\). Thus \(s_p(E)\) is the rank of \(E(\mathbf{Q})\) plus the corank of \(\text{Ш}(E/\mathbf{Q})[p^\infty]\). A \(p\)-converse seeks to recover analytic rank from this Selmer corank.

The first converse paper [OA1] announces the following for every \(E/\mathbf{Q}\) and every prime \(p\), treating odd \(p\) and importing the case \(p=2\) from a companion paper:

The following is my account of the proposed odd-prime argument. The correctness of the announced proofs remains under investigation.

The companion [2-converse] also announces Goldfeld’s density conjecture. The leading-term paper [BSD1] treats the odd-primary components of the BSD formula and imports the \(2\)-primary component from [BSD2]. Together they announce the full leading-term formula whenever \(s_p(E)\le1\) for some prime \(p\).

The Goldfeld paper states that \(50\%\) of the quadratic twists \(E^{(d)}\) of each \(E/\mathbf{Q}\) have analytic rank zero and \(50\%\) have analytic rank one, with non-zero squarefree \(d\) ordered by \(|d|\). Smith’s theorem [Selmer-distribution] gives these proportions for \(s_2(E^{(d)})\). The announced \(2\)-converse and leading-term results would yield full BSD for a density-one set of twists of every \(E/\mathbf{Q}\) (see also [rank-zero-converse]).

Another application concerns Sylvester’s 1879 conjecture: every prime \(\ell\equiv4,7,8\pmod9\) is a sum of two rational cubes. The recent resolution by Yin [Sylvester1] for \(4,7\pmod9\) and Burungale–Tian [Sylvester2] for \(8\pmod9\) construct rational points explicitly. For \(E_\ell:X^3+Y^3=\ell Z^3\), we have \(s_3(E_\ell)=1\), so the announced \(3\)-converse would also imply these cube-sum expressions, albeit inexplicitly.

Prior work

The above questions have been studied systematically since the mid-1980s. The \(p\)-converses are now largely known for odd primes \(p\) of good reduction. The survey [BSD-survey], Sections 3–4, describes some of these developments.

In rank zero, Rubin’s main-conjecture methods give CM converses (see also [rank-zero-converse], for every prime). For general elliptic curves, Kato’s Euler system [Euler-system] gives one divisibility in the cyclotomic main conjecture, while Skinner–Urban’s Eisenstein congruences [main-conjecture] give the reverse divisibility for ordinary primes under their hypotheses, leading to rank-zero converses and cases of BSD (see also [supersingular]). The first general rank-one converses grew out of work of Skinner [converse1] and Zhang [converse2] around 2012. Both approaches rely on Iwasawa theory, more directly in Skinner’s approach.

The rank-one goal is to show that a suitable Heegner point \(y\in E(K)\) has infinite order. Here \(K\) is imaginary quadratic, every conductor prime splits in \(K\), and the quadratic twist \(E^{(K)}\) has analytic rank zero.

In Skinner’s approach, suppose \(p=v\bar v\) splits in \(K\) and put \(V=T_pE\otimes_{\mathbf{Z}_p}\mathbf{Q}_p\). Write \(H_f^1(K,V)\) for the associated Bloch–Kato Selmer group, defined here by the local Kummer images. Put

The Bertolini–Darmon–Prasanna (BDP) formula [BDP] relates the trivial-character value of an anticyclotomic \(p\)-adic \(L\)-function to the Heegner logarithm:

Let \(X_v\) be the Pontryagin dual of the corresponding discrete Selmer group over the anticyclotomic \(\mathbf{Z}_p\)-extension \(K_\infty/K\). Under suitable hypotheses at an odd prime \(p\) of good reduction, Wan’s divisibility [divisibility] gives

Wan constructs congruences between Klingen Eisenstein series and cusp forms on \(\mathrm{GU}(3,1)\), then extracts Galois extensions. If \(H_f^1(K,V)=\mathbf{Q}_p a\) and \(\mathop{\mathrm{loc}}\nolimits_v(a)\ne0\), Poitou–Tate duality gives \(S_v=0\). Specialization control makes \(\mathop{\mathrm{char}}\nolimits_\Lambda(X_v)\) nonzero at the trivial character, and the divisibility, with the BDP formula imply that \(\log_v y\ne0\) and so \(y\) is non-torsion.

To get around the possible vanishing of \(\mathop{\mathrm{loc}}\nolimits_v(a)\), a key idea is to employ Iwasawa theory of norm-compatible Heegner points, together with Wan’s divisibility, to produce a class surviving specialization. See also [Heegner-main-conjecture1], [CM-converse], and [Heegner-main-conjecture2], as well as the supersingular [supersingular-converse] and Eisenstein [Eisenstein-converse] cases. These methods require control of the local condition at \(p\) along \(K_\infty\), which is not yet available at primes of additive reduction. This has been the essential difficulty in establishing the rank-one \(p\)-converse at additive primes.

The Sylvester strategy

The Sylvester proof [Sylvester2] for primes \(\ell\equiv8\pmod9\) is not Iwasawa-theoretic. An auxiliary prime \(q\) split in \(\mathbf{Q}(\omega)\) is chosen, with \(\omega\) a primitive cube root of unity. It provides an orbit \(Z\) of CM points on an elliptic curve \(A\). Its ordinary trace under a cyclic group \(C\) vanishes. The idea is to divide before taking the trace: writing \(N_C=\sum_{g\in C}g\) and \(\lambda=1-\omega\), one chooses \([\lambda]R=Z\) and proves

A key step is to prove non-vanishing of this division boundary \(N_{C}(R)\), using Frobenius at \(\ell\). Cubic-character projections of the divided orbit, with the requisite integrality, give a non-torsion point.

The rank-one strategy in OA1

We illustrate the rank-one case for odd \(p\). Rather than varying along an anticyclotomic extension at \(p\), a new idea of OA1 is to employ ring-class characters ramified at auxiliary primes away from \(p\). These primes are chosen to obstruct lifts of Selmer classes. Theta lifts connect weighted logarithms of Heegner points to congruences with cusp forms, whose Galois representations give a lower bound for the Selmer group.

Assume \(s_p(E)=1\), and let \(N\) be the conductor of \(E\). Choose \(K\) so that every prime dividing \(2Np\) splits in \(K\) and \(E^{(K)}\) has analytic rank zero. Then \(H_f^1(K,V)=\mathbf{Q}_p a\). The aim is to prove that the conductor-one Heegner trace \(y\in E(K)\) has infinite order. Duality gives

Tame deformation. Choose primes \(q_i\not\mid Np\) split in \(K\), whose ring-class groups (of conductor \(q_i\)) admit cyclic quotients \(\Delta_i=\langle X_i\rangle\) of order \(p^{n_i}\), with \(n_i\to\infty\); see OA1, Proposition 3.2. Write \(H_{q_i}\) for the corresponding ring-class fields. The quotient character \(\tau_i:G_K\to\mathbf{Z}/p^{n_i}\mathbf{Z}\), where \(G_K=\mathop{\mathrm{Gal}}\nolimits(\overline K/K)\), is chosen to vanish on decomposition groups above \(p\) and at bad primes. Put \(\nu_i(g)=X_i^{\tau_i(g)}\) and \(t=X_i-1\), the augmentation parameter at the trivial character.

For a finite extension \(k/\mathbf{Q}_p\) with ring of integers \(\mathcal{O}\), the finite-stage substitution

is valid for odd \(p\), since \(p^{n_i}\mid\binom{p^{n_i}}2\). On a Galois-stable lattice the twisted action is \(g\mapsto\rho_E(g)(1+t)^{\tau_i(g)}\).

For \(b=1,2,3\), keep only the first \(b\) coefficients in \(t\). OA1 takes coefficientwise limits of the finite-stage cocycles on sequences of Galois elements, using a fixed ultrafilter and uniformly bounded denominators. The resulting coefficient ring is \(R_b=k[t]/(t^b)\). Let \(S_b\) be the modified Selmer module, relaxed at \(v\), strict at \(\bar v\), and allowing auxiliary ramification. Note that \(S_1=S_v\otimes_{\mathbf{Q}_p}k\). For \(b=2,3\), the specialization property in OA1 identifies the kernel of \(S_b\to S_1\) with the image of \(t:S_{b-1}\to S_b\).

The congruence bound. The goal is \(\dim_k S_b\ge b\) when the first \(b\) weighted logarithmic coefficients below tend to zero. For a Heegner point \(y_{q_i}\in E(H_{q_i})\) and a fixed \(p\)-adic embedding above \(v\), form

where \(r_i(g)\) represents \(\tau_i(g)\) in \(\{0,\ldots,p^{n_i}-1\}\). At \(t=0\), the Hecke trace relation gives \(\ell_{i,0}=(a_{q_i}-2)\log_v y\), with \(a_q=q+1-\#E(\mathbf{F}_q)\).

For the theta construction (OA1, §4), OA1 fixes a real quadratic field \(F\), using the Goldfeld announcement, so that the additional quadratic-twist summand of the modified Selmer group over \(L=KF\) vanishes. The auxiliary primes \(q_i\) are chosen to split also in \(F\). Let \(A\) be a two-dimensional Hermitian space over \(L\), definite at both real places, and \(W\) a four-dimensional skew-Hermitian space whose unitary group has signature \((3,1)\) at both places. Base change and Jacquet–Langlands transfer attach to \(E\) a fixed representation \(\pi\) of \(U(A)\). Fix a vector \(\varphi\in\pi^\vee\). The theta family on \(U(W)\) has the form

Here \(\theta_{\Phi_{i,m}}\) is the theta kernel determined by a Schwartz function \(\Phi_{i,m}\), and \(\mathcal{C}\) is a fixed algebraic character. The index \(i\) refers to the tame character \(\nu_i\) and the local Schwartz function at \(q_i\); the other finite local data stay fixed. The character \(\nu_i\) is pulled back to the norm-one torus of \(L/F\) through \(N_{L/K}\). The integer \(m\) changes the algebraic twist and the archimedean Schwartz function, hence the weight. Thus the untwisted representation \(\pi\) is fixed, while its twists and the theta kernels vary.

Cuspidality requires vanishing of the constant Fourier–Jacobi coefficient at every boundary component. Each is a theta lift to a definite rank-two unitary group. With local intertwiners fixed, these lifts are controlled by one global invariant \(S_{i,m}\). The Rallis and Waldspurger formulas give the period relation

in the tame group algebra (OA1, Lemma 5.2). Here \(A_{i,m},B_{i,m}\) collect the local factors, while \(P_{0,i,m},P_{1,i,m}\) are toric periods for \(E\) and an auxiliary quadratic twist. For \(m_i\to\infty\) with \(m_i\to0\) \(p\)-adically, OA1 relates \(P_{0,i,m_i}\) to \(\mathscr{L}_i(t)\) by a BDP-type comparison.

Integral control and cuspidal congruences. In OA1, \(\ell_{i,j}\to0\) for \(j<b\) implies that every constant Fourier–Jacobi coefficient of \(\Theta_i=\Theta_{i,m_i}\) tends to zero modulo \(t^b\) (§5, Proposition 5.1). The integral estimates are uniform in conductor and weight: bounded denominators for the periods and \(S_{i,m}\), bounded valuation of \(A_{i,m}(0)\ne0\), and uniform bounds for the local maps expressing boundary restrictions in terms of \(S_{i,m}\) (§4, Lemma 4.5 and Lemma 4.6). At the same time, a positive-index Fourier–Jacobi coefficient has valuation bounded above at \(t=0\) in the same integral lattice (§6, Proposition 6.5). Thus the whole form does not vanish along with its boundary. Multiplication by a lift \(h_i\) of a power of the Hasse invariant on the ordinary locus, followed by lifting to higher weight, gives (§7)

The surviving coefficient is used to preserve the cuspidal Hecke congruence over all of \(R_b\). The four-dimensional Galois representations are compared with a twist of \(V\) and two characters. An off-diagonal \(R_b\)-module \(D_b\) records extensions of one character by the \(V\)-block. Its dual gives classes satisfying the modified Selmer conditions, including splitting at \(\bar v\), and the resulting bound is (§9, Proposition 9.7)

Suppose now, for a contradiction, that \(y\) is torsion. If \(\mathop{\mathrm{loc}}\nolimits_v(a)\ne0\), the trace relation gives \(\ell_{i,0}=0\). Then (6) with \(b=1\) contradicts \(S_1=0\) from (2). It remains to treat the case of zero localization.

Zero localization and auxiliary primes. Suppose \(\mathop{\mathrm{loc}}\nolimits_v(a)=0\), so \(\dim_k S_1=2\). Here \(E(\mathbf{Q})\) is finite: a non-torsion rational point would have nonzero local logarithm. Thus the Selmer corank comes entirely from \(\text{Ш}(E/\mathbf{Q})[p^\infty]\). The aim is to prove \(\dim_k S_3\le2\).

A first-order lift of \(x\in S_1\) is an element of \(S_2\) reducing to \(x\) at \(t=0\). Global reciprocity gives two linear conditions on \(x\) that must hold for such a lift to exist. In a basis \((a,h)\) of \(S_1\), write \(x=\alpha a+\beta h\). OA1 chooses the limiting auxiliary Frobenius \(\gamma\) so that these conditions are

A commutator calculation and Chebotarev supply such auxiliary primes (OA1, Lemmas 3.5–3.6). Since \(H_\gamma\) is invertible, only \(x=0\) can lift. Thus \(S_2\to S_1\) is the zero map, as is \(S_3\to S_1\), which factors through it. The specialization property above gives \(S_2=tS_1\) and \(S_3=tS_2\), hence

Division of the Heegner class. To obtain the opposite bound \(\dim_k S_3\ge3\) from (6), it suffices to show \(\ell_{i,0},\ell_{i,1},\ell_{i,2}\to0\). OA1 does this by dividing the weighted Heegner Kummer class twice (OA1, Proposition 3.9). Its limit \(\mathscr{K}_3\) over \(R_3\) satisfies Kummer conditions at both primes over \(p\). These conditions are preserved by division, since the tame character is trivial locally at \(p\).

The first division follows from torsion-ness: the trace relation gives \(\mathscr{K}_3(0)=0\) in rational Kummer cohomology, hence \(\mathscr{K}_3=t\mathscr{K}’\) with \(\mathscr{K}’\) over \(R_2\). Its specialization lies in \(H_f^1(K,V)\otimes k=ka\), by the Kummer conditions. Complex conjugation \(c\) fixes \(ka\): the negative Selmer eigenspace comes from the analytic-rank-zero twist \(E^{(K)}\) and vanishes.

The second division follows by comparing linear coefficients in the Heegner conjugation relation. This relation has sign \(-w(E)=+1\), since \(w(E)=-1\) by \(p\)-parity, and a group-ring factor congruent to \(1\) modulo \(t\). But conjugation sends \(t\) to \((1+t)^{-1}-1\equiv-t\pmod{t^2}\). Since \(\mathscr{K}_3\equiv t\mathscr{K}'(0)\pmod{t^2}\), the factor congruent to \(1\) does not change its linear coefficient. Comparing the two sides modulo \(t^2\) gives

Thus \(\mathscr{K}'(0)^c=-\mathscr{K}'(0)\). On the other hand, \(\mathscr{K}'(0)\in ka\) is fixed by conjugation, so \(\mathscr{K}'(0)=0\). A second division yields

In turn, OA1’s uniform Kummer-to-logarithm estimate (Lemma 3.8) gives \(\ell_{i,0},\ell_{i,1},\ell_{i,2}\to0\). Equation (6) then contradicts (8):

Relation to earlier work. The first half of the argument may be viewed as a tame analogue of Skinner’s converse strategy: an automorphic congruence turns vanishing of a Heegner logarithm into non-zero Selmer classes. (For a different use of tame automorphic congruences, see [Brumer–Stark].) A key novelty is the division of the Heegner class and its connection, through a BDP-type formula, with this tame congruence argument. This supplies the extra logarithmic vanishing needed when \(\mathop{\mathrm{loc}}\nolimits_v(a)=0\). As in Sylvester, division recovers information lost by the ordinary trace.

Acknowledgements. The author thanks Frank Calegari, Marco Sangiovanni Vincentelli, Chris Skinner and Ye Tian for helpful discussions.