February 26, 2024: Article on the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction

Why this matters. Iwasawa theory studies arithmetic invariants of an elliptic curve simultaneously along an infinite tower of number fields, which often reveals structure invisible at a single level. Its Main Conjecture predicts that an analytic object (built from a $p$ -adic $L$ -function) and an algebraic one (built from Galois cohomology) coincide. Establishing it has strong consequences: as the examples below show, it yields a $p$ -converse theorem, new cases of the Birch–Swinnerton-Dyer conjecture, and improved proportions towards Goldfeld’s conjecture on the distribution of ranks in families. The new case here is that we work at Eisenstein primes, where the mod- $p$ Galois representation is reducible and the standard tools fail.

Let $E/\mathbf{Q}$ be an elliptic curve of analytic rank $0$ or $1$ such that the mod- $p$ Galois representation $\rho_p\colon G_\mathbf{Q}\to \operatorname{Aut}(E[p](\overline{\mathbf{Q}}))$ is reducible (“Eisenstein prime”). In our recent preprint “On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction” with Mulun Yin (University of California Santa Barbara), we prove (a) the Iwasawa Main Conjecture and Perrin-Riou’s Heegner point Main Conjecture over a Heegner field in which the odd prime $p$ of good or bad multiplicative reduction is split, and, as applications, (b) a $p$ -converse theorem (also for modular abelian varieties) and (c) the $p$ -part of strong BSD if $p > 2$ has good reduction.

Here are the elliptic curves $E$ over $\mathbf{Q}$ in the LMFDB with $E(\mathbf{Q})_\mathrm{tors}\cong\mathbf{Z}/p$ , rank $0$ or $1$ , and $p$ a prime of good reduction: $p=3$ , $p=5$ , and $p = 7$ .

As an application of (b), we obtain better proportions for the Goldfeld conjecture ( $50\,\%$ of the quadratic twists have rank $0$ and $50\,\%$ have rank $1$ ) for families of elliptic curves having a $3$ -isogeny where $3$ is a prime of good or bad multiplicative reduction. This follows from a combination with results of Ari Shnidman et al. on the average $3$ -Selmer rank.

Let us now show that, even though our results on the $p$ -part of strong BSD are for elliptic curves only (the reason is that [Jetchev–Skinner–Wan] for $\rho_p$ irreducible work over imaginary quadratic fields that do not satisfy the Heegner condition), we can nevertheless use them to treat many more cases for modular abelian varieties with the article “Complete verification of strong BSD for many modular abelian surfaces over $\mathbf{Q}$ ” with Michael Stoll:

Consider the genus $2$ curve $X\colon y^2 = 5x^6 + 10x^5 - 13x^4 - 30x^3 + 9x^2 + 20x - 8$ with absolutely simple semistable RM Jacobian $J$ of squarefree level $5 \cdot 13$ and rank $0$ associated to the newform 65.2.a.c. Note that this curve is not covered by op. cit. The algorithms from op. cit. prove the prime-to- $3$ part of strong BSD for $J/\mathbf{Q}$ with $\#\mathrm{Ш}(J/\mathbf{Q})_\mathrm{an}= 2$ . Our results allow us to also prove the $3$ -part: Let the good ordinary prime $3$ split as $\mathfrak{p}\mathfrak{p}'$ in $\mathbf{Z}[f]=\mathbf{Z}[\sqrt{3}]$ such that $\rho_\mathfrak{p}$ is irreducible and $\rho_{\mathfrak{p}'}$ is reducible with a trivial $1$ -dimensional subrepresentation coming from the $3$ -torsion. The results and algorithms of op. cit. prove that the Heegner index $I_K$ for $D_K = -131$ (so $3$ is split in $\mathcal{O}_K$ ) equals $\mathfrak{p}'$ times some ideal lying above $2$ , and that the Tamagawa product considered as an ideal in $\mathbf{Z}[f]$ equals $\mathfrak{p}'$ . Hence [loc. cit., Theorem 5.2.2] proves that $\mathrm{Ш}(J/\mathbf{Q})[\mathfrak{p}] = 0$ . Our IMC (a) shows that $v_{\mathfrak{p}'}(\#\mathrm{Ш}(J/K)[\mathfrak{p}'^\infty]) = 2v_{\mathfrak{p}'}(I_K) - 2v_{\mathfrak{p}'}(\operatorname{Tam}(J/\mathbf{Q})) = 2 - 2 = 0$ . Since $\mathfrak{p}' \nmid 2$ , we have $\mathrm{Ш}(J/K)[\mathfrak{p}'] \simeq\mathrm{Ш}(J/\mathbf{Q})[\mathfrak{p}'] \oplus \mathrm{Ш}(J^K/\mathbf{Q})[\mathfrak{p}']$ , hence we get $\mathrm{Ш}(J/\mathbf{Q})[\mathfrak{p}'] = 0$ , so strong BSD holds for $J/\mathbf{Q}$ with $\mathrm{Ш}(J/\mathbf{Q}) \cong\mathbf{Z}/2$ .