A $p$-Converse theorem for Real Quadratic Fields

Technical Summary: A p-Converse Theorem for Real Quadratic Fields

Problem Statement
The paper addresses the "p-converse" problem for elliptic curves defined over real quadratic fields ($F$). Specifically, it seeks to establish the converse of the Gross-Zagier and Kolyvagin theorem in the context of rank 1. While the classical theorem states that if the analytic rank is 1, then the algebraic rank is 1 and the Tate-Shafarevich group is finite, the converse asks: if the algebraic rank is 1 and the $p$-part of the Tate-Shafarevich group is finite, does it follow that the analytic rank is 1?

Prior to this work, such converse results were largely established for elliptic curves over $\mathbb{Q}$ (e.g., by Skinner, Zhang, and Venerucci) or for specific reduction types. This paper extends these results to the setting of real quadratic fields, where $p > 5$ is a rational prime inert in $F$, and the elliptic curve $E/F$ has split multiplicative reduction at the prime $\mathfrak{p}$ lying above $p$.

Methodology
The authors employ a sophisticated synthesis of Hida theory, Iwasawa theory, and the theory of Selmer complexes to bridge the gap between algebraic and analytic ranks. The proof strategy relies on the following key ingredients:

1. Hida Families and Big Galois Representations: The elliptic curve $E$ is associated with a Hilbert modular form $f$ of parallel weight 2. The authors utilize the Hida family $\mathbf{f}$ passing through $f$ and the associated big Galois representation $T_f$. They work within a localized Hida family over a discrete valuation ring $I_P$.
2. Iwasawa Main Conjecture (Wan's Theorem): The proof relies heavily on the Iwasawa Main Conjecture for Hilbert modular forms over a totally imaginary quadratic extension $K/F$ (where $p$ splits in $K$), as established by Wan [Wan15]. This provides an inclusion of the characteristic ideal of the Selmer group into the ideal generated by a $p$-adic L-function.
3. Three-Variable and Four-Variable $p$-adic L-functions: The authors construct and utilize central critical $p$-adic L-functions. They relate the order of vanishing of a central critical $p$-adic L-function $L^{cc}_p(\mathbf{f}_\infty/K, k)$ at $k=2$ to the length of the Pontryagin dual of a specific Selmer group.
4. Nekovář's Selmer Complexes: To analyze the structure of the Selmer groups, the authors use Nekovář's theory of Selmer complexes. This allows for the definition of a $p$-adic weight pairing on the cohomology of these complexes.
5. Control Theorems and Pairing Computations: A crucial step involves proving a control theorem that relates the Selmer group over the infinite extension to the strict Selmer group over the base field. By computing the $p$-adic weight pairing (specifically relating it to $p$-adic logarithms of points on the elliptic curve), the authors determine the exact length of the Selmer module.

Key Contributions and Results
The paper establishes the following main theorem (Theorem 1.3 / Theorem 9.2):

• Main Theorem: Let $F$ be a real quadratic field and $p > 5$ a prime inert in $F$. Let $E/F$ be an elliptic curve with split multiplicative reduction at the prime $\mathfrak{p} | p$. Assuming the residual representation is absolutely irreducible (irred) and a minimal modular lifting exists (MML), if:

  1. $\text{rank}_{\mathbb{Z}} E(F) = 1$, and
  2. The $p$-part of the Tate-Shafarevich group $\text{Ш}(E/F)_{p^\infty}$ is finite,
     Then the analytic rank is 1: $\text{ord}_{s=1} L(E/F, s) = 1$.

• Equivalence: Combining this with Zhang's generalization of the Gross-Zagier theorem, the authors establish an equivalence for the rank 1 case:
  $$\text{rank}_{\mathbb{Z}} E(F) = 1 \text{ and } \#\text{Ш}(E/F) < \infty \iff \text{ord}_{s=1} L(E/F, s) = 1$$

• Application to $\mathbb{Q}$: As a consequence, the authors prove a strengthened $p$-converse theorem for elliptic curves over $\mathbb{Q}$ (Theorem 1.5). They remove a technical hypothesis (the existence of a specific prime $q$ with certain properties) required in Venerucci's earlier work [Ven16], provided $p > 5$ and $E$ has split multiplicative reduction at $p$.

Significance and Claims
The paper claims to be the first to establish a rank 1 $p$-converse theorem for elliptic curves over a real quadratic field. The significance lies in:

• Extension of Scope: Moving beyond the rational field $\mathbb{Q}$ to real quadratic fields, which presents additional complexities due to the nature of Hilbert modular forms and the behavior of primes in quadratic extensions.
• Refinement of Existing Results: The application to $\mathbb{Q}$ demonstrates that the strategy developed for real quadratic fields can simplify and strengthen existing results over $\mathbb{Q}$ by removing restrictive hypotheses.
• Methodological Synthesis: The work successfully integrates the Iwasawa Main Conjecture of Wan with Nekovář's Selmer complexes and $p$-adic weight pairings to handle the split multiplicative reduction case in a non-CM setting over a totally real field.

The authors note that their results hold for $p=5$ if a specific exceptional case regarding the projective image of the Galois representation is excluded, a limitation imposed by the reliance on Fujiwara's results used in Wan's theorem. The paper concludes by suggesting that these techniques could potentially be extended to general totally real number fields where Leopoldt's conjecture holds, though it acknowledges that the CM case would likely require different methods.