Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$E/\textbf{Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo stretchy="false">/</mo> <mi mathvariant="bold">Q</mi> </mrow> </math></EquationSource> </InlineEquation> be an elliptic curve of conductor N and let f be the cuspidal eigenform on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Gamma _0(N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> associated to E by the modularity theorem. Denote by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K_\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> the anticyclotomic <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textbf{Z}_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="bold">Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-extension of an imaginary quadratic field K, where p is a prime number unramified in K. Under appropriate arithmetic assumptions we prove the main conjectures of Iwasawa theory for E over <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$K_\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>. Our results cover both the cases where p is good ordinary and supersingular for E, and both the definite and indefinite settings. This leaves out a single case, which we term exceptional, for which we establish one of the two expected divisibilities.

The anticyclotomic main conjectures for elliptic curves

期刊论文
发表日期：

2026年4月17日

查看全文

Massimo Bertolini,
Matteo Longo,
Rodolfo Venerucci

摘要

Let $E/\textbf{Q}$ $E / Q$ be an elliptic curve of conductor N and let f be the cuspidal eigenform on $\Gamma _0(N)$ $Γ_{0} (N)$ associated to E by the modularity theorem. Denote by $K_\infty $ $K_{\infty}$ the anticyclotomic $\textbf{Z}_p$ $Z_{p}$ -extension of an imaginary quadratic field K, where p is a prime number unramified in K. Under appropriate arithmetic assumptions we prove the main conjectures of Iwasawa theory for E over $K_\infty $ $K_{\infty}$ . Our results cover both the cases where p is good ordinary and supersingular for E, and both the definite and indefinite settings. This leaves out a single case, which we term exceptional, for which we establish one of the two expected divisibilities.