Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textsf{E}/\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">E</mi> <mo stretchy="false">/</mo> <mi mathvariant="double-struck">Q</mi> </mrow> </math></EquationSource> </InlineEquation> be an elliptic curve. By the modularity theorem, it admits a surjection from a modular curve <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$X_0(N) \rightarrow \textsf{E}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="sans-serif">E</mi> </mrow> </math></EquationSource> </InlineEquation>, and the minimal degree among such maps is called the modular degree of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textsf{E}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">E</mi> </math></EquationSource> </InlineEquation>. By the Mordell–Weil Theorem, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textsf{E}(\mathbb {Q})\simeq \mathbb {Z}^r \oplus T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">E</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">)</mo> </mrow> <mo>≃</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>r</mi> </msup> <mo>⊕</mo> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> for some nonnegative integer r and some finite group T. Watkins’ Conjecture predicts that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$2^r$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation> divides the modular degree, thus suggesting an intriguing link between these geometrically- and algebraically-defined invariants. We offer some new cases of Watkins’ Conjecture, specifically for elliptic curves with additive reduction at 2, good reduction outside of at most two odd primes, and a rational point of order two.

Elliptic curves of conductor $2^m p$, quadratic twists, and Watkins’ conjecture

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发表日期：

2025年9月16日

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Jeffrey Hatley,
Debanjana Kundu

摘要

Let $\textsf{E}/\mathbb {Q}$ $E / Q$ be an elliptic curve. By the modularity theorem, it admits a surjection from a modular curve $X_0(N) \rightarrow \textsf{E}$ $X_{0} (N) \to E$ , and the minimal degree among such maps is called the modular degree of $\textsf{E}$ $E$ . By the Mordell–Weil Theorem, $\textsf{E}(\mathbb {Q})\simeq \mathbb {Z}^r \oplus T$ $E (Q) ≃ Z^{r} \oplus T$ for some nonnegative integer r and some finite group T. Watkins’ Conjecture predicts that $2^r$ $2^{r}$ divides the modular degree, thus suggesting an intriguing link between these geometrically- and algebraically-defined invariants. We offer some new cases of Watkins’ Conjecture, specifically for elliptic curves with additive reduction at 2, good reduction outside of at most two odd primes, and a rational point of order two.

Elliptic curves of conductor \(2^m p\), quadratic twists, and Watkins’ conjecture

摘要