论文标题
苏尔维斯特的猜想和戈德菲尔德的猜想
Supersingular main conjectures, Sylvester's conjecture and Goldfeld's conjecture
论文作者
论文摘要
我们证明了椭圆曲线的$ p $ converse定理$ e/\ mathbb {q} $，通过整数$ \ mathcal {o} _k $ ytiminary quadratic字段$ k $进行复杂的乘法，其中$ p $在其中损坏了$ k $。也就是说，让$ r_p = \ mathrm {corank} _ {\ mathbb {z} _p} _p} \ mathrm {sel} _ {p^{\ infty}}}（e/\ mathbb {q}）$ \ Mathrm {rank} _ {\ Mathbb {z}} e（\ MathBb {q}）= \ Mathrm {ord} _ {s = 1} l（e/\ mathbb {q}，s），s）= r_p $ and $ \ \＃\＃\ \ \ \ \＃iff}特别是，这在两个经典的二芬太汀问题上都有应用。首先，它解决了Sylvester的猜想，这表明对于所有Primes $ \ ell \ equiv 4,7,8 \ pmod {9} $，存在$（x，y）\ in \ in \ in \ mathbb {q}^{q}^{\ oplus 2} $ x^$ x^3 + y^3 + y^3 + y^3 + y^= \ Ell。其次，与史密斯的工作相结合，它解决了100％的案件中的一致数量问题，并在戈德菲尔德（Goldfeld）的猜想中建立了一致数字家族的二次曲折等级的猜想。显示上述$ p $ converse定理的方法依赖于iwasawa理论在非split Primes的假想二次次数和相对$ p $ -Adic Hodge理论之间的相互作用。特别是，我们表明某个de rham期间$ q _ {\ mathrm {dr}} $可用于构造Hecke角色和新形式的反气旋$ P $ -ADIC $ l $ l $ functions，在中央关键范围内插值反环球曲折的积极体重。此外，可以通过新的“ Coleman Map”将椭圆形单元的Iwasawa模块与这些反风速$ P $ -ADIC $ L $ functions相关联，这是大概的，即$ Q _ {\ Mathrm {dr}} $ - Coleman Power Series Map的扩展。使用此功能，我们为椭圆形单元制定并证明了一种新的Rubin型主要猜想，最终与Heegner点有关，以证明$ p $ - converse定理。
We prove a $p$-converse theorem for elliptic curves $E/\mathbb{Q}$ with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$ in which $p$ is ramified. Namely, letting $r_p = \mathrm{corank}_{\mathbb{Z}_p}\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})$, we show that $r_p \le 1 \implies \mathrm{rank}_{\mathbb{Z}}E(\mathbb{Q}) = \mathrm{ord}_{s = 1}L(E/\mathbb{Q},s) = r_p$ and $\#\mathrm{Sha}(E/\mathbb{Q}) < \infty$. In particular, this has applications to two classical Diophantine problems. First, it resolves Sylvester's conjecture on rational sums of cubes, showing that for all primes $\ell \equiv 4,7,8 \pmod{9}$, there exists $(x,y) \in \mathbb{Q}^{\oplus 2}$ such that $x^3 + y^3 = \ell$. Second, combined with work of Smith, it resolves the congruent number problem in 100\% of cases and establishes Goldfeld's conjecture on ranks of quadratic twists for the congruent number family. The method for showing the above $p$-converse theorem relies on new interplays between Iwasawa theory for imaginary quadratic fields at nonsplit primes and relative $p$-adic Hodge theory. In particular, we show that a certain de Rham period $q_{\mathrm{dR}}$ can be used to construct anticyclotomic $p$-adic $L$-functions for Hecke characters and newforms, interpolating anticyclotomic twists of positive Hodge-Tate weight in the central critical range. Moreover, one can relate the Iwasawa module of elliptic units to these anticyclotomic $p$-adic $L$-functions via a new "Coleman map", which is, roughly speaking, the $q_{\mathrm{dR}}$-expansion of the Coleman power series map. Using this, we formulate and prove a new Rubin-type main conjecture for elliptic units, which is eventually related to Heegner points in order to prove the $p$-converse theorem.
