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Creators/Authors contains: "Rubin, Karl"

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  1. ; ; ; (, Mathematics of Computation)
    Let E E be an elliptic curve over Q \mathbb {Q} with Mordell–Weil rank 2 2 and p p be an odd prime of good ordinary reduction. For every imaginary quadratic field K K satisfying the Heegner hypothesis, there is (subject to the Shafarevich–Tate conjecture) a line, i.e., a free Z p \mathbb {Z}_p -submodule of rank 1 1 , in E ( K ) ⊗<#comment/> Z p E(K)\otimes \mathbb {Z}_p given by universal norms coming from the Mordell–Weil groups of subfields of the anticyclotomic Z p \mathbb {Z}_p -extension of K K ; we call it theshadow line. When the twist of E E by K K has analytic rank 1 1 , the shadow line is conjectured to lie in E ( Q ) ⊗<#comment/> Z p E(\mathbb {Q})\otimes \mathbb {Z}_p ; we verify this computationally in all our examples. We study the distribution of shadow lines in E ( Q ) ⊗<#comment/> Z p E(\mathbb {Q})\otimes \mathbb {Z}_p as K K varies, framing conjectures based on the computations we have made. 
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    Free, publicly-accessible full text available July 31, 2026
  2. ; ; (, Journal of Number Theory)
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  3. ; ; (, Acta Arithmetica)
    We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $$\Q$$, paying attention to the uniformity of definitions. The uniformity follows from the simplicity of our first-order definition of $$\Z$$. Namely, we prove that for a large collection of algebraic extensions $$K/\Q$$, $$ \{x \in \oo_K : \text{$$\forall \e \in \oo_K^\times \;\exists \delta \in \oo_K^\times$ such that $$\delta-1 \equiv (\e-1)x \pmod{(\e-1)^2}$$}\} = \Z $$ where $$\oo_K$$ denotes the ring of integers of $$K$$. One of the corollaries of our results is undecidability of the field of constructible numbers, a question posed by Tarski in 1948. \end{abstract} 
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  4. ; ; ; ; (, Algebra & Number Theory)
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