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Creators/Authors contains: "Mazur, Barry"

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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. ; ; (, Bulletin of the American Mathematical Society)
    Andrew Ogg’s mathematical viewpoint has inspired an increasingly broad array of results and conjectures. His results and conjectures have earmarked fruitful turning points in our subject, and his influence has been such a gift to all of us. Ogg’s celebrated torsion conjecture—as it relates to modular curves—can be paraphrased as saying that rational points (on the modular curves that parametrize torsion points on elliptic curves) exist if and only if there is a good geometric reason for them to exist. We give a survey of Ogg’s torsion conjecture and the subsequent developments in our understanding of rational points on modular curves over the last fifty years. 
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    Free, publicly-accessible full text available April 1, 2026
  3. ; ; (, Journal of Number Theory)
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  4. ; ; (, 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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  5. ; ; (, Notices of the International Consortium of Chinese Mathematicians)
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