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Award ID contains: 1945452

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  1. ; ; ; (, Research in Number Theory)
    Abstract We complete the computation of all$$\mathbb {Q}$$ Q -rational points on all the 64 maximal Atkin-Lehner quotients$$X_0(N)^*$$ X 0 ( N ) ∗ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levelsN, we classify all$$\mathbb {Q}$$ Q -rational points as cusps, CM points (including their CM field andj-invariants) and exceptional ones. We further indicate how to use this to compute the$$\mathbb {Q}$$ Q -rational points on all of their modular coverings. 
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  2. ; ; ; (, 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
  3. ; ; (, 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
  4. ; ; (, Mathematics of Computation)
    Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over Q \mathbb {Q} . Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields Q ( d ) \mathbb {Q}(\sqrt {d}) with | d | > 10 4 |d| > 10^4 we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over 19 19 quadratic fields, including Q ( 213 ) \mathbb {Q}(\sqrt {213}) and Q ( −<#comment/> 2289 ) \mathbb {Q}(\sqrt {-2289}) . To make this procedure work, we determine all of the finitely many quadratic points on the modular curves X 0 ( 125 ) X_0(125) and X 0 ( 169 ) X_0(169) , which may be of independent interest. 
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  5. ; ; ; (, Advances in Mathematics)
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  6. ; ; ; (, Advances in Mathematics)
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  7. ; ; ; ; (, Compositio Mathematica)
    We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus$$g>1$$whose Jacobians have Mordell–Weil rank$$g$$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients$$X_0^+(N)$$of prime level$$N$$, the curve$$X_{S_4}(13)$$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve$$X_{\scriptstyle \mathrm { ns}} ^+ (17)$$. 
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  8. (, Snapshots of modern mathematics from Oberwolfach)
    Accepted Manuscript Full Text Available
  9. ; ; (, La Matematica)
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