scholarly journals The Brauer–Manin obstruction for integral points on curves

2010 ◽  
Vol 149 (3) ◽  
pp. 413-421 ◽  
Author(s):  
DAVID HARARI ◽  
JOSÉ FELIPE VOLOCH
Keyword(s):  
Elliptic Curves ◽  
Negative Answer ◽  
Rational Curves ◽  
Positive Answer ◽  
Hasse Principle ◽  
Strong Version ◽  
Integral Points ◽  
Local Conditions

AbstractWe discuss the question of whether the Brauer–Manin obstruction is the only obstruction to the Hasse principle for integral points on affine hyperbolic curves. In the case of rational curves we conjecture a positive answer, we prove that this conjecture can be given several equivalent formulations and we relate it to an old conjecture of Skolem. Finally, we show that for elliptic curves minus one point a strong version of the question (describing the set of integral points by local conditions) has a negative answer.

scholarly journals Points of low height on elliptic surfaces with torsion

2010 ◽  
Vol 13 ◽  
pp. 370-387
Author(s):  
Sonal Jain
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Open Subset ◽  
Function Field ◽  
Rational Point ◽  
Torsion Point ◽  
Integral Points ◽  
Elliptic Surfaces ◽  
Canonical Height

AbstractWe determine the smallest possible canonical height$\hat {h}(P)$for a non-torsion pointPof an elliptic curveEover a function field(t) of discriminant degree 12nwith a 2-torsion point forn=1,2,3, and with a 3-torsion point forn=1,2. For eachm=2,3, we parametrize the set of triples (E,P,T) of an elliptic curveE/with a rational pointPandm-torsion pointTthat satisfy certain integrality conditions by an open subset of2. We recover explicit equations for all elliptic surfaces (E,P,T) attaining each minimum by locating them as curves in our projective models. We also prove that forn=1,2 , these heights are minimal for elliptic curves over a function field of any genus. In each case, the optimal (E,P,T) are characterized by their patterns of integral points.

scholarly journals CM RELATIONS IN FIBERED POWERS OF ELLIPTIC FAMILIES

2017 ◽  
Vol 18 (5) ◽  
pp. 941-956 ◽  
Author(s):  
Fabrizio Barroero
Keyword(s):  
Elliptic Curves ◽  
Complex Multiplication ◽  
Positive Answer ◽  
Complex Numbers ◽  
Linearly Independent

Let $E_{\unicode[STIX]{x1D706}}$ be the Legendre family of elliptic curves. Given $n$ points $P_{1},\ldots ,P_{n}\in E_{\unicode[STIX]{x1D706}}(\overline{\mathbb{Q}(\unicode[STIX]{x1D706})})$, linearly independent over $\mathbb{Z}$, we prove that there are at most finitely many complex numbers $\unicode[STIX]{x1D706}_{0}$ such that $E_{\unicode[STIX]{x1D706}_{0}}$ has complex multiplication and $P_{1}(\unicode[STIX]{x1D706}_{0}),\ldots ,P_{n}(\unicode[STIX]{x1D706}_{0})$ are linearly dependent over End$(E_{\unicode[STIX]{x1D706}_{0}})$. This implies a positive answer to a question of Bertrand and, combined with a previous work in collaboration with Capuano, proves the Zilber–Pink conjecture for a curve in a fibered power of an elliptic scheme when everything is defined over $\overline{\mathbb{Q}}$.

Neural constraints on cognitive modularity?

1997 ◽  
Vol 20 (4) ◽  
pp. 575-576 ◽  
Author(s):  
Brian J. Scholl
Keyword(s):  
Negative Answer ◽  
Positive Answer ◽  
Crucial Issue ◽  
Cognitive Modularity

Is (some) innate cognitive modularity consistent with a lack of innate neural modularity? Quartz & Sejnowski's (Q&S's) implicit negative answer to his question fuels their antinativist and antimodular cognitive conclusions. I attempt here to suggest a positive answer and to solicit discussion of this crucial issue.

scholarly journals Bounds on $2$-torsion in class groups of number fields and integral points on elliptic curves

2020 ◽  
Vol 33 (4) ◽  
pp. 1087-1099 ◽  
Author(s):  
M. Bhargava ◽  
A. Shankar ◽  
T. Taniguchi ◽  
F. Thorne ◽  
J. Tsimerman ◽  
...  
Keyword(s):  
Elliptic Curves ◽  
Number Fields ◽  
Class Groups ◽  
Integral Points

scholarly journals Knowledge of future contingents

2021 ◽  
Author(s):  
Andrea Iacona
Keyword(s):  
Negative Answer ◽  
Positive Answer ◽  
Future Contingents ◽  
Epistemic Access

AbstractThis paper addresses the question whether future contingents are knowable, that is, whether one can know that things will go a certain way even though it is possible that things will not go that way. First I will consider a long-established view that implies a negative answer, and draw attention to some endemic problems that affect its credibility. Then I will sketch an alternative line of thought that prompts a positive answer: future contingents are knowable, although our epistemic access of them is limited in some important respects.

scholarly journals One-cycles on rationally connected varieties

2014 ◽  
Vol 150 (3) ◽  
pp. 396-408 ◽  
Author(s):  
Zhiyu Tian ◽  
Hong R. Zong
Keyword(s):  
Complete Intersection ◽  
Finite Fields ◽  
Chow Group ◽  
Rational Curves ◽  
Positive Answer ◽  
Tate Conjecture ◽  
Rationally Connected ◽  
Rationally Connected Varieties

AbstractWe prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. Applying the same technique, we also show that the Chow group of 1-cycles on a separably rationally connected Fano complete intersection of index at least 2 is generated by lines. As a consequence, we give a positive answer to a question of Professor Totaro about integral Hodge classes on rationally connected 3-folds. And by a result of Professor Voisin, the general case is a consequence of the Tate conjecture for surfaces over finite fields.

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