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Search: MSC category 11D41 ( Higher degree equations; Fermat's equation )

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1. CJM 2011 (vol 64 pp. 282)

Dahmen, Sander R.; Yazdani, Soroosh
Level Lowering Modulo Prime Powers and Twisted Fermat Equations
We discuss a clean level lowering theorem modulo prime powers for weight $2$ cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations $ax^n+by^n+cz^n=0$.

Keywords:modular forms, level lowering, Diophantine equations
Categories:11D41, 11F33, 11F11, 11F80, 11G05

2. CJM 2011 (vol 63 pp. 481)

Baragar, Arthur
The Ample Cone for a K3 Surface
In this paper, we give several pictorial fractal representations of the ample or Kähler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be $1.296 \pm .010$.

Keywords:Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamics
Categories:14J28, , , , 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05

3. CJM 2004 (vol 56 pp. 23)

Bennett, Michael A.; Skinner, Chris M.
Ternary Diophantine Equations via Galois Representations and Modular Forms
In this paper, we develop techniques for solving ternary Diophantine equations of the shape $Ax^n + By^n = Cz^2$, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters $A$, $B$ and $C$. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan--Nagell type.

Categories:11D41, 11F11, 11G05

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