594 research outputs found
On the Fermat-type Equation
We prove that the Fermat-type equation has no solutions
satisfying and when is not a square
mod~. This improves to approximately the Dirichlet density of the
set of prime exponents to which the previous equation is known to not have such
solutions. For the proof we develop a criterion of independent interest to
decide if two elliptic curves with certain type of potentially good reduction
at 2 have symplectically or anti-symplectically isomorphic -torsion modules
Base change for Elliptic Curves over Real Quadratic Fields
Let E be an elliptic curve over a real quadratic field K and F/K a totally
real finite Galois extension. We prove that E/F is modular.Comment: added a short proof of Proposition 2.1 and a few more small changes
to improve readabilit
Criteria for irreducibility of mod p representations of Frey curves
Let K be a totally real Galois number field and let A be a set of elliptic
curves over K. We give sufficient conditions for the existence of a finite
computable set of rational primes P such that for p not in P and E in A, the
representation on E[p] is irreducible. Our sufficient conditions are often
satisfied for Frey elliptic curves associated to solutions of Diophantine
equations; in that context, the irreducibility of the mod p representation is a
hypothesis needed for applying level-lowering theorems. We illustrate our
approach by improving on an existing result for Fermat-type equations of
signature (13, 13, p).Comment: Some minor misprints have been corrected. The paper will appear in
Journal de Th\'eorie des Nombres de Bordeau
The Asymptotic Fermat's Last Theorem for Five-Sixths of Real Quadratic Fields
Let be a totally real field. By the asymptotic Fermat's Last Theorem over
we mean the statement that there is a constant such that for prime
exponents the only solutions to the Fermat equation with , , in are the trivial ones satisfying . With
the help of modularity, level lowering and image of inertia comparisons we give
an algorithmically testable criterion which if satisfied by implies the
asymptotic Fermat's Last Theorem over . Using techniques from analytic
number theory, we show that our criterion is satisfied by for a subset of having density among the
squarefree positive integers. We can improve this to density 1 if we assume a
standard "Eichler-Shimura" conjecture.Comment: 20 pages. New title. The paper is rewritten and reorganized (a second
time). The proofs are substantially shorter and more efficien
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