1,608 research outputs found
Point counting on reductions of CM elliptic curves
We give explicit formulas for the number of points on reductions of elliptic
curves with complex multiplication by any imaginary quadratic field. We also
find models for CM -curves in certain cases. This generalizes
earlier results of Gross, Stark, and others.Comment: Minor corrections. To appear in Journal of Number Theor
On certain finiteness questions in the arithmetic of modular forms
We investigate certain finiteness questions that arise naturally when
studying approximations modulo prime powers of p-adic Galois representations
coming from modular forms. We link these finiteness statements with a question
by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms.
Specifically, we conjecture that for fixed N, m, and prime p with p not
dividing N, there is only a finite number of reductions modulo p^m of
normalized eigenforms on \Gamma_1(N). We consider various variants of our basic
finiteness conjecture, prove a weak version of it, and give some numerical
evidence.Comment: 25 pages; v2: one of the conjectures from v1 now proved; v3:
restructered parts of the article; v4: minor corrections and change
Magnetic and structural quantum phase transitions in CeCu6-xAux are independent
The heavy-fermion compound CeCuAu has become a model system for
unconventional magnetic quantum criticality. For small Au concentrations , the compound undergoes a structural transition from
orthorhombic to monoclinic crystal symmetry at a temperature with
for . Antiferromagnetic order sets in
close to . To shed light on the interplay between quantum
critical magnetic and structural fluctuations we performed neutron-scattering
and thermodynamic measurements on samples with . The
resulting phase diagram shows that the antiferromagnetic and monoclinic phase
coexist in a tiny Au concentration range between and . The
application of hydrostatic and chemical pressure allows to clearly separate the
transitions from each other and to explore a possible effect of the structural
transition on the magnetic quantum critical behavior. Our measurements
demonstrate that at low temperatures the unconventional quantum criticality
exclusively arises from magnetic fluctuations and is not affected by the
monoclinic distortion.Comment: 5 pages, 3 figure
Endomorphisms of superelliptic jacobians
Let K be a field of characteristic zero, n>4 an integer, f(x) an irreducible
polynomial over K of degree n, whose Galois group is doubly transitive simple
non-abelian group. Let p be an odd prime, Z[\zeta_p] the ring of integers in
the p-th cyclotomic field,
C_{f,p}:y^p=f(x) the corresponding superelliptic curve and J(C_{f,p}) its
jacobian. Assuming that either n=p+1 or p does not divide n(n-1), we prove that
the ring of all endomorphisms of J(C_{f,p}) coincides with Z[\zeta_p].Comment: Several typos have been correcte
Virtually abelian K\"ahler and projective groups
We characterise the virtually abelian groups which are fundamental groups of
compact K\"ahler manifolds and of smooth projective varieties. We show that a
virtually abelian group is K\"ahler if and only if it is projective. In
particular, this allows to describe the K\"ahler condition for such groups in
terms of integral symplectic representations
Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture
Most, if not all, unconditional results towards the abc-conjecture rely
ultimately on classical Baker's method. In this article, we turn our attention
to its elliptic analogue. Using the elliptic Baker's method, we have recently
obtained a new upper bound for the height of the S-integral points on an
elliptic curve. This bound depends on some parameters related to the
Mordell-Weil group of the curve. We deduce here a bound relying on the
conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable
quantities. We then study which abc-type inequality over number fields could be
derived from this elliptic approach.Comment: 20 pages. Some changes, the most important being on Conjecture 3.2,
three references added ([Mas75], [MB90] and [Yu94]) and one reference updated
[BS12]. Accepted in Bull. Brazil. Mat. So
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