220 research outputs found
Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4
We work out a non-trivial example of lifting a so-called weak eigenform to a
true, characteristic 0 eigenform. The weak eigenform is closely related to
Ramanujan's tau function whereas the characteristic 0 eigenform is attached to
an elliptic curve defined over . We produce the lift by showing
that the coefficients of the initial, weak eigenform (almost all) occur as
traces of Frobenii in the Galois representation on the 4-torsion of the
elliptic curve. The example is remarkable as the initial form is known not to
be liftable to any characteristic 0 eigenform of level 1. We use this example
as illustrating certain questions that have arisen lately in the theory of
modular forms modulo prime powers. We give a brief survey of those questions
Odd values of the Klein j-function and the cubic partition function
In this note, using entirely algebraic or elementary methods, we determine a
new asymptotic lower bound for the number of odd values of one of the most
important modular functions in number theory, the Klein -function. Namely,
we show that the number of integers such that the Klein -function
--- or equivalently, the cubic partition function --- is odd is at least of the
order of for large. This improves
recent results of Berndt-Yee-Zaharescu and Chen-Lin, and approaches
significantly the best lower bound currently known for the ordinary partition
function, obtained using the theory of modular forms. Unlike many works in this
area, our techniques to show the above result, that have in part been inspired
by some recent ideas of P. Monsky on quadratic representations, do not involve
the use of modular forms.
Then, in the second part of the article, we show how to employ modular forms
in order to slightly refine our bound. In fact, our brief argument, which
combines a recent result of J.-L. Nicolas and J.-P. Serre with a classical
theorem of J.-P. Serre on the asymptotics of the Fourier coefficients of
certain level 1 modular forms, will more generally apply to provide a lower
bound for the number of odd values of any positive power of the generating
function of the partition function.Comment: A few minor revisions in response to the referees' comments. To
appear in the J. of Number Theor
Non-Standard Numeration Systems
We study some properties of non-standard numeration systems with an irrational base ß >1, based on the so-called beta-expansions of real numbers [1]. We discuss two important properties of these systems, namely the Finiteness property, stating whether the set of finite expansions in a given system forms a ring, and then the problem of fractional digits arising under arithmetic operations with integers in a given system. Then we introduce another way of irrational representation of numbers, slightly different from classical beta-expansions. Here we restrict ourselves to one irrational base – the golden mean ? – and we study the Finiteness property again.
First steps towards -adic Langlands functoriality
By the theory of Colmez and Fontaine, a de Rham representation of the Galois
group of a local field roughly corresponds to a representation of the
Weil-Deligne group equipped with an admissible filtration on the underlying
vector space. Using a modification of the classical local Langlands
correspondence, we associate with any pair consisting of a Weil-Deligne group
representation and a type of a filtration (admissible or not) a specific
locally algebraic representation of a general linear group. We advertise the
conjecture that this pair comes from a de Rham representation if and only if
the corresponding locally algebraic representation carries an invariant norm.
In the crystalline case, the Weil-Deligne group representation is unramified
and the associated locally algebraic representation can be studied using the
classical Satake isomorphism. By extending the latter to a specific norm
completion of the Hecke algebra, we show that the existence of an invariant
norm implies that our pair, indeed, comes from a crystalline representation. We
also show, by using the formalism of Tannakian categories, that this latter
fact is compatible with classical unramified Langlands functoriality and
therefore generalizes to arbitrary split reductive groups
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