47 research outputs found
Galois action on special theta values
For a primitive Dirichlet character χ of conductor N set θχ(τ) = ∑n ∈ℤ n∈ χ(n) eπin2τ/N (where ∈ = 0 for even χ, ∈ = 1 for odd χ) the associated theta series. Its value at its point of symmetry under the modular transformation τ(image found)−1/τ is related by θχ(i) = W(χ)θ(image found) (i) to the root number of the L-series of χ and hence can be used to calculate the latter quickly if it does not vanish. Using Shimura’s reciprocity law, we calculate the Galois action on these special values of theta functions with odd N normalised by the Dedekind eta function. As a consequence, we prove some experimental results of Cohen and Zagier and we deduce a partial result on the non-vanishing of these special theta values with prime N
Badly approximable points in twisted Diophantine approximation and Hausdorff dimension
For any j_1,...,j_n>0 with j_1+...+j_n=1 and any x \in R^n, we consider the set of points y \in R^n for which max_{1\leq i\leq n}(||qx_i-y_i||^{1/j_i})>c/q for some positive constant c=c(y) and all q\in N. These sets are the `twisted' inhomogeneous analogue of Bad(j_1,...,j_n) in the theory of simultaneous Diophantine approximation. It has been shown that they have full Hausdorff dimension in the non-weighted setting, i.e provided that j_i=1/n, and in the weighted setting when x is chosen from Bad(j_1,...,j_n). We generalise these results proving the full Hausdorff dimension in the weighted setting without any condition on x
Representation of integers by sparse binary forms
We will give new upper bounds for the number of solutions to the inequalities
of the shape , where is a sparse binary form,
with integer coefficients, and is a sufficiently small integer in terms of
the absolute value of the discriminant of the binary form . Our bounds
depend on the number of non-vanishing coefficients of . When is
really sparse, we establish a sharp upper bound for the number of solutions
that is linear in terms of the number of non-vanishing coefficients. This work
will provide affirmative answers to a number of conjectures posed by Mueller
and Schmidt in 1988, for special but important cases