A reduction algorithm for Hilbert modular groups

A Fast Algorithm for Modular Reduction

We present an algorithm for computing the residue R = X mod M . The algorithm is based on a sign estimation technique that estimates the sign of a number represented by a carry-sum pair produced by a carry save adder. Given the (n + k)-bit X and the n-bit M , the modular reduction algorithm computes the n-bit residue R in O(k + log n) time, and is particularly useful when the operand size is la...

Density Results for Automorphic Forms on Hilbert Modular Groups Ii

We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for SL2 over a totally real number field F , with a discrete subgroup of Hecke type Γ0(I) for a non-zero ideal I in the ring of integers of F . The weights are products of Fourier coefficients. This implies in particular the existence of infinitely many cuspidal automorphic representations with mu...

Deformations of Hilbert Modular Galois Representations and Adjoint Selmer Groups

We prove the vanishing of the geometric Bloch–Kato Selmer group for the adjoint representation of a Galois representation associated to a Hilbert modular form under mild assumptions on the residual image. In particular, we do not assume the residual image satisfies the Taylor–Wiles hypothesis. Using this, we deduce that the localization and completion of the universal deformation ring for the r...

Modular Symbols for Q-rank One Groups and Voronoǐ Reduction

Let G be a reductive algebraic group of Q-rank one associated to a self-adjoint homogeneous cone defined over Q, and let Γ ⊂ G be a torsion-free arithmetic subgroup. Let d be the cohomological dimension of Γ. We present an algorithm to compute the action of the Hecke operators on H(Γ;Z). This generalizes the classical modular symbol algorithm, when Γ ⊂ SL2(Z), to a setting including Bianchi gro...

Tate Conjectures for Hilbert Modular Surfaces