1,720,982 research outputs found
A zig-zag conjecture and local constancy for Galois representations (Algebraic Number Theory and Related Topics 2018)
No full text
A zig-zag conjecture and local constancy for Galois representations (Algebraic Number Theory and Related Topics 2018)
Get PDFAlgebraic Number Theory and Related Topics 2018. November 26-30, 2018. edited by Takao Yamazaki and Shuji Yamamoto. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We make a zig-zag conjecture describing the reductions of irreducible crystalline twodimensional representations of GQp of half-integral slopes and exceptional weights. Such weights are two more than twice the slope mod (p − 1). We explain how zig-zag can be deduced from known results for half-integral slopes at most 3 2 . We then explore the connection between zig-zag and local constancy results in the weight. We show that known cases of zig-zag force local constancy to fail for small weights, and explain how local constancy forces zig-zag to fail for some small weights and half-integral slopes at least 2. However, we expect zig-zag to be qualitatively true in general. We end with some compatibility results between zig-zag and other results
Critical values of the twisted tensor L-function in the imaginary quadratic case
No full textThis article does not have an abstract
Zig-zag for Galois Representations
Get PDFThe zig-zag conjecture says that the reductions of two-dimensional
crystalline representations of the Galois group of of large
exceptional weights and half-integral slopes up to vary through
an alternating sequence of irreducible and reducible mod representations.
We prove this conjecture in smoothly varying families of such representations
for . The proof uses a limiting argument due to Chitrao-Ghate-Yasuda
to reduce to the case of semi-stable representations of weights at most ,
and then appeals to the work of Breuil-M\'ezard, Guerberoff-Park and
Chitrao-Ghate.Comment: Updated version: title changed to reflect that this version contains
a proof of the conjecture on the full Galois group not just on the inertia
subgroup; also includes a proof for the top two slope
The arithmetic and geometry of Salem numbers
Get PDFA Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and arithmetic, and discuss the possible implications for the 'minimization problem'. This is an old question in number theory which asks whether the set of Salem numbers is bounded away from 1
Modular representations of using calculus
Get PDFWe show that certain modular induced representations of can be written as cokernels of operators acting on symmetric power representations of . When the induction is from the Borel subgroup, respectively the anisotropic torus, the operators involve multiplication by newly defined twisted Dickson polynomials, respectively, twisted Serre operators. Our isomorphisms are explicitly defined using differential operators. As a corollary, we improve some periodicity results for quotients in the theta filtration.To appear in Forum Mathematicu
Supercuspidal ramification of modular endomorphism algebras
No full textThe endomorphism algebra Χf attached to a non-CM primitive cusp form f of weight at least two is a 2-torsion element in the Brauer group of a number field F. We give formulas for the ramification of Χf locally at primes lying above the odd supercuspidal primes of f. We show that the local Brauer class is determined by the underlying local Galois representation together with an auxiliary Fourier coefficient
Reductions of Galois representations via the mod p local Langlands correspondence
No full textWe describe the semisimplification of the mod p reduction of certain crystalline two dimensional local Galois representations of weights bounded by p2−p and slopes in (1,2). This builds on previous results, for weights bounded by 2p+1, for large slopes, and for slopes in (0,1). In proving our results we give a complete description of the submodules generated by the top two monomials in the mod p symmetric power representation of GL2(Fp)in the above range
Reductions of semi-stable representations using the Iwahori mod Local Langlands Correspondence
Get PDFWe determine the mod reductions of all two-dimensional semi-stable
representations of the Galois group of of
weights and -invariants for
primes . In particular, we describe the constants appearing in the
unramified characters completely. The proof involves computing the reduction of
Breuil's -Banach space ,
by studying certain logarithmic functions using background material developed
by Colmez, and then applying an Iwahori theoretic version of the mod Local
Langlands Correspondence.Comment: Complete proofs of solutions to some matrix equations are now
provided in an appendi
- …
