8,963 research outputs found
Invariant dimensions and maximality of geometric monodromy action
Let X be a smooth separated geometrically connected variety over F_q and
f:Y-> X a smooth projective morphism. We compare the invariant dimensions of
the l-adic representation V_l and the F_l-representation \bar V_l of the
geometric \'etale fundamental group of X arising from the sheaves R^wf_*Q_l and
R^wf_*Z/lZ respectively. These invariant dimension data is used to deduce a
maximality result of the geometric monodromy action on V_l whenever \bar V_l is
semisimple and l is sufficiently large. We also provide examples for \bar V_l
to be semisimple for l>>0
On the rationality of algebraic monodromy groups of compatible systems
Let E be a number field and X be a smooth geometrically connected variety
defined over a characteristic p finite field F_q. Given an n-dimensional pure
E-compatible system of semisimple \lambda-adic representations \rho_\lambda of
the fundamental group \pi_1(X) with connected algebraic monodromy groups
G_\lambda, we construct a common E-form G of all the groups G_\lambda. In the
absolutely irreducible case, we construct a common E-form i:G->GL_{n,E} of all
the tautological representations G_\lambda->GL_{n,E_\lambda} and a G-valued
adelic representation \rho_A^G of \pi_1(X) such that their composition is
isomorphic to the product representation of all \rho_\lambda. Moreover, if X is
a curve and the (absolute) outer automorphism group of G^der is trivial, then
the \lambda-components of \rho_A^G form an E-compatible system of
G-representations. Analogous rationality results in characteristic zero,
predicted by the Mumford-Tate conjecture, are obtained under some conditions
including ordinariness.Comment: 35 pages. Thm. 1.1(ii) is improved so that G sits in GL_{n,E
Specialization of monodromy group and l-independence
Let be an abelian scheme over a geometrically connected variety
defined over , a finitely generated field over . Let be
the generic point of and a closed point. If and
are the Lie algebras of the -adic Galois
representations for abelian varieties and , then
is embedded in by specialization. We
prove that the set closed point is independent of and confirm Conjecture 5.5 in [2].Comment: 4 page
Entropy-vanishing transition and glassy dynamics in frustrated spins
In an effort to understand the glass transition, the dynamics of a
non-randomly frustrated spin model has been analyzed. The phenomenology of the
spin model is similar to that of a supercooled liquid undergoing the glass
transition. The slow dynamics can be associated with the presence of extended
string-like structures which demarcate regions of fast spin flips. An
entropy-vanishing transition, with the string density as the order parameter,
is related to the observed glass transition in the spin model.Comment: 4 pages,5 figures, accepted in PRL January 200
Gradient methods for convex minimization: better rates under weaker conditions
The convergence behavior of gradient methods for minimizing convex
differentiable functions is one of the core questions in convex optimization.
This paper shows that their well-known complexities can be achieved under
conditions weaker than the commonly accepted ones. We relax the common gradient
Lipschitz-continuity condition and strong convexity condition to ones that hold
only over certain line segments. Specifically, we establish complexities
and for the ordinary and
accelerate gradient methods, respectively, assuming that is
Lipschitz continuous with constant over the line segment joining and
for each x\in\dom f. Then we improve them to
and
for function that also
satisfies the secant inequality
for each x\in \dom f and its projection to the minimizer set of .
The secant condition is also shown to be necessary for the geometric decay of
solution error. Not only are the relaxed conditions met by more functions, the
restrictions give smaller and larger than they are without the
restrictions and thus lead to better complexity bounds. We apply these results
to sparse optimization and demonstrate a faster algorithm.Comment: 20 pages, 4 figures, typos are corrected, Theorem 2 is ne
Adelic openness without the Mumford-Tate conjecture
Let be a non-singular projective variety over a number field , a
non-negative integer, and V_{\A}, the etale cohomology of with
coefficients in the ring of finite adeles \A_f over \Q. Assuming the
Mumford-Tate conjecture, we formulate a conjecture (Conjecture 1.2) describing
the largeness of the image of the absolute Galois group in H(\A_f)
under the adelic Galois representation \rho_{\A}: G_K ->
\Aut(V_{\A})=\GL_n(\A_f), where is the Hodge group. The motivating example
is a celebrated theorem of Serre, which asserts that if is an elliptic
curve without complex multiplication over and , then
\rho_{\A}(G_K) is an open subgroup of \GL_2(\hat \Z)\subset \GL_2(\A_f). We
state and in some cases prove a weaker conjecture which does not require
Mumford-Tate but which, together with Mumford-Tate, implies Conjecture 1.2. We
also relate our conjectures to Serre's conjectures on maximal motives.Comment: Section 5 is ne
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