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 $E$ be an abelian scheme over a geometrically connected variety $X$ defined over $k$, a finitely generated field over $\mathbb{Q}$. Let $\eta$ be the generic point of $X$ and $x\in X$ a closed point. If $\mathfrak{g}_l$ and $(\mathfrak{g}_l)_x$ are the Lie algebras of the $l$-adic Galois representations for abelian varieties $E_{\eta}$ and $E_x$, then $(\mathfrak{g}_l)_x$ is embedded in $\mathfrak{g}_l$ by specialization. We prove that the set $\{x\in X$ closed point $| (\mathfrak{g}_l)_x\subsetneq \mathfrak{g}_l\}$ is independent of $l$ 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 $O(\frac{R}{\epsilon})$ and $O(\sqrt{\frac{R}{\epsilon}})$ for the ordinary and accelerate gradient methods, respectively, assuming that $\nabla f$ is Lipschitz continuous with constant $R$ over the line segment joining $x$ and $x-\frac{1}{R}\nabla f$ for each x\in\dom f. Then we improve them to $O(\frac{R}{\nu}\log(\frac{1}{\epsilon}))$ and $O(\sqrt{\frac{R}{\nu}}\log(\frac{1}{\epsilon}))$ for function $f$ that also satisfies the secant inequality $\ \ge \nu\|x-x^*\|^2$ for each x\in \dom f and its projection $x^*$ to the minimizer set of $f$. 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 $R$ and larger $\nu$ 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 $X$ be a non-singular projective variety over a number field $K$, $i$ a non-negative integer, and V_{\A}, the etale cohomology of $\bar X$ 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 $G_K$ in H(\A_f) under the adelic Galois representation \rho_{\A}: G_K -> \Aut(V_{\A})=\GL_n(\A_f), where $H$ is the Hodge group. The motivating example is a celebrated theorem of Serre, which asserts that if $X$ is an elliptic curve without complex multiplication over $\bar K$ and $i=1$, 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