On a convexity property of tensor products of irreducible, rational representations of $SL(n)$

The aim of this note is to point out a convexity property with respect to the root lattice for the support of the highest weights that occur in a tensor product of irreducible rational representations of $SL(n)$ over the complex numbers. The observation is a consequence of the convexity properties of the saturation cone and the validity of the saturation conjecture for $SL(n)$.

Finiteness theorems for potentially equivalent Galois representations: extension of Faltings finiteness criteria

We study the relationship between potential equivalence and character theory; we observe that potential equivalence of a representation $rho$ is determined by an equality of an $m$-power character $gmapsto Tr(rho(g^m))$ for some natural number $m$. Using this, we extend Faltings finiteness criteria to determine the equivalence of two $ell$-adic, semisimple representations of the absolute Galois group of a number field, to the context of potential equivalence. We also discuss finiteness results for twist unramified representations.

A uniform bound for inertially equivalent, pure $ell$-adic representations: an extension of Faltings theorem

We introduce a notion of inertial equivalence for integral $ell$-adic representation of the Galois group of a global field. We show that the collection of continuous, semisimple, pure $ell$-adic representations of the absolute Galois group of a global field lifting a fixed absolutely irreducible residual representation and with given inertial type outside a fixed finite set of places is uniformly bounded independent of the inertial type.

Singular Gauss sums, Polya-Vinogradov inequality for $GL(2)$ and growth of primitive elements

We establish an analogue of the classical Polya-Vinogradov inequality for $GL(2, F_p)$, where $p$ is a prime. In the process, we compute the `singular Gauss sums for $GL(2, F_p)$. As an application, we show that the collection of elements in $GL(2,Z)$ whose reduction modulo $p$ are of maximal order in $GL(2, F_p)$ and whose matrix entries are bounded by $x$ has the expected size as soon as $xgg p^{1/2+ep}$ for any $ep>0$. In particular, there exist elements in $GL(2,Z)$ with matrix entries that are of the order $O(p^{1/2+ep})$ whose reduction modulo $p$ are primitive elements.

On the structure of locally potentially equivalent Galois representations

Suppose $rho_1, rho_2$ are two $ell$-adic Galois representations of the absolute Galois group of a number field, such that the algebraic monodromy group of one of the representations is connected and the representations are locally potentially equivalent at a set of places of positive upper density. We classify such pairs of representations and show that up to twisting by some representation, it is given by a pair of representations one of which is trivial and the other abelian. Consequently, assuming that the first representation has connected algebraic monodromy group, we obtain that the representations are potentially equivalent, provided one of the following conditions hold: (a) the first representation is absolutely irreducible; (b) the ranks of the algebraic monodromy groups are equal; (c) the algebraic monodromy group of the second representation is also connected and (d) the commutant of the image of the second representation remains the same upon restriction to subgroups of finite index of the Galois group.

Solvable base change

We determine the image and fibres for solvable base change.

A universal Torelli theorem for elliptic surfaces

Given two semistable, non potentially isotrivial elliptic surfaces over a curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the N{e}ron-Severi lattices of the base changed elliptic surfaces for all finite separable maps $Bto C$ arises from an isomorphism of the elliptic surfaces. Without the effectivity hypothesis, we show that the two elliptic surfaces are isomorphic. We also determine the group of universal automorphisms of a semistable elliptic surface. In particular, this includes showing that the Picard-Lefschetz transformations corresponding to an irreducible component of a singular fibre, can be extended as universal isometries. In the process, we get a family of homomorphisms of the affine Weyl group associated to $tilde{A}_{n-1}$ to that of $tilde{A}_{dn-1}$, indexed by natural numbers $d$, which are closed under composition.

Distinguishing Galois representations by their normalized traces

Suppose ( rho_1 ) and ( rho_2 ) are two pure Galois representations of the absolute Galois group of a number field $K$ of weights ( k_1 ) and ( k_2 ) respectively, having equal normalized Frobenius traces ( Tr(rho_1(sigma_v)) /Nv^{k_1/2}) and ( Tr(rho_2(sigma_v)) /Nv^{k_2/2}) at a set of primes ( v) of $K$ with positive upper density. Assume further that the algebraic monodromy group of $rho_1$ is connected and the repesentation is absolutely irreducible. We prove that ( rho_1 ) and ( rho_2 ) are twists of each other by power of a Tate twist times a character of finite order. We apply this to modular forms and deduce a result proved by Murty and Pujahari.

Hermitian symmetric space, flat bundle and holomorphicity criterion

Let $Kbackslash G$ be an irreducible Hermitian symmetric space of noncompact type and $Gamma ,subset, G$ a closed torsionfree discrete subgroup. Let $X$ be a compact Kahler manifold and $rho, :, pi_1(X, x_0),longrightarrow, Gamma$ a homomorphism such that the adjoint action of $rho(pi_1(X, x_0))$ on $text{Lie}(G)$ is completely reducible. A theorem of Corlette associates to $rho$ a harmonic map $X, longrightarrow, Kbackslash G/Gamma$. We give a criterion for this harmonic map to be holomorphic. We also give a criterion for it to be anti--holomorphic.

On the splitting fields of generic elements in Zariski dense subgroups

Let $G$ be a connected, absolutely almost simple, algebraic group defined over a finitely generated, infinite field $K$, and let $Gamma$ be a Zariski dense subgroup of $G(K)$. We show, apart from some few exceptions, that the commensurability class of the field $mathcal{F}$ given by the compositum of the splitting fields of characteristic polynomials of generic elements of $Gamma$ determines the group $G$ upto isogeny over the algebraic closure of $K$.