Related papers: Explicit growth and expansion for SL_2

Let p be an odd prime number, E an elliptic curve over a number field k, and F/k a Galois extension of degree twice a power of p. We study the Z_p-corank rk_p(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for…

This work, following the outline set in [B2], presents an explicit construction of a family of monotone expanders. The family is essentially defined by the Mobius action of SL_2(R) on the real line. For the proof, we show a product-growth…

We give a new proof of Gromov's theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally…

We study the growth of the local $L^2$-norms of the unitary Eisenstein series for reductive groups over number fields, in terms of their parameters. We derive a \emph{poly-logarithmic} bound on an average, for a large class of reductive…

Taking the Iwaniec explicit formula as a starting point, we give a short proof of a more precise $\frac{2}{3}$ bound for the exponent in the error term of the Gallagher-type prime geodesic theorem for the modular surface.

We prove an analog of the classical Hartogs extension theorem for CR $L^{2}$ functions defined on boundaries of certain (possibly unbounded) domains on coverings of strongly pseudoconvex manifolds. Our result is related to a problem posed…

Let $p$ be an odd prime number. In this paper, we study the growth of the Sylow $p$-subgroups of the even $K$-groups of rings of integers in a $p$-adic Lie extension. Our results generalize previous results of Coates and Ji-Qin, where they…

It was studied the growth of Temperley-Lieb type algebras with orthogonal and commutative relations associated with 2-colored edges graphs. Depending on the structure of the graphs they can be finite-dimensional or to have linear or…

We shall give an explicit version of Bombieri-Vinogradov Theorem for moduli not divisible by an exceptional modulus.

We compute the Galois group of the maximal 2-ramified and complexified pro-2-extension of any 2-rational number field.

Given a periodic quotient of a torsion-free hyperbolic group, we provide a fine lower estimate of the growth function of any sub-semi-group. This generalizes results of Razborov and Safin for free groups.

We provide explicit series expansions for the exponential and logarithm functions attached to a rank r Drinfeld module that generalize well known formulas for the Carlitz exponential and logarithm. Using these results we obtain a procedure…

We prove a quantitative refinement of the statement that groups of polynomial growth are finitely presented. Let $G$ be a group with finite generating set $S$ and let $\operatorname{Gr}(r)$ be the volume of the ball of radius $r$ in the…

We show that some classical results on expander graphs imply growth results on normal subsets in finite simple groups. As one application, it is shown that given a nontrivial normal subset $ A $ of a finite simple group $ G $ of Lie type of…

The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism $\natural$ from the universal Hahn algebra $\mathcal H$ into…

Let $p\geq 5$ be a prime number. Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$. Let $K$ be an imaginary quadratic field where $p$ splits, and such that the generalized Heegner hypothesis holds. Under…

Let $\Gamma < G := \operatorname{SO}(d+1, 1)$ for $d \geq 1$ be a Zariski dense, geometrically finite, discrete subgroup with critical exponent strictly greater than $d/2$. We show that $L^2(\Gamma\backslash G)$ admits a strong spectral…

We prove that if $f:X \rightarrow A$ is a morphism from a smooth projective variety $X$ to an abelian variety $A$ over a number field $K$, and $G$ is a subgroup of automorphisms of $X$ satisfying certain properties, and if a prime $p$…

We study the analytic behavior of the restriction of a Siegel modular form to $\mathbb{H} \times \mathbb{H}$ in the case that the Siegel form is a Saito-Kurokawa lift. A formula of Ichino links this behavior to a family of $GL_3 \times…

For $E/K$ an elliptic curve without complex multiplication we bound the index of the image of $\operatorname{Gal}(\bar{K}/K)$ in $\operatorname{GL}_2(\hat{\mathbb{Z}})$, the representation being given by the action on the Tate modules of…