In this paper we show that, if T is an area-minimizing 2-dimensional integral current with ∂T=Q〚Γ〛, where Γ a C1,α curve for α>0 and Q arbitrary integer, then has unique tangent cone at every boundary point, polynomial convergence rate. The proof simple reduction to the case Q=1, studied by Hirsch Marini (2019).

The number A(q) is the upper limit of the ratio of the maximum number of points of a curve defined over Fq to the genus. By constructing class field towers with good parameters we present improvements of lower bounds of A(q) for q an odd power of a prime.

Let p be a prime number and M a quadratic number field, M , Q( √ p) if p ≡ 1 mod 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D2p and an elliptic curve E/Q such that F contains M and the p-Selmer group of E/F has size at least pd.

We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields. First, we show that if L/K is a finite Galois extension of number fields such that Gal(L/K) does not have an index 2 subgroup and A/K is an Abelian variety, then rkA(L)− rkA(K) can never be 1. We obtain more precise results when Gal(L/K) is of odd order, alternating, SL2(F...

We recently reported the analysis of the frequency noise in the frequency modulation atomic force microscopy (FM-AFM) both in high-Q and low-Q environments [Rev. Sci. Instrum. 80, 043708 (2009)]. We showed in the paper that the oscillator noise, the frequency fluctuation of the oscillator, becomes prominent in the modulation frequency lower than f(0)∕2Q, where f(0) and Q are the resonance frequ...

Abstract We describe an algorithm for computing a $${\mathbb {Q}}$$ Q -rational model the quotient of modular curve by automorphism group, under mild assumptions on and automorphisms, determining q -expansions basis corresponding space cusp forms. also give moduli interpretation general morphisms between curves.

Construction methods are presented that generate hermite interpolation quaternion curves on SO(3). Two circular curves C 1 (t) and C 2 (t), 0 t 1, are generated that interpolate two orientations q 1 and q 2 , and have boundary angular velocities: C 0 1 (0) = ! 1 and C 0 2 (1) = ! 2 , respectively. They are smoothly blended together on SO(3) to generate a hermite quaternion curve Q(t) 2 SO(3), 0...

Let E/K be an elliptic curve defined over a number field K and without complex multiplication (CM). For a rational prime , let K(E[ ]) be the th division field of E, which we know is a finite Galois extension of K. By a celebrated result of Serre [18], there exists a positive constant c(E,K), depending on E and K, such that Gal(K(E[ ])/K) GL2(Z/ Z) for all ≥ c(E,K). In [18, 19], Serre asked whe...

Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)tors and the torsion subgroup E(K)tors, where K is a cubic number field. In particular, we study the number of cubic number fields K such that E(Q)tors ̸= E(K)tors.

The Jacobian J0(23) of the modular curve X0(23) is a semi-stable abelian variety over Q with good reduction outside 23. It is simple. We prove that every simple semi-stable abelian variety over Q with good reduction outside 23 is isogenous over Q to J0(23). 2010 Mathematics Subject Classification. Primary 14L15; Secondary 11G18, 11R37.