Let E be an elliptic curve defined over Q and (sic)E-p denote the reduction of E modulo a prime p. Due to work of Serre, Katz, and Mazur, it is known that the integer gcd(p subset of S) |(sic)E-p (F-p)| is an element of {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 12; 16} where S is a set of primes of density one containing primes of good reduction of E. In this note, we consider the integer gcd(p subset of S) |(sic)E-p (F-p)| when S is taken to be the set of all rational primes. We show that gcdpES Ep(Fp) <4. We also show that there are elliptic curves for which gcd(p subset of S) |(sic)E-p (F-p)| is either 2, 3 or 4.
We determine all pairs (p, n), where p is a prime and n a positive integer, such that there exists a reduced fraction u/v > 1 with u+ v= n and u/v has a nonterminating Schneider's p-adic continued fraction expansion. We also prove a bound on the length of the preperiod in the p-adic continued fraction of u/v when max{|u|,|v|} < p.
An imaginary biquadratic field K is a number field of the form Q(root-m(1,)root-m(2), where m(1) and m(2) are distinct positive squarefree integers. In this paper, we present an algorithm for determining all the imaginary biquadratic fields with small odd class numbers. We use this algorithm to determine all imaginary biquadratic fields K with odd class number h(K) satisfying 1 <= h(K) <50. Moreover, we note that many of these fields are not found in the L-functions and modular forms database (LMFDB) as of writing.
Let X 1/4 G/H be a homogeneous space, where G superset of H are reductive Lie groups. We ask: in the setting where FnG/H is a standard quotient in the sense of Kassel-Kobayashi [9], to what extent can the discrete subgroup F be deformed while preserving the proper discontinuity of the F-action on X? We provide several classification results, including: conditions under which local rigidity holds for compact standard quotients FnX; criteria for when a standard quotient can be deformed into a nonstandard one; a characterization of the maximal Zariski-closure of discontinuous groups under small deformations; and conditions under which Zariski-dense deformations occur.
We consider a surface embedded in the Euclidean 3-space and fix a tangential vector v at a given point p on the surface. In this paper, we first review a history of the formula obtained by Mannheim, d'Ocagne and Koenderink, which asserts that the Gaussian curvature of the surface at p can be obtained if one knows "the normal curvature at p with respect to v" and "the curvature of the contour line L of the surface at p" with respect to the orthogonal projection induced by v. Unfortunately, this formula does not work when v points in an asymptotic direction. When v is just the case, we give anlogues of the formula, which include an invariant of cusp singular points of L.
In proving Rellich inequalities in the framework of equalities, N. Bez, S. Machihara, and T. Ozawa obtained some interesting norm inequalities in the spirit of Evans and Lewis that compare the standard Laplacian with its radial and spherical components. In this paper we give a simple unified proof and a strict improvement of these Evans-Lewis inequalities in the subtle dimension three case. Our approach is robust and explains clearly the occurrence of the sharp constant.
In this paper, we aim to implement the forgetful maps from X-1(N) to X-0(N) and from X-0(N) to X(1) using the Hauptmoduls of X-1(N), X-0(N), and X(1). This approach is similar to the work of Kim and Koo [KK4], who constructed the Hauptmoduls for X-1(N) through various methods and utilized the known eta quotients as the Hauptmoduls for X-0(N). However we use other Hauptmoduls for X-1(N) and X-0(N) which can be constructed by using the generators of the function field of X-1(N) introduced by Baaziz [B].
Let j is an element of Z(<= 1), k is an element of N->= 2 and f(n) := (1 - 2(1 - k))) + (- 1)(n) . In the present paper, we show that the series expression of L-*(j; k;1;f) := Sigma(m>n>0)m(-j)n(-k )f(n) converges when j + k >= 3 as a counter example of [1, Proposition 1.1].
Let G be a locally compact group and (H, L) a pair of closed non-compact subgroups of G. The main result of this paper is that the L-action on the homogeneous space X = G/H is proper if and only if the pair (H, L) is asymptotically disjoint in G with respect to a suitable coarse structure on G. It is worth noting that in the case where G is linear reductive, an algebraic criterion for the properness of the L-action on X was established by T. Kobayashi [Math. Ann. 1989, J. Lie Theory 1996] and Y. Benoist [Ann. Math. 1996]. We discuss that our result mentioned above gives a coarse geometric explanation of their properness criterion. As another application of our result, a relation between affine proper actions on the affine space Rn and the space of normal distributions on Rn equipped with the Fisher metric is also given.
In [3], we gave a presentation of the PGL2 Hecke algebra of level 4. We now use this presentation to describe the finite dimensional representations of the Hecke algebra and compute the Whittaker function for newforms of level 4.
A theorem due to Kaup and Upmeier states that two bounded balanced pseudoconvex domains are biholomorphic if and only if they are linearly equivalent. In this article, we prove that this theorem can occur for possibly unbounded non-hyperbolic Reinhardt domains under certain Bergman theoretic conditions. We also find explicit unbounded non-hyperbolic examples of our theorem.
We consider quasi-linear second order elliptic differential equations with gradient terms and study a convergence property of supersolutions of the equation. As an application of the convergence property we investigate the relation between supersolutions and superharmonic functions.
Let Ln be a simplest cubic field with Galois group G 1/4 Gal & eth;Ln=Q & THORN;, OLn the ring of integers of Ln. Leopoldt showed that OLn' ALn=Q as ALn=Q-modules, where ALn=Q is the associated order for Ln=Q. In this paper, we give a generator of the ALn=Q-module OLn explicitly using the roots of Shanks' cubic polynomial.
In spite of the geometric and analytic importance of hypocycloids, there are little systematic studies of geometric inequalities characterizing hypocycloids. Recently, Kwong--Lee obtained a countable family of geometric inequalities for convex curves by virtue of higher order Wirtinger-type inequalities. This family contains the isoperimetric inequality and Lin-Tsai's inequality. In this paper, generalizing the Kwong--Lee's family of inequalities to curves with admissible singularities (called ℓ-convex Legendre curves), we derive a countable family of inequalities whose equality conditions are characterized by hypocycloids. 2020 Mathematics Subject Classification. Primary 53A04; Secondary 52A40, 26D10.
We define a length function for a perfect crystal. As an application, we derive a variant of the Rogers-Ramanujan identities which involves (a q-analog of) the Fibonacci numbers.
We discuss connections of toroidal compactifications and Borel–Serre compactifications in view of the fundamental diagram of extended period domains. We give a complement to a work of Goresky–Tai.
We construct modular forms on Gamma(0)(2) in which all of their zeros in the fundamental domain for Gamma(0)(2) are on a part of a specific geodesic inside the domain not on the boundary of the domain.
Geometric aspects of effectively hyperbolic critical points on time $t=0$ are discussed assuming that the characteristic roots are real on one side of time $t$, namely time is positive. In particular, we aim to elucidate the differences in the geometric aspects of effectively hyperbolic critical points on time $t=0$ when the characteristic roots are real on both the positive and negative sides of time.
