Research in Number Theory最新文献

Let E and $E^{\text{'}}$ be elliptic curves over $Q$ with complex multiplication by the ring of integers of an imaginary quadratic field K and let $Y=\text{Kum}(E\times E^{\text{'}})$ be the minimal desingularisation of the quotient of $E\times E^{\text{'}}$ by the action of $-1$ . We study the Brauer groups of such surfaces Y and use them to furnish new examples of transcendental Brauer-Manin obstructions to weak approximation.

We investigate a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms F, G for orthogonal groups of signature $(2,n+2)$ . In the case when F is a Hecke eigenform and G is a Maass lift of a Poincaré series, we establish a connection with the standard L-function attached to F. What is more, we find explicit choices of orthogonal groups, for which we obtain a clear-cut Euler product expression for this Dirichlet series. Through our considerations, we recover a classical result for Siegel modular forms, first introduced by Kohnen and Skoruppa, but also provide a range of new examples, which can be related to other kinds of modular forms, such as paramodular, Hermitian, and quaternionic.

Recently, Amdeberhan, Griffin, Ono, and Singh started the study of "traces of partition Eisenstein series" and used it to give explicit formulas for many interesting functions. In this note we determine the precise spaces in which they lie, find modular completions, and show how they are related via operators.

Mazur and Rubin introduced the notion of n-Selmer companion elliptic curves and gave several examples of pairs of non-isogenous Selmer companions. We construct several pairs of families of elliptic curves, each parameterised by $t\in Z$ , such that the two curves in a pair corresponding to a given t are non-isogenous 3-Selmer companions, possibly provided that t satisfies a simple congruence condition.

Friedberg-Jacquet proved that if $\pi$ is a cuspidal automorphic representation of ${\text{GL}}_{2n}\left(A\right)$ , then $\pi$ is a functorial transfer from ${\text{GSpin}}_{2n+1}$ if and only if a global zeta integral $Z_H$ over $H={\text{GL}}_n\times {\text{GL}}_n$ is non-vanishing on $\pi$ . We conjecture a p-refined analogue: that any P-parahoric p-refinement ${\tilde{\pi }}^P$ is a functorial transfer from ${\text{GSpin}}_{2n+1}$ if and only if a P-twisted version of $Z_H$ is non-vanishing on the ${\tilde{\pi }}^P$ -eigenspace in $\pi$ . This twisted $Z_H$ appears in all constructions of p-adic L-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the ${\text{GL}}_{2n}$ eigenvariety, and-by proving upper bounds on the dimensions of such families-obtain various results towards the conjecture.

In this paper we give an algorithm to find the 3-torsion subgroup of the Jacobian of a smooth plane quartic curve with a marked rational point. We describe $3-$ torsion points in terms of cubics which triply intersect the curve, and use this to define a system of equations whose solution set corresponds to the coefficients of these cubics. We compute the points of this zero-dimensional, degree 728 scheme first by approximation, using homotopy continuation and Newton-Raphson, and then using continued fractions to obtain accurate expressions for these points. We describe how the Galois structure of the field of definition of the 3-torsion subgroup can be used to compute local wild conductor exponents, including at $p=2$ .

Denote by $N_{\ell }\left(n\right)$ the number of $\ell$ -tuples of elements in the symmetric group $S_n$ with commuting components, normalized by the order of $S_n$ . In this paper, we prove asymptotic formulas for $N_{\ell }\left(n\right)$ . In addition, general criteria for log-concavity are shown, which can be applied to $N_{\ell }\left(n\right)$ among other examples. Moreover, we obtain a Bessenrodt-Ono type theorem which gives an inequality of the form $c\left(a\right)c\left(b\right)>c(a+b)$ for certain families of sequences c(n).

In this paper, we prove that the number of unimodal sequences of size n is log-concave. These are coefficients of a mixed false modular form and have a Rademacher-type exact formula due to recent work of the second author and Nazaroglu on false theta functions. Log-concavity and higher Turán inequalities have been well-studied for (restricted) partitions and coefficients of weakly holomorphic modular forms, and analytic proofs generally require precise asymptotic series with error term. In this paper, we proceed from the exact formula for unimodal sequences to carry out this calculation. We expect our method applies to other exact formulas for coefficients of mixed mock/false modular objects.

In this paper, we consider the Diophantine equation $V_n-b^m=c$ for given integers b, c with $b\ge 2$, whereas $V_n$ varies among Lucas-Lehmer sequences of the second kind. We prove under some technical conditions that if the considered equation has at least three solutions (n, m) , then there is an upper bound on the size of the solutions as well as on the size of the coefficients in the characteristic polynomial of $V_n$.