Symmetric power functoriality for holomorphic modular forms, II

Let $f$ f be a cuspidal Hecke eigenform without complex multiplication. We prove the automorphy of the symmetric power lifting $\operatorname{Sym}^{n} f$ Sym n f for every $n \geq 1$ n ≥ 1 .

A Coherent Accumulation Detection Method Based on SA-DPT for Highly Maneuvering Target

The emergence of highly maneuverable weak targets has led to a serious degradation or even failure of traditional radar detection. In this paper, a coherent accumulation algorithm based on combination of scaling algorithm(SA) and discrete polynomial-phase Transform(DPT) is proposed in terms of the calculation burden and detection performance, which can, firstly, perform fewer times of speed parameter compensation based on SA for the transmitted signal; secondly, use segmented FFT to estimate the time delay range of the target initial distance for the echo signal, and then use the DPT algorithm to complete the parameter estimation such as target speed and acceleration within the estimated time delay unit, while analyzing the effects of the number of segments, compensation speed and delay unit on the detection performance, and giving out the output SNR and the amount of complex multiplication of the proposed algorithm. Finally, the proposed algorithm has been verified by experimental data for its effectiveness in accumulation gain and parameter estimation. This method is for sub-optimal estimation, requiring much less computation than full-parameter search methods, but performs better than non-parametric search methods in detecting weak signals.

Explicit characterization of the torsion growth of rational elliptic curves with complex multiplication over quadratic fields

In a series of papers we classify the possible torsion structures of rational elliptic curves base-extended to number fields of a fixed degree. In this paper we turn our attention to the question of how the torsion of an elliptic curve with complex multiplication defined over the rationals grows over quadratic fields. We go further and we give an explicit characterization of the quadratic fields where the torsion grows in terms of some invariants attached to the curve.

Primitive divisors of sequences associated to elliptic curves with complex multiplication

Let P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

Integrality of Seshadri constants and irreducibility of principal polarizations on products of two isogenous elliptic curves

In this paper we consider the question of when all Seshadri constants on a product of two isogenous elliptic curves $$E_1\times E_2$$ E 1 × E 2 without complex multiplication are integers. By studying elliptic curves on $$E_1\times E_2$$ E 1 × E 2 we translate this question into a purely numerical problem expressed by quadratic forms. By solving that problem, we show that all Seshadri constants on $$E_1\times E_2$$ E 1 × E 2 are integers if and only if the minimal degree of an isogeny $$E_1\rightarrow E_2$$ E 1 → E 2 equals 1 or 2. Furthermore, this method enables a characterization of irreducible principal polarizations on $$E_1\times E_2$$ E 1 × E 2 .

An inverse Jacobian algorithm for Picard curves

We study the inverse Jacobian problem for the case of Picard curves over $${\mathbb {C}}$$ C . More precisely, we elaborate on an algorithm that, given a small period matrix $$\varOmega \in {\mathbb {C}}^{3\times 3}$$ Ω ∈ C 3 × 3 corresponding to a principally polarized abelian threefold equipped with an automorphism of order 3, returns a Legendre–Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike–Weng (Math Comput 74(249):499–518, 2005) which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most $$4$$ 4 . In particular, we obtain the complete list of maximal CM Picard curves defined over $${\mathbb {Q}}$$ Q . In the appendix, Vincent gives a correction to the generalization of Takase’s formula for the inverse Jacobian problem for hyperelliptic curves given in [Balakrishnan–Ionica–Lauter–Vincent, LMS J. Comput. Math., 19(suppl. A):283-300, 2016].

A note on moduli spaces of conformal classes for flat tori of higher dimension and on their conformal multiplication

Motivated by the theory of complex multiplication of abelian varieties, in this paper we study the conformality classes of flat tori in $${\mathbb {R}}^{n}$$ R n and investigate criteria to determine whether a n-dimensional flat torus has non trivial (i.e. bigger than $${\mathbb {Z}}^{*}={\mathbb {Z}}{\setminus }\{0\}$$ Z ∗ = Z \ { 0 } ) semigroup of conformal endomorphisms (the analogs of isogenies for abelian varieties). We then exhibit several geometric constructions of tori with this property and study the class of conformally equivalent lattices in order to describe the moduli space of the corresponding tori.