Complex multiplication
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Quadratic points on modular curves Open
In this paper we determine the quadratic points on the modular curves X0(N), where the curve is non-hyperelliptic, the genus is 3, 4 or 5, and the Mordell–Weil group ofJ0(N)is finite. The values of Nare 34, 38, 42, 44, 45, 51, 52, 54, 55, …
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Weierstrass mock modular forms and elliptic curves Open
Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves $E/\mathbb {Q}$ . W…
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Finiteness theorems for K3 surfaces and abelian varieties of CM type Open
We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational …
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-adic images of Galois for elliptic curves over (and an appendix with John Voight) Open
We discuss the $\ell $ -adic case of Mazur’s ‘Program B’ over $\mathbb {Q}$ : the problem of classifying the possible images of $\ell $ -adic Galois representations attached to elliptic curves E over $\mathbb {Q}$ , equivalently, classifyi…
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Galois representations attached to elliptic curves with complex\n multiplication Open
The goal of this article is to give an explicit classification of the\npossible $p$-adic Galois representations that are attached to elliptic curves\n$E$ with CM defined over $\\mathbb{Q}(j(E))$. More precisely, let $K$ be an\nimaginary qu…
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Examples of CM curves of genus two defined over the reflex field Open
Van Wamelen [ Math. Comp. 68 (1999) no. 225, 307–320] lists 19 curves of genus two over $\mathbf{Q}$ with complex multiplication (CM). However, for each curve, the CM-field turns out to be cyclic Galois over $\mathbf{Q}$ , and the generic …
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Composite images of Galois for elliptic curves over $\mathbf {Q}$ and entanglement fields Open
Let $E$ be an elliptic curve defined over $\mathbf {Q}$ without complex multiplication. For each prime $\ell$, there is a representation $\rho _{E,\ell }\colon \operatorname {Gal}(\overline {\mathbf {Q}}/\mathbf {Q}) \rightarrow \operatorn…
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Height pairings on orthogonal Shimura varieties Open
Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$ . We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of s…
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Galois representations attached to elliptic curves with complex multiplication Open
The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary quadra…
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No Singular Modulus Is a Unit Open
A result of the 2nd-named author states that there are only finitely many complex multiplication (CM)-elliptic curves over $\mathbb{C}$ whose $j$-invariant is an algebraic unit. His proof depends on Duke’s equidistribution theorem and is h…
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Kronecker’s Limit Formula, Holomorphic Modular Functions, and<i>q</i>-Expansions on Certain Arithmetic Groups Open
For any square-free integer $N$ such that the "moonshine group"\n$\\Gamma_0(N)^+$ has genus zero, the Monstrous Moonshine Conjectures relate the\nHauptmoduli of $\\Gamma_0(N)^+$ to certain McKay-Thompson series associated to\nthe represent…
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CM values of regularized theta lifts and harmonic weak Maaß forms of weight 1 Open
We study special values of regularized theta lifts at complex multiplication (CM) points. In particular, we show that CM values of Borcherds products can be expressed in terms of finitely many Fourier coefficients of certain harmonic weak …
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Faltings heights of abelian varieties with complex multiplication Open
Let M be the Shimura variety associated with the group of spinor similitudes of a rational quadratic space over of signature (n,2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of specia…
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Entanglement in the family of division fields of elliptic curves with complex multiplication Open
For every elliptic curve $E$ which has complex multiplication (CM) and is\ndefined over a number field $F$ containing the CM field $K$, we prove that the\nfamily of $p^{\\infty}$-division fields of $E$, with $p \\in \\mathbb{N}$ prime,\nbe…
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Rankin–Selberg L-functions and the reduction of CM elliptic curves Open
Let q be a prime and $$K={\mathbb Q}(\sqrt{-D})$$ be an imaginary quadratic field such that q is inert in K. If $$\mathfrak {q}$$ is a prime above q in the Hilbert class field of K, there is a reduction map $$\begin{aligned} r_{\mathfrak q…
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Generalized Heegner cycles at Eisenstein primes and the Katz p-adic L-function Open
We consider normalized newforms [math] whose nonconstant term Fourier coefficients are congruent to those of an Eisenstein series modulo some prime ideal above a rational prime [math] . In this situation, we establish a congruence between …
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Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp Open
Using a combination of several powerful modularity theorems and class field\ntheory we derive a new modularity theorem for semistable elliptic curves over\ncertain real abelian fields. We deduce that if $K$ is a real abelian field of\ncond…
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Katz type p-adic L-functions for primes p non-split in the CM field Open
For every triple F,K,p where F is a classical elliptic eigenform, K is a quadratic imaginary field and p> 3 is a prime integer which is not split in K, we attach a p-adic L function which interpolates the algebraic parts of the special val…
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Fields of definition of elliptic $k$-curves and the realizability of all genus 2 Sato–Tate groups over a number field Open
Let A/Q be an abelian variety of dimension g = 1 that is isogenous over Q to Eg, where E is an elliptic curve. If E does not have complex multiplication (CM), by results of Ribet and Elkies concerning fields of definition of elliptic Q-cur…
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Non‐vanishing theorems for central L‐values of some elliptic curves with complex multiplication Open
The paper uses Iwasawa theory at the prime p=2 to prove non‐vanishing theorems for the value at s=1 of the complex L‐series of certain quadratic twists of the Gross family of elliptic curves with complex multiplication by the field K=Q(−q)…
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Primes dividing invariants of CM Picard curves Open
We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Our proof is simpler than the previous proofs for genus 2 and 3 and, unlike previous bounds for genus 3, our bou…
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High Speed RADIX-2 Butterfly Structure Using Novel Wallace Multiplier Open
In Signal Processing applications the arithmetic units mainly consists of adders and multipliers. These arithmetic units are used in to enhance the performance of Fast Fourier Transform (FFT) Butterfly structure implementation. This paper …
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Cyclic Isogenies for Abelian Varieties with Real Multiplication Open
We study quotients of principally polarized abelian varieties with real multiplication by finite Galois-stable subgroups and describe when these quotients are principally polarizable. We use this character-ization to provide an algorithm t…
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New Parallel Approaches for Scalar Multiplication in Elliptic Curve over Fields of Small Characteristic Open
We present two new strategies for parallel implementation of scalar multiplication over elliptic curves. We first introduce a Montgomery-halving algorithm which is a variation of the original Montgomery-ladder for point multiplication. Thi…
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CM RELATIONS IN FIBERED POWERS OF ELLIPTIC FAMILIES Open
Let $E_{\unicode[STIX]{x1D706}}$ be the Legendre family of elliptic curves. Given $n$ points $P_{1},\ldots ,P_{n}\in E_{\unicode[STIX]{x1D706}}(\overline{\mathbb{Q}(\unicode[STIX]{x1D706})})$ , linearly independent over $\mathbb{Z}$ , we p…
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CM periods, CM regulators and hypergeometric functions, I Open
We study the $H^2$ of certain surfaces with complex multiplication by a cyclotomic field. The periods are written in terms of values of the gamma function and the conjecture of Gross-Deligne is verified. The regulators of certain $K_1$-ele…
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Reductions of abelian surfaces over global function fields Open
Let $A$ be a non-isotrivial ordinary abelian surface over a global function field of characteristic $p>0$ with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a mu…
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Quickly constructing curves of genus 4 with many points Open
The "defect" of a curve over a finite field is the difference between the\nnumber of rational points on the curve and the Weil-Serre bound for the curve.\nWe present a construction for producing genus-4 double covers of genus-2 curves\nove…
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Frobenius Additive Fast Fourier Transform Open
In ISSAC 2017, van der Hoeven and Larrieu showed that evaluating a polynomial P ın Fq [x] of degree <n at all n -th roots of unity in Fqd can essentially be computed d times faster than evaluating Q ın Fqd x at all these roots, assuming Fq…
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Simultaneous supersingular reductions of CM elliptic curves Open
We study the simultaneous reductions at several supersingular primes of elliptic curves with complex multiplication. We show – under additional congruence assumptions on the CM order – that the reductions are surjective (and even become eq…