Alina Carmen Cojocaru, Matthew Fitzpatrick
Let $E/\mathbb{Q}$ be a non-CM elliptic curve. Assuming GRH, we prove that, for a set of primes $p$ of density $1$, the absolute discriminant of the $\mathbb{F}_p$-endomorphism ring of the reduction of $E$ modulo $p$ is close to maximal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Let $ψ: A \to F\{τ\}$ be a Drinfeld $A$-module over $F$ of rank 2 and without complex multiplication, where $A = {\mathbb{F}}_q[T]$, $F = {\mathbb{F}}_q(T)$, and $q$ is an odd prime power. For a prime $\mathfrak{p} = p A$ of $A$ of good reduction for $ψ$ and with residue field ${\mathbb{F}}_{\mathfrak{p}}$, we study the growth of the absolute value $|Δ_{\mathfrak{p}}|$ of the discriminant of the ${\mathbb{F}}_{\mathfrak{p}}$-endomorphism ring of the reduction of $ψ$ modulo $\mathfrak{p}$. We prove that for all $\mathfrak{p}$, $|Δ_{\mathfrak{p}}|$ grows with $|p|$. Moreover, we prove that for a…
The Brezing-Weng method is a general framework to generate families of pairing-friendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endomorphism rings with small class number.
Let $E$ be an elliptic curve over $\Q$. It is well known that the ring of endomorphisms of $E_p$, the reduction of $E$ modulo a prime $p$ of ordinary reduction, is an order of the quadratic imaginary field $Q(π_p)$ generated by the Frobenius element $π_p$. When the curve has complex multiplication (CM), this is always a fixed field as the prime varies. However, when the curve has no CM, very little is known, not only about the order, but about the fields that might appear as algebra of endomorphisms varying the prime. The ring of endomorphisms is obviously related with the arithmetic of $a_p^2…
We give an improvement of a result of J. Martinet on Stickelberger's congruences for the absolute norms of relative discriminants of number fields, by using classical arguments of class field theory.
We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field F_q. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the second algorithm depends primarily on log |D_E|, where D_E is the discriminant of the order isomorphic to End(E). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed.
We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a constant times X. In contrast with the quartic case, we also show that a density of 100% of quintic fields, when ordered by absolute discriminant, have Galois closure with full Galois group $S_5$. The analogues of these results are also proven for orders in quintic fields. Finally, we give an interpretation of the various constants appearing in these theorems in terms of local masses of quintic ri…
Computing the endomorphism ring of an elliptic curve over a number field
本文提出确定性和概率性算法，可判断首一不可约多项式是否为希尔伯特类多项式、代数整数是否为CM椭圆曲线的j不变量，还能计算数域上椭圆曲线的自同态环，且实现简单、速度优于现有方法。
Consider an absolutely simple abelian variety X over a number field K. If the absolute endomorphism ring of X is commutative and satisfies certain parity conditions, then the reduction X_p is absolutely simple for almost all p. Conversely, if the absolute endomorphism ring of X is noncommutative, then X_p is reducible for p in a set of positive density.
Let E be an elliptic curve with complex multiplication by R, where R is an order of discriminant D<-4 of an imaginary quadratic field K . If a prime number p is decomposed completely in the ring class field associated with R, then E has good reduction at a prime ideal P of K dividing p and there exist positive integers u and v such that 4p=u^2-Dv^2. It is well known that square of the trace a_P of Frobenius endomorphism of the reduction of E modulo P is equal to u^2. We determine whether a_P=u or a_P=-u in the case the class number of R is 2 or 3 and D is divided by 3,4 or 5. This article is a…
Computing supersingular endomorphism rings using inseparable endomorphisms
提出一种仅需单条路径计算超奇异椭圆曲线不可分自同态的算法，在GRH假设下达到最优复杂度，结合现有算法可更快计算自同态环基。
Let $K$ be a number field and $d_K$ the absolute value of the discrimant of $K/\mathbb{Q}$. We consider the root discriminant $d_L^{\frac{1}{[L:\mathbb{Q}]}}$ of extensions $L/K$. We show that for any $N>0$ and any positive integer n, the set of length n solvable extensions of $K$ with root discriminant less than $N$ is finite. The result is motivated by the study of class field towers.
In this work, we compute the perfect forms for all imaginary quadratic fields of absolute discriminant up to $5000$ and study the number and types of the polytopes that arise. We prove a bound on the combinatorial types of polytopes that can arise regardless of discriminant and give a volumetric argument for a lower bound on the number of perfect forms as well as a heuristic for a better lower bound for imaginary quadratic fields of sufficiently large absolute discriminant.
We prove an improvement on Schmidt's upper bound on the number of number fields of degree $n$ and absolute discriminant less than X for $6 \leq n \leq 94$. We carry this out by improving and applying a uniform bound on the number of monic integer polynomials, having bounded height and discriminant divisible by a large square, that we proved in a previous work.
Let d1 and d2 be discriminants of distinct quadratic imaginary orders O_d1 and O_d2 and let J(d1,d2) denote the product of differences of CM j-invariants with discriminants d1 and d2. In 1985, Gross and Zagier gave an elegant formula for the factorization of the integer J(d1,d2) in the case that d1 and d2 are relatively prime and discriminants of maximal orders. To compute this formula, they first reduce the problem to counting the number of simultaneous embeddings of O_d1 and O_d2 into endomorphism rings of supersingular curves, and then solve this counting problem. Interestingly, this counti…
Endomorphism Rings of Supersingular Elliptic Curves and Ternary Quadratic Forms
本文研究了超奇异椭圆曲线的自同态环，提出一种通过求解有限域平方根，从 Eichler order 高效计算 $B_{p,\infty}$ 中极大阶的算法，并给出了核理想右阶的基。
We consider the problem of checking whether an elliptic curve defined over a given number field has complex multiplication. We study two polynomial time algorithms for this problem, one randomized and the other deterministic. The randomized algorithm can be adapted to yield the discriminant of the endomorphism ring of the curve.
Tabulating Absolute Lucas Pseudoprimes
研究固定判别式下绝对 Lucas 伪素数的枚举问题，提出受 Carmichael 数启发的枚举算法及渐近分析，证明给定素因子集合下伪素数有限，并首次给出判别式 5 的枚举结果。
Rational Points and Zeta Functions of Humbert Surfaces with Square Discriminant
本文研究有限域上具(n,n)分裂雅可比矩阵的2类曲线轨迹算术性质，计算多有限域有理点并推导n=2、3时的泽塔函数，据此提出分裂雅可比矩阵高效检测法，推动后量子2类系统的曲线选择、安全分析与协议设计。
Some thoughts on the congruence D=0,1(mod 4) for the absolute discriminant D of a number field
We prove inequalities that compare the regulator of a number field with its absolute discriminant. We refine some ideas in Silverman's work in 1984 where such general inequalities are first proven. In order to prove our main theorems, we combine these refinements with the authors' recent results on bounding the product of heights of relative units in a number field extension.
We prove a local analogue of a theorem of J. Martinet about the absolute norm of the relative discriminant ideal of an extension of number fields. The result can be seen as a statement about 2-primary units. We also prove a similar statement about the absolute norms of p-primary units, for all primes p.
Connecting Kani's Lemma and path-finding in the Bruhat-Tits tree to compute supersingular endomorphism rings
本文提出了一种基于Kani引理和布鲁hat-蒂茨树路径寻找的确定性多项式时间算法，用于计算奇异超椭圆曲线的自同态环。
We determine the density of monic integer polynomials of given degree $n>1$ that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials $f(x)$, such that $f(x)$ is irreducible and $\mathbb Z[x]/(f(x))$ is the ring of integers in its fraction field, is positive, and is in fact given by $ζ(2)^{-1}$. It also follows from our methods that there are $\gg X^{1/2+1/n}$ monogenic number fields of degree $n$ having associated Galois group $S_n$ and absolute discrim…
Building on recent work of Ardakov and Wadsley, we prove Schur's lemma for absolutely irreducible admissible p-adic Banach space (respectively locally analytic) representations of p-adic Lie groups. We also prove finiteness results for the endomorphism algebra of an irreducible admissible representation.
Weighted Distributions of Complex Multiplication Orders in Ordinary Isogeny Classes
本文基于Deuring对应，研究有限域上普通椭圆同源类内端射影环的加权分布，通过类数公式得到全局分布，并揭示了同源图几何与类域理论的联系。
The Endomorphism Rings of Supersingular Elliptic Curves over $\mathbb{F}_p$ and the Binary Quadratic Forms
本文建立了 $\mathbb{F}_p$ 上超奇异椭圆曲线同构类与特定判别式的本原既约二元二次型间的一一对应，证明同源运算与二次型复合相容，并借此将 CSIDH 密码系统的安全性归约到显式计算该对应关系上。
Let $E$ be an elliptic curve without CM that is defined over a number field $K$. For all but finitely many nonarchimedean places $v$ of $K$ there is the reduction $E(v)$ of $E$ at $v$ that is an elliptic curve over the residue field $k(v)$ at $v$. The set of $v$'s with ordinary $E(v)$ has density 1 (Serre). For such $v$ the endomorphism ring $End(E(v))$ of $E(v)$ is an order in an imaginary quadratic field. We prove that for any pair of relatively prime positive integers $N$ and $M$ there are infinitely many nonarchimedean places $v$ of $K$ such that the discriminant $Δ(v)$ of $End(E(v))$ is d…
We prove that the jacobian of a hyperelliptic curve y^2=f(x) is absolutely simple if deg(f)=q+1 where q is a power prime congruent to 5 modulo 8, the polynomial f(x) is irreducible over the ground field of characteristic zero and its Galois group is L_2(q).
Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders
本文研究了寻找超奇异椭圆曲线定向的困难度，证明判断曲线可定向性的预言机能为计算对应自同态提供非平凡信息，并提供了显式算法、复杂度分析及四元数代数中的嵌入算法与高效代码实现。
We present here a formula for expressing the trace of the Frobenius endomorphism of an elliptic curve $E$ over $\mathbb{F}_q$ satisfying $j(E)\neq 0,1728$ and $q\equiv 1 \pmod{12}$ in terms of special values of Gaussian hypergeometric series. This paper uses methods introduced in Fuselier's work for one parameter families of curves to express the trace of Frobenius of $E$ as a function of its $j$-invariant and discriminant instead of a parameter, which are more intrinsic characteristics of the curve.
The Hilbert class polynomial has as roots the j-invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a prescribed number of points. Since its coefficients are typically rather large, there has been continued interest in finding alternative modular functions whose corresponding class polynomials are smaller. Best known are Weber's functions, that reduce the size by a factor of 72 for a positive density subset of imaginary quadratic discriminants. On the other hand, Bröker and Stevenhagen sho…
Genus 2 curves with bad reduction at one odd prime
本文研究了由 $y^2=xh(x)$ 描述的亏格2曲线，证明了当 $h(x)$ 可约时绝对判别式为奇素数的曲线仅有两族，并在 $h(x)$ 不可约时给出了显式曲线族示例。
In this paper we will give a global description of the Frobenius for the division fields of an elliptic curve E which is strictly analogous to the cyclotomic case. This is then applied to determine the splitting of a prime p in subfields of such a division field. Such fields include a large class of non-solvable quintic extensions and our application provides an arithmetic counterpart to Klein's "solution" of quintic equations using elliptic functions. A central role is played by the discriminant of the ring of endomorphisms of the elliptic curve reduced modulo p.
We consider polynomials of degree $d$ with only real roots and a fixed value of discriminant, and study the problem of minimizing the absolute value of polynomials at a fixed point off the real line. There are two explicit families of polynomials that turn out to be extremal in terms of this problem. The first family has a particularly simple expression as a linear combination of $d$-th powers of two linear functions. Moreover, if the value of the discriminant is not too small, then the roots of the extremal polynomial and the smallest absolute value in question can be found explicitly. The se…
A lower bound for the distance between CM points on Shimura curves
本文针对志村曲线上的CM点建立定量丢番图逼近结果，通过富克斯一致化等方法，依据CM点自同态环判别式，证明了收敛到固定CM点的一列CM点与该固定点距离的下界，给出哈贝格尔结果的志村曲线类似物。
Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as $S$. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves that appear over the field with smallest absolute value of its discriminant and having Galois group $S_3$ and over the field with smallest absolute value of its discriminant and having Galois group $\Z…
Higher rank elliptic sieve via CM endomorphisms
本文针对带复乘的椭圆曲线，将椭圆筛法应用于椭圆网，证明几乎所有非挠点倍点的y坐标不属于阿贝尔扩张的范数群，拓展了椭圆筛法的应用场景。
We prove that the jacobian of a hyperelliptic curve $y^2=(x-t)h(x)$ has no nontrivial endomorphisms over an algebraic closure of the ground field $K$ of characteristic zero if $t \in K$ and the Galois group of the polynomial $h(x)$ over $K$ is "very big" and $deg(h)$ is an even number >8. (The case of odd $deg(h)>3$ follows easily from previous results of the author.)
For $n=3$, 4, and 5, we prove that, when $S_n$-number fields of degree $n$ are ordered by their absolute discriminants, the lattice shapes of the rings of integers in these fields become equidistributed in the space of lattices.
Capitulation discriminants of genus one curves
本文将$n$次亏格1模型概念推广到任意奇数$n$，证明$\mathbb{Q}$上处处局部可解的亏格1曲线存在极小积分模型，并将结果应用于椭圆曲线Tate-Shafarevich群的capitulation问题，得到相关分裂数域的判别式上界。
Let $F(x, y)$ be a binary form with integer coefficients, degree $n\geq 3$ and irreducible over the rationals. Suppose that only $s + 1$ of the $n + 1$ coefficients of $F$ are nonzero. We show that the Thue inequality $|F(x,y)|\leq m$ has $\ll sm^{2/n}$ solutions provided that the absolute value of the discriminant $D(F)$ of $F$ is large enough. We also give a new upper bound for the number of solutions of $|F(x,y)|\leq m$, with no restriction on the discriminant of $F$ that depends mainly on $s$ and $m$, and slightly on $n$. Our bound becomes independent of $m$ when $m<|D(F)|^{2/(5(n-1))}$, a…
Consider an absolutely simple abelian variety A defined over a number field K. For most places v of K, we study how the reduction A_v of A modulo v splits up to isogeny. Assuming the Mumford-Tate conjecture for A and possibly increasing K, we will show that A_v is isogenous to the m-th power of an absolutely simple abelian variety for all places v of K away from a set of density 0, where m is an integer depending only on the endomorphism ring End(A_Kbar). This proves many cases, and supplies justification, for a conjecture of Murty and Patankar. Under the same assumptions, we will also describ…
Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$. Let $R_F(Z)$ denote the number of integers of absolute value at most $Z$ which are represented by $F$. We prove that there is a positive number $C_F$ such that $R_F(Z)$ is asymptotic to $C_F Z^{\frac{2}{d}}$.
A result of the second-named author states that there are only finitely many CM-elliptic curves over $\mathbb{C}$ whose $j$-invariant is an algebraic unit. His proof depends on Duke's Equidistribution Theorem and is hence non-effective. In this article, we give a completely effective proof of this result. To be precise, we show that every singular modulus that is an algebraic unit is associated with a CM-elliptic curve whose endomorphism ring has discriminant less than $10^{15}$. Through further refinements and computer-assisted computations, we eventually rule out all remaining cases, showing…
Let $F/\mathbb{Q}$ be a totally real field and $A$ a modular $\GL_2$-type abelian variety over $F$. Let $K/F$ be a CM quadratic extension. Let $χ$ be a class group character over $K$ such that the Rankin-Selberg convolution $L(s,A,χ)$ is self-dual with root number $-1$. We show that the number of class group characters $χ$ with bounded ramification such that $L'(1, A, χ) \neq 0$ increases with the absolute value of the discriminant of $K$. We also consider a rather general rank zero situation. Let $π$ be a cuspidal cohomological automorphic representation over $\GL_{2}(\BA_{F})$. Let $χ$ be a …
Let K be a fixed number field and G its absolute Galois group. We give a bound C(K), depending only on the degree, the class number and the discriminant of K, such that for any elliptic curve E defined over K and any prime number p strictly larger than C(K), the image of the representation of G attached to the p-torsion points of E contains a subgroup of homotheties of index smaller than 12.
Improved bounds for Szpiro's conjecture
本文改进了Szpiro猜想的界，对一般椭圆曲线证明log Δ ≪ N log log N，对半稳定情形证明log Δ ≪ N，方法基于Shimura曲线理论。
An abelian variety over a number field is called L-abelian variety if, for any element of the absolute Galois group of a number field L, the conjugated abelian variety is isogenous to the given one by means of an isogeny that preserves the Galois action on the endomorphism rings. We can think of them as generalizations of abelian varieties defined over L with endomorphisms also defined over L. In the one dimensional case, an elliptic curve defined over L gives rise to a Galois representation provided by the Galois action on its Tate module. This classical Galois representation has been a centr…
The purpose of this note is to suggest an analogue for genus 2 curves of part of Gross and Zagier's work on elliptic curves. Experimentally, for genus 2 curves with CM by a quartic CM field K, it appears that primes dividing the denominators of the discriminants of the Igusa class polynomials all have the property 1) that they are bounded by d, the absolute value of the discriminant of K, and 2) that they divide d-x^2, for some integer x whose square is less than d. A slightly stronger condition is given in Section 3. Such primes are primes of bad reduction for the genus 2 curve and primes of …
The geometry of secondary terms in arithmetic statistics
本文证明了函数域三次扩张计数中次级项的存在，利用几何参数化方法对光滑曲线计数，推广了有理数域上的结果。
A weighted projective stack is a stacky quotient $\mathscr P(\mathbf a)=(\mathbf A^n-\{0\})/\mathbb G_m$, where the action of $\mathbb G_m$ is with weights $\mathbf a\in\mathbb Z^n_{>0}$. Examples are: the compactified moduli stack of elliptic curves $\mathscr P(4,6)$ and the classifying stack of $μ_m$-torsors $Bμ_m=\mathscr P(m)$. We define heights on the weighted projective stacks. The heights generalize the naive height of an elliptic curve and the absolute discriminant of a torsor. We use the heights to count rational points. We find the asymptotic behaviour for the number of rational poin…
For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r=6) to over 100 (for r=10 and r=11). We describe our search methods, and tabulate, for each r=5,6,...,11, the five curves of lowest conductor, and (except for r=11) also the five of lowest absolute discriminant, that we found.
We use the Taylor-Wiles-Kisin patching method to investigate the multiplicities with which Galois representations occur in the mod $\ell$ cohomology of Shimura curves over totally real number fields. Our method relies on explicit computations of local deformation rings done by Shotton, which we use to compute the Weil class group of various deformation rings. Exploiting the natural self-duality of the cohomology groups, we use these class group computations to precisely determine the structure of a patched module in many new cases in which the patched module is not free (and so multiplicity on…
Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be an odd prime and $ι: \overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ an embedding. Let $K$ be an imaginary quadratic field and $H_{K}$ the corresponding Hilbert class field. For a class group character $χ$ over $K$, let $\mathbb{Q}(χ)$ be the field generated by the image of $χ$ and $\mathfrak{p}_χ$ the prime of $\mathbb{Q}(χ)$ above $p$ determined via $ι_p$. Under mild hypotheses, we show that the number of class group characters $χ$ such that the $χ$-isotypic Tate--Shafarevich group of $E$ over $H_{K}$ is finite with trivial $\math…
Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field \(K=\mathbb{Q}(\sqrt{d})\), p-ring spaces \(V_p(c)\) modulo c are introduced by defining a morphism \(ψ:\,f\mapsto V_p(f)\) from the divisor lattice \(\mathbb{N}\) of positive integers to the lattice S of subspaces of the direct product \(V_p\) of the p-elementary class group \(C/C^p\) and unit group \(U/U^p\) of K. Their properties admit an exact count of all extension fields N over K, having the dihedral group of order 2p as absolute Galois group \(Gal(N | \mathbb{Q})\) and sharing a common discriminan…
Let $n \geq 6$ be an integer. We prove that the number of number fields with Galois group $A_n$ and absolute discriminant at most $X$ is asymptotically at least $X^{1/8 + O(1/n)}$. For $n \geq 8$ this improves upon the previously best known lower bound of $X^{(1 - \frac{2}{n!})/(4n - 4) - ε}$, due to Pierce, Turnage-Butterbaugh, and Wood.
Tate's question, Standard conjecture D, semisimplicity and Dynamical degree comparison conjecture
本文在标准猜想D假设下证明动力次数比较猜想的较弱形式，建立Frobenius自同态与极化自同态半单性的等价性，并给出Tate问题/Serre猜想的新证明。
Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d \geq 3$. Let $R_F(Z)$ denote the number of integers of absolute value at most $Z$ which are represented by $F$. In 2019 Stewart and Xiao proved that $R_F(Z) \sim C_FZ^{2/d}$ for some positive number $C_F$. We compute $C_{R_n}$ and $C_{J_n}$ for the binary forms $R_n(x, y)$ and $J_n(x, y)$ defined by means of the relation $ (x + yi)^n = R_n(x, y) + J_n(x, y)i$, where the variables $x$ and $y$ are real.
Curves on Compact Arithmetic Quotients of Hyperbolic 2-ball
本文研究双曲2-球紧算术商上的曲线，利用体积估计及特殊子簇与奇点几何，证明大判别式时不存在任意固定亏格的复曲线。
Computing Isomorphisms between Products of Supersingular Elliptic Curves
研究有限域上超奇异椭圆曲线乘积间同构的计算方法，利用德林对应将问题转化为四元数代数问题，通过求解相关方程组，结合ℓ - 进技术，给出高效概率算法，在广义黎曼假设下有期望多项式时间复杂度。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。