Generalized class polynomials

We give a new algorithm for constructing Picard curves over a finite field with a given endomorphism ring. This has important applications in cryptography since curves of genus 3 allow for smaller key sizes than elliptic curves. For a sextic CM-field $K$ containing the cube roots of unity, we define and compute certain class polynomials modulo small primes and then use the Chinese Remainder Theorem to construct the class polynomials over the rationals. We also give some examples.

Let H_{D}(T) denote the Hilbert class polynomial of the imaginary quadratic order of discriminant D . We study the rate of growth of the greatest common divisor of H_{D}(a) and H_{D}(b) as |D| \to \infty for a and b belonging to various Dedekind domains. We also study the modular support problem: if for all but finitely many D every prime ideal dividing H_{D}(a) also divides H_{D}(b) , what can we say about a and b ? If we replace H_{D}(T) by T^{n}-1 and the Dedekind domain is a ring of S -integers in some number field, then these are classical questions that have been investigated by Bugeaud–Corvaja–Zannier, Corvaja–Zannier, and Corrales-Rodrigáñez–Schoof.

We present algorithms for computing j j -invariants, modular polynomials and explicit isogenies for ordinary rank 2 Drinfeld modules over finite fields and describe how Drinfeld modular polynomials can be used to compute isogeny graphs and endomorphism rings of ordinary rank 2 Drinfeld modules. Our technique for computing Drinfeld modular polynomials is based on the traditional analytic approach for obtaining classical modular polynomials. Our ideas for generating isogeny graphs and finding endomorphism rings for rank 2 Drinfeld modules closely follows the work of Kohel and Fouquet. All our algorithms were implemented in SAGE and numerical examples are included.

Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

The aim of this paper is to justify the common cryptographic practice of selecting elliptic curves using their order as the primary criterion. We can formalize this issue by asking whether the discrete log problem (DLOG) has the same difficulty for all curves over a given finite field with the same order. We prove that this is essentially true by showing polynomial time random reducibility of DLOG among such curves, assuming the Generalized Riemann Hypothesis (GRH). We do so by constructing certain expander graphs, similar to Ramanujan graphs, with elliptic curves as nodes and low degree isogenies as edges. The result is obtained from the rapid mixing of random walks on this graph. Our proof works only for curves with (nearly) the same endomorphism rings. Without this technical restriction such a DLOG equivalence might be false; however, in practice the restriction may be moot, because all known polynomial time techniques for constructing equal order curves produce only curves with nearly equal endomorphism rings.

We extend many results on Selmer groups for elliptic curves and modular forms to the non-ordinary setting. More precisely, we study the signed Selmer groups defined using the machinery of Wach modules over ℤ p -cyclotomic extensions. First, we provide a definition of residual and non-primitive Selmer groups at non-ordinary primes. This allows us to extend techniques developed by Greenberg (for p-ordinary elliptic curves) and Kim (p-supersingular elliptic curves) to show that if two p-non-ordinary modular forms are congruent to each other, then the Iwasawa invariants of their signed Selmer groups are related in an explicit manner. Our results have several applications. First of all, this allows us to relate the parity of the analytic ranks of such modular forms generalizing a recent result of the first-named author for p-supersingular elliptic curves. Second, we can prove a Kida-type formula for the signed Selmer groups generalizing results of Pollack and Weston.

Using the reduction theory of Nrron we give necessary conditions for the existence of points of order q on elliptic curves E rational over global fields. An application is the determination of all elliptic cu rves /Q with integer j and torsion points, generalizing Olson [8]. Another application is a theorem about semistable reduction whose consequences generalize a theorem of Olson [9] ( K = Q) and give divisibility conditions for the discriminant and the coefficients of E related with the paper of Zimmer [13] as well as diophantine equations related with Fermat's equation that are discussed for K Q and K a function field. We are interested in elliptic curves over global fields K (i.e. : K is a finite number field or K is a function field of one variable over a finite field) and especially in the torsion group of E(K), where E(K) is the group of K-rational points of E. It is well known that E(K) is finitely generated, it is conjectured that if K is a number field then the order of the torsion group of E(K) is bounded by some number depending only on K (cf. Demjanenko [1]). In any case in order to handle with E(K) the first step is to determine the torsion group. In principle this is not so difficult; if one uses the results of Lutz [6] and Zimmer [13], one sees immediately that for every E there exist points of q-power-order only for a finite number of primes q, as the equations for points of order q are known (in principle) one has only t ~ test what orders really occur. But as the computational work grows very rapidly with q it is usefull to look for sharper necessary conditions, and this shall be done in this paper.

Factorization of a class of polynomials over finite fields

This thesis investigates three distinct questions concerning the asymptotic average order of certain group theoretic objects. The three main chapters can be read independently. In chapter zero, we give a high level overview of our results. For a polynomial f [is an element of] [finite field]q[x] with nonzero constant term, define the order of f to be the order of x in the quotient ring [finite field]q[x]/(f). In Chapter 1, we consider the average order of several classes of polynomials as the degree n [right arrow] [infinity]. We show that the average order of an irreducible polynomial is asymptotically [theta]([phi](q^n - 1)) with [phi] the Euler Totient function. Building on work of Stong [32], we show that the asymptotic average order of both squarefree and nonsquarefree polynomials is q^n / n^(1+o(1)). As an application we relate this to the expected orbit size of a vector under iteration by a random invertible matrix. In chapter two, we count the number of compositions of n with k parts whose pairwise gcds are coprime to a given finite set P of primes. We obtain asymptotic results when n [right arrow] [infinity], showing that the number of such compositions is ([product](p[is an element of]P|(n)) h[k](p) + (-1)^(k-1)(k-1)/p^(k-1) (1-1/p)) x ([product](p[is an element of P[crossed pipe](n)) h[k](p) + (-1)^k(k-1)/p^k) n^(k-1)/(k-1)! + O(n^(k-2)) where h[k](p) := (1-1/p)^(k-1)(1+(k-1)/p). Building on work of Bubbolini, Luca and Spiga [6], we also consider the set of compositions whose parts are pairwise coprime. We use a geometric argument to strengthen their result when k = 3. We apply their results to obtain bounds on the average order of a permutation with a given number of cycles. Chapter three is joint work with Huseyin Acan, Charles Burnette, Sean Eberhard, and Eric Schmutz. Motivated by a question of Thibault Godin [17], we show that the probability P₁ that two random permutations in the symmetric group S[n] have the same order is asymptotically n^(-2+o(1)). We compare this to the probability P₂ that two permutations are conjugate, showing by construction that lim sup(n) P₁/P₂ = [infinity]. Additionally, we show by more direct methods that with probability O(log log n / log n) there exists some prime dividing the order of one of the permutations but not the other.

Hardware acceleration of cryptographic algorithms is beneficial because considerable performance improvements can be attained compared to software implementations. Thus, hardware implementations can be used in critical applications requiring high encryption or decryption speeds. Parallel architecture with efficient hardware implementation of Galois field arithmetic operations is used to produce high speed computation time for the scalar multiplication operation which is the main operation in Elliptic Curve Cryptography (ECC) system. This work proposed a modification in karatsuba-ofman algorithm which is one of the best algorithms used to perform multiplication operation over Galois field. The modification contrasted on truncating karatsuba-ofman algorithm in a low level and using the classic polynomial multiplication algorithm. In addition, this work proposed architecture for implementing ECC on hardware using Montgomery algorithm in projective coordinates. The results show that the proposed architecture is able to compute GF(2^191) elliptic curve scalar multiplication operations in 72.939 μs on Xilinx Virtex-II XC2V6000 FPGA device and 100.68 μs on Xilinx VirtexE 2600. Also, the proposed architecture can be changed to be suitable for any arbitrary Galois field size with little modifications.KeywordsGalois fieldElliptic curve cryptographyKaratsuba-Ofman multiplierfield programmable gate arrayspolynomial multiplicationpolynomial inversion

For a complex abelian surface A with endomorphism ring isomorphic to the maximal order in a quartic CM field K , the Igusa invariants j 1 (A), j 2 (A), j 3 (A) generate an unramified abelian extension of the reflex field of K .In this paper we give an explicit geometric description of the Galois action of the class group of this reflex field on j 1 (A), j 2 (A), j 3 (A).Our description can be expressed by maps between various Siegel modular varieties, and we can explicitly compute the action for ideals of small norm.We use the Galois action to modify the CRT method for computing Igusa class polynomials, and our run time analysis shows that this yields a significant improvement.Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby implying that the 'isogeny volcano' algorithm to compute endomorphism rings of ordinary elliptic curves over finite fields does not have a straightforward generalization to computing endomorphism rings of abelian surfaces over finite fields.Gruenewald thanks Microsoft Research, where this research was undertaken, for its hospitality.

As typically the quantum Rabi model, particular attention has been paid recently to studying the spectrum of self-adjoint operators with non-commutative coefficients, not only in mathematics but also in theoretical/experimental physics, e.g. aiming at an application to quantum information processing. The non-commutative harmonic oscillator (NcHO) is a self-adjoint operator, which is a generalization of the harmonic oscillator, having an interaction term. The Rabi model is shown to be obtained by a second order element of the universal enveloping algebra of the Lie algebra \(\mathfrak {sl}_2\), which is arising from NcHO through the oscillator representation. Precisely, an equivalent picture of the model is obtained as a confluent Heun equation derived from the Heun operator defined by that element via another representation. Though the spectrum of NcHO is not fully known, it has a rich structure. In fact, one finds interesting arithmetics/geometry described by e.g. elliptic curves, modular forms in the study of the spectral zeta function of NcHO. In this article, we draw this picture, which may give a better understanding of interacting quantum models.KeywordsEichler integralHeun ODENon-commutative harmonic oscillatorOscillator representationRabi modelSpectral zeta functionUniversal enveloping algebraZeta regularization.

The Apery-like numbers J2(n) associated to the special value ζQ(2) of the spectral zeta function ζQ(s) for the non-commutative harmonic oscillator Q have remarkable modular form interpretation. In fact, we show that the differential equation satisfied by the generating function w2(t) of J2(n) is the Picard-Fuchs equation for the universal family of elliptic curves equipped with rational 4-torsion. The parameter t of this family can be regarded as a modular function for the congruent subgroup Γ0(4). Further, we see that the function w2(t) is regarded as a Γ0(4)-modular form of weight 1 in the variable τ by taking t as the classical Legendre modular function λ(τ).

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus hypergeometric function example. We then focus on identities relating the same hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale’s paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to , hypergeometric functions, and show that one just reduces to the previous cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

In this article, I discuss material which is related to the recent proof of Fermat’s Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey’s construction, and the conjectures of Serre and of Taniyama-Shimura.