pinecone/core-2020-05-10-deduplication huggingface dataset:
The Rank of Abelian Varieties over Infinite Galois Extensions

AbstractG. Frey and M. Jarden (1974, Proc. London Math. Soc.28, 112–128) asked if every Abelian variety A defined over a number field k with dimA>0 has infinite rank over the maximal Abelian extension kab of k. We verify this for the Jacobians of cyclic covers of P1, with no hypothesis on the Weierstrass points or on the base field. We also derive an infinite rank criterion by analyzing the ramification of division points of an Abelian variety. As an application, we show that any d -dimensional Abelian variety A over k with a degree n projective embedding over k has infinite rank over the compositum of all extensions of k of degree <n(4d+2)
Cohen–Lenstra Heuristics and the Spiegelungssatz: Number Fields

AbstractIn this paper we study the compatibility of Cohen–Lenstra heuristics with Leopoldt's Spiegelungssatz (the reflection theorem). We generalize Dutarte's ([1983, in “Théorie des nombres, Besançon, 1983–1984”]) work to every prime number p: He proved the compatibility of the Cohen–Lenstra conjectures with the Spiegelungssatz in the case p=3. We also show that the Spiegelungssatz is compatible with the conjectural probabilities on the p-rank of some subgroups of the class group of a cyclic extension of degree q over Q, where q is a prime number dividing p−1
On the Evaluation of Weil Sums of Dembowski–Ostrom Polynomials

AbstractLet Fq denote the finite field of q elements, q=pe odd, let χ1 denote the canonical additive character of Fq where χ1(c)=e2πiTr(c)/p for all c∈Fq, and let Tr represent the trace function from Fq to Fp. We are interested in evaluating Weil sums of the form S=S(a1, …, an)=∑x∈Fqχ1(D(x)) where D(x)=∑ni=1aixpαi+pβi, αi⩾βi for each i, is known as a Dembowski–Ostrom polynomial (or as a D-O polynomial). Coulter has determined the value of S when D(x)=axpα+1; in this note we show how Coulter's methods can be generalized to determine the absolute value of S for any D-O polynomial. When e is even, we give a subclass of D-O polynomials whose Weil sums are real-valued, and in certain cases we are able to resolve the sign of S. We conclude by showing how Coulter's work for the monomial case can be used to determine a lower bound on the number of Flq-solutions to the diagonal-type equation ∑li=1xpγ+1i+(xi+λ)pγ+1=0, where l is even, e/gcd(γ, e) is odd, and h (X)=λpe−γXpe−γ+λpγX is a permutation polynomial over Fq
Discrepancy of Arithmetic Progressions in Higher Dimensions

AbstractK. F. Roth (1964, Acta. Arith.9, 257–260) proved that the discrepancy of arithmetic progressions contained in [1, N]={1, 2, …, N} is at least cN1/4, and later it was proved that this result is sharp. We consider the d-dimensional version of this problem. We give a lower estimate for the discrepancy of arithmetic progressions on [1, N]d and prove that this result is nearly sharp. We use our results to give an upper estimate for the discrepancy of lines on an N×N lattice, and we also give an estimate for the discrepancy of a related random hypergraph
The Kernel of the Eisenstein Ideal

AbstractLet N be a prime number, and let J0(N) be the Jacobian of the modular curve X0(N). Let T denote the endomorphism ring of J0(N). In a seminal 1977 article, B. Mazur introduced and studied an important ideal I⊆T, the Eisenstein ideal. In this paper we give an explicit construction of the kernel J0(N)[I] of this ideal (the set of points in J0(N) that are annihilated by all elements of I). We use this construction to determine the action of the group Gal(Q/Q) on J0(N)[I]. Our results were previously known in the special case where N−1 is not divisible by 16
Explicit Action of Hecke Operators on Siegel Modular Forms

AbstractWe develop an algorithm for determining an explicit set of coset representatives (indexed by lattices) for the action of the Hecke operators T(p), Tj(p2) on Siegel modular forms of fixed degree and weight. This algorithm associates each coset representative with a particular lattice Ω, pΛ⊆Ω⊆1pΛ where Λ is a fixed reference lattice. We then evaluate the action of the Hecke operators on Fourier series. Since this evaluation yields incomplete character sums for Tj(p2), we complete these sums by replacing this operator with a linear combination of Tℓ(p2), 0⩽ℓ⩽j. In all cases, this yields a clean and simple description of the action on Fourier coefficients
A Note on Algebraic Γ-Monomials and Double Coverings

AbstractWe discuss algebraic Γ-monomials of Deligne. Deligne used the theory of Hodge Cycles to show that algebraic Γ-monomials generate Kummer extensions of certain cyclotomic fields. Das, using a double complex of Anderson and Deligne's results, showed that certain powers of algebraic Γ-monomials and certain square roots of sine monomials generate abelian extensions of Q. Das also gave one example of a nonabelian double covering of a cyclotomic field generated by the square root of a sine monomial. In this note, we will produce infinitely many examples of non-abelian double coverings of cyclotomic fields of Das type. The construction of the examples depends in an interesting way on a lemma of Gauss figuring in an elementary proof of quadratic reciprocity
Isogeny Classes of Hilbert–Blumenthal Abelian Varieties over Finite Fields

AbstractThis paper gives an explicit formula for the size of the isogeny class of a Hilbert–Blumenthal abelian variety over a finite field. More precisely, let OL be the ring of integers in a totally real field dimension g over Q, let N0 and N be relatively prime square-free integers, and let k be a finite field of characteristic relatively prime to both N0N and disc(L, Q). Finally, let (X/k, ι, α) be a g-dimensional abelian variety over k equipped with an action by OL and a Γ0(N0, N)-level structure. Using work of Kottwitz, we express the number of (X′/k, ι′, α′) which are isogenous to (X, ι, α) as a product of local orbital integrals on GL(2); then, using work of Arthur and Clozel and the affine Bruhat decomposition we evaluate all the relevant orbital integrals, thereby finding the cardinality of the isogeny class
Finite Submodules and Iwasawa μ-Invariants

AbstractGiven a Zp-extension of number fields K∞/K and a GK-module A which is cofree as a Zp-module, we can define a Selmer group SA(K∞). If A satisfies certain ordinariness conditions, then SA(K∞) is a cofinitely generated, cotorsion Λ-module. In this paper, we study the effect that finite submodules of A can have on the μ-invariant of SA(K∞). For each finite submodule α⊂A, we use the Galois cohomology of α to define an invariant δ(α) which is a lower bound for μ(SA(K∞). Then we define an invariant m(A) that organizes the information that we get from all of the finite α⊂A. These invariants will lead to an elegant way of explaining some previously known examples where the μ-invariant is positive and they will provide us with new kinds of examples where μ is positive
Theta Series of Quaternion Algebras over Function Fields

AbstractLet K be the function field over a finite field of odd order, and let H be a definite quaternion algebra over K. If Λ is an order of level M in H, we define theta series for each ideal I of Λ using the reduced norm on H. Using harmonic analysis on the completed algebra H∞ and the arithmetic of quaternion algebras, we establish a transformation law for these theta series. We also define analogs of the classical Hecke operators and show that in general, the Hecke operators map the theta series to a linear combination of theta series attached to different ideals, a generalization of the classical Eichler Commutation Relation
Twisted Zeta-Functions and Abelian Stark Conjectures

AbstractWe present a new version “at s=1” of Rubin's refined, higher order Stark conjecture at s=0 for an abelian extension of number fields (K. Rubin, 1996, Ann. Inst. Fourier46, No. 1, 33–62). The key idea is to introduce a formalism of “twisted zeta-functions” to replace the L-functions underlying Rubin's conjecture. This achieves certain simplifications, notably eliminating Gauss sums in a natural way from our version at s=1. It also facilitates some further developments, including an important motivation of the present paper: the formulation of an analogous p-adic conjecture to be presented in a sequel
On the Arithmetic of the Curves y2=xℓ+A, II

AbstractThis paper continues the investigation of the arithmetic of the curves CA:y2=xℓ+A and their Jacobians JA, where ℓ is an odd prime and A is an integer not divisible by ℓ, which was begun in an earlier paper. In the first part, we sketch how to extend the formula for the dimension of a certain Selmer group of JA to the case when A is a (non-zero) square modℓ. The second part deals with the L-series of JA. We determine the corresponding Hecke character and find a formula for the root number of the L-series. This formula is then used to show the “Birch and Swinnerton–Dyer conjecture mod2”ords=1L(JA, s)≡rankJA(Q)mod2 for those A that are covered by the result of the first part, assuming the ℓ-part of Ш(Q, JA) to be finite
Measures of Simultaneous Approximation for Quasi-Periods of Abelian Varieties

AbstractWe examine various extensions of a series of theorems proved by Chudnovsky in the 1980s on the algebraic independence (transcendence degree 2) of certain quantities involving integrals of the first and second kind on elliptic curves; these extensions include generalizations to abelian varieties of arbitrary dimensions, quantitative refinements in terms of measures of simultaneous approximation, as well as some attempt at unifying the aforementioned theorems. In the process we develop tools that might prove useful in other contexts, revolving around explicit “algebraic” theta functions on the one hand, and Eisenstein's theorem and G-functions on the other hand
Non-Archimedean Integration and Elliptic Curves over Function Fields

AbstractLet F be a global function field of characteristic p and E/F an elliptic curve with split multiplicative reduction at the place ∞: then E can be obtained as a factor of the Jacobian of some Drinfeld modular curve. This fact is used to associate to E a measure μE on P1(F∞). By choosing an appropriate embedding of a quadratic unramified extension K/F into the matrix algebra M2(F), μE is pushed forward to a measure on a p-adic group G, isomorphic to an anticyclotomic Galois group over the Hilbert class field of K. Integration on G then yields a Heegner point on E when ∞ is inert in K and an analogue of the L-invariant if ∞ is split. In the last section, the same methods are extended to integration on a geometric cyclotomic Galois group
A Generating Function Technique for Beatty Sequences and Other Step Sequences

AbstractLet g(x,n), with x∈R+, be a step function for each n. Assuming certain technical hypotheses, we give a constant α and function f such that ∑∞n=1:g(x,n) can be written in the form α+∑0<r<xf(r), where the summation is extended over all points in (0,x) at which some g(ċ,n) is not continuous. A typical example is ∑∞n=1z⌊n/x⌋=(1z−1)∑:zq/(1−zq), with the summation extending over all pairs p,q of positive integers satisfying 0<p/q<x and gcd(p,q)=1. We then apply such representations to prove identities such as ζ(z)=∑∞n=1φ(n)nz(ζ(z)−ζ(z,1+1n)), the Lambert series for Euler's totient function, and ∑∞n=0(−1)nσz(2n+1)2n+1=π4z1+z2, where ζ(z) and ζ(z,a) are the Riemann and Hurwitz zeta functions and σz(n)=∑d∣ndzd. We also give a generalization of the Rayleigh–Beatty theorem and a new result of a similar nature for the sequences (⌊2nα⌋−⌊nα⌋)∞n=1
An Effective Version of Belyi's Theorem

AbstractWe compute bounds on covering maps that arise in Belyi's Theorem. In particular, we construct a library of height properties and then apply it to algorithms that produce Belyi maps. Such maps are used to give coverings from algebraic curves to the projective line ramified over at most three points. The computations here give upper bounds on the degree and coefficients of polynomials and rational functions over the rationals that send a given set of algebraic numbers to the set {0,1,∞} with the additional property that the only critical values are also contained in {0,1,∞}
Degrees of the Elliptic Teichmüller Lift

AbstractThe purpose of this paper is to find upper bounds for the degrees, or equivalently, for the order of the poles at O, of the coordinate functions of the elliptic Teichmüller lift of an ordinary elliptic curve over a perfect field of characteristic p. We prove the following bounds:ord0(xn)⩾−(n+2)pn+npn−1, ord0(yn)⩾−(n+3)pn+npn−1. Also, we prove that the bound for xn is not the exact order if, and only if, p divides (n+1), and the bound for yn is not the exact order if, and only if, p divides (n+1)(n+2)/2. Finally, we give an algorithm to compute the reduction modulo p3 of the canonical lift for p≠2,3
Diophantine Definability and Decidability in Large Subrings of Totally Real Number Fields and Their Totally Complex Extensions of Degree 2

AbstractLet M be a number field. Let W be a set of non-archimedean primes of M. LetOM,W={x∈M∣ordpx⩾0 ∀p∉W}. The author continues her investigation of Diophantine definability and decidability in rings OM,W where W is infinite. In this paper, she improves her previous density estimates and extends the results to the totally complex extensions of degree 2 of the totally real fields. In particular, the following results are proved: (1) Let M be a totally real field or a totally complex extension of degree 2 of a totally real field. Then, for any ε>0, there exists a set WM of primes of M whose density is greater than 1−[M:Q]−1−ε and such that Z has a Diophantine definition over OM,WM. (Thus, Hilbert's Tenth Problem is undecidable in OM,WM.) (2) Let M be as above and let ε>0 be given. Let SQ be the set of all rational primes splitting in M. (If the extension is Galois but not cyclic, SQ contains all the rational primes.) Then there exists a set of M-primes WM such that the set of rational primes WQ below WM differs from SQ by a set contained in a set of density less than ε and such that Z has a Diophantine definition over OM,WM. (Again this will imply that Hilbert's Tenth Problem is undecidable in OM,WM.
On Orbits of SL(2,Z)+ and Values of Binary Quadratic Forms on Positive Integral Pairs

AbstractWe consider actions of SL(2,Z) and SL(2,Z)+ (semigroup of matrices with nonnegative integral entries) on the projective space P and on P×P. Results are obtained on orbit-closures under these actions and they are applied to describe a class of binary quadratic forms Q such that the sets Q(Z2) or Q(Z+2) are dense in R. We prove also a result generalising a theorem of Troessaert and Valette (On values at integer points of some irrational quadratic forms, preprint) associating to certain binary quadratic forms Q rational quadratic forms Q′ such that the closure of Q(Z2) contains Q′(Z2)
Factoring Polynomials and the Knapsack Problem

AbstractFor several decades the standard algorithm for factoring polynomials f with rational coefficients has been the Berlekamp–Zassenhaus algorithm. The complexity of this algorithm depends exponentially on n, where n is the number of modular factors of f. This exponential time complexity is due to a combinatorial problem: the problem of choosing the right subsets of these n factors. In this paper, this combinatorial problem is reduced to a type of Knapsack problem that can be solved with lattice reduction algorithms. The result is a practical algorithm that can factor polynomials that are far out of reach for previous algorithms. The presented solution to the combinatorial problem is different from previous lattice-based factorizers; these algorithms avoided the combinatorial problem by solving the entire factorization problem with lattice reduction. This led to lattices of large dimension and coefficients, and thus poor performance. This is why lattice-based algorithms, despite their polynomial time complexity, did not replace Berlekamp–Zassenhaus as the standard method. That is now changing; new versions of computer algebra systems such as Maple, Magma, NTL and Pari have already switched to the algorithm presented here
On The Ono Invariants of Imaginary Quadratic Fields

AbstractLet Ed(x) denote the “Euler polynomial” x2+x+(1−d)/4 if d≡1 (mod 4) and x2−d if d≡2,3 (mod 4). Set Ω(n)=the number of prime factors (counting multiplicity) of the positive integer n. The Ono invariantOnod of K is defined to be max{Ω(Ed(b)): b=0,1,…∣Δd∣/4−1∣ except when d=−1,−3 in which case Onod is defined to be 1. Finally, let hd=hK denote the class number of K=Q(d). It is known that hd=1⇔Onod=1 for all negative integers d (Frobenius–Rabinowitch). We improve a result of Chowla–Cowles–Cowles and characterize imaginary quadratic fields of class number one in terms of least prime quadratic residues. This yields an elementary characterization of imaginary quadratic fields with Ono invariant equal to one. It is also known that hd=2⇔Onod=2 for all negative integers d (Sasaki). Sasaki also proved that hd⩾Onod for all negative integers d. If hd=3, then necessarily Onod=3 and −d is a prime p≡3 (mod 4). Computer calculations support the conjecture that the converse holds. We prove in this paper that this conjecture fails for at most finitely many d. We will show in fact that there are only finitely many d with −d a prime p≡3 (mod 4) and Onod=3. Moreover, using a result of Bach (which assumes the extended Riemann hypothesis), we find that the conjecture holds for all −d=prime p≡3 (mod 4) greater than 1017. Computer calculations so far show that the conjecture holds up to 1.5×107. Another conditional result, derived from Lagarias–Odlyzko (Effective versions of the Chebotarev density theorem, in: “Algebraic Number Fields” (A. Frohlich, Ed.), Academic Press, London, 1977), which assumes GRH, is limd→−∞Onod=∞
On a Conjecture of Nicolas–Sárközy about Partitions

AbstractLet N be the set of positive integers, B={b1<…<bk}⊂N, N∈N, and N⩾ bk. For I=0 or 1, A=AI(B,N) is the set (introduced by Nicolas, Ruzsa, and Sárközy, J. Number Theory73 (1998), 292–317) such that A∩{1,…,N}=B and p(A,nm)≡I(mod2) for n∈N,n>N, where p(A,n) denotes the number of partitions of n with parts in A. Let us denote by σ(A,n) the sum of the divisors of n belonging to A. In this paper, we prove that σ(A, 2n) mod 4 is periodic with period q2 multiple of q period of σ(A,n) mod 2; we also give the sets B⊂{1,…,5} and the values of N, N⩽10, for which q2≠q. Moreover, we show that if A(x) is the counting function of A then for A=A0({1,2,3},3),lim̄}x→∞A(x)/x⩽1/4
On the Number of Sparse RSA Exponents

NOTICE: this is the author's version of a work that was accepted for publication in the Journal of Number Theory. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Number Theory, Volume 95, Issue 2,(2002), Pages 340-350. doi:10.1006/jnth.2001.2775. http://www.elsevier.com/locate/jntAn RSA modulus is a product M = pl of two primes p and l. We show that for almost all RSA moduli M, the number of sparse exponents e (which allow for fast RSA encryption) with the property that gcd(e,φ(M)) = 1 (hence RSA decryption can also be performed) is very close to the expected value
Geometry of Numbers Proof of Götzky's Four-Squares Theorem

AbstractThe totally positive algebraic integers of certain number fields have been shown to be the sums of four squares of integers from their respective fields. The case of Q(5) was demonstrated by Götzky and the cases of Q( 2) and Q( 3) were demonstrated by Cohn. In the latter situation, only those integers with even coefficient on the radical term could possibly be represented by sums of squares. These results utilized modular functions in order to get the exact number of representations. Here a method of Grace is adapted to show the existence of a four-squares representation for Q( 5) without, however, obtaining the number of these. Also, results about the representation of primes by sums of two squares are obtained for Q( 5)
Fast Computation of the Biquadratic Residue Symbol

AbstractThis article describes an asymptotically fast algorithm for the computation of the biquadratic residue symbol. The algorithm achieves a running time of O(n(log n)2log log n) for Gaussian integers bounded by 2n in the norm. Our algorithm is related to an asymptotically fast GCD computation in Z[i] which uses the technique of a controlled Euclidean descent in Z[i]. At first, we calculate a Euclidean descent with suitable Euclidean steps xj−1=qjxj+xj+1 storing the qj's for later use. Then we calculate the biquadratic residue symbol of x0, x1 from the quotient sequence in linear time in the length of the qj's
The Frobenius Problem, Rational Polytopes, and Fourier–Dedekind Sums

AbstractWe study the number of lattice points in integer dilates of the rational polytope P={(x1,…,xn)∈R⩾0n:∑k=1nxkak⩽1}, where a1,…,an are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a1,…,an, find the largest value of t (the Frobenius number) such that m1a1+···+mnan=t has no solution in positive integers m1,…,mn. This is equivalent to the problem of finding the largest dilate tP such that the facet {∑k=1nxkak=t} contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials L(P,t)≔#(tP∩Zn) and L(P°,t)≔#(tP°∩Zn). Within the computations a Dedekind-like finite Fourier sum appears. We obtain a reciprocity law for these sums, generalizing a theorem of Gessel. As a corollary of our formulas, we rederive the reciprocity law for Zagier's higher-dimensional Dedekind sums. Finally, we find bounds for the Fourier–Dedekind sums and use them to give new bounds for the Frobenius number
Mordell–Weil Ranks of Quadratic Twists of Pairs of Elliptic Curves

AbstractMotivated by a conjecture of Mazur, Kuwata and Wang proved that for elliptic curves E1 and E2 whose j-invariants are not simultaneously 0 or 1728, there exist infinitely many square-free integers d for which the rank of the Mordell–Weil group of the d-quadratic twists of E1 and E2 satisfy: rk(Ed1, Q)>0 and rk(Ed2, Q)>0. Here we present results for the related questions: Are there infinitely many square-free integers d for which: rk(Ed1, Q)=0 and rk(Ed2, Q)=0? And, are there infinitely many square-free integers d for which: rk(Ed1, Q)=0 and rk(Ed2, Q)>0
The Stationary-Phase Method for Exponential Sums with Multiplicative Characters

AbstractLet f(x) be a polynomial with p-adic coefficients, and defineS(f;χ;Z/pmZ)=∑x∈(Z/pmZ)×χ(x)e2πif(x)/pm, where χ is a character of (Z/pmZ)×. For example, f(x)=x leads to the Gauss sum of χ. An explicit formula for S(f;χ;Z:/pmZ) is given if f′(x) has distinct roots in Z/pmZ and m⩾2, or if f′(x) has distinct roots p-adically and m is sufficiently large. In either case, the formula involves one term for each root of f′(x). If the derivative does not have distinct roots, there is a less explicit formula, requiring one term for each approximate root. Analogous results are proved for sums involving several variables. As applications of these results, Mauclaire's evaluation of the Gauss sum is re-derived, and new results on n-variable Kloosterman sums are obtained
On a Normal Integral Basis Problem over Cyclotomic Zp-extensions, II

AbstractLet p be an odd prime number, K an imaginary abelian field with ζp∈K×, and K∞/K the cyclotomic Zp-extension with its nth layer Kn. In the previous paper, we showed that for any n and any unramified cyclic extension L/Kn of degree p, LKn+1/Kn+1 does have a normal integral basis (NIB) even if L/Kn has no NIB, under the assumption that p does not divide the class number of the maximal real subfield K+ (and some additional assumptions on K). In this paper, we show that similar but more delicate phenomena occur for a certain class of tamely ramified extensions of degree p