Inspired by the analogy between the group of units $\mathbb{F}_p^{\times}$ of the finite field with $p$ elements and the group of points $E(\mathbb{F}_p)$ of an elliptic curve $E/\mathbb{F}_p$, E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curve sums analogous to the Titchmarsh divisor sum $\sum_{p \leq x} τ(p + a) \sim C x$. In this paper, we present a comprehensive study of the constants $C(E)$ emerging in the asymptotic study of these elliptic curve divisor sums. Specifically, by analyzing the division fields of an elliptic curve $E/\mathbb{Q}$, we prove upper bounds for the constants $C(E)$ and, in the generic case of a Serre curve, we prove explicit closed formulae for $C(E)$ amenable to concrete computations. Moreover, we compute the moments of the constants $C(E)$ over two-parameter families of elliptic curves $E/\mathbb{Q}$. Our methods and results complement recent studies of average constants occurring in other conjectures about reductions of elliptic curves by addressing not only the average behaviour, but also the individual behaviour of these constants, and by providing explicit tools towards the computational verifications of the expected asymptotics.
We consider a binary quadratic variant of the Titchmarsh divisor problem and give an asymptotic formula for $\sum_{p^2+q^2\leq N} τ(p^2+q^2+1)$, where $p,q$ are primes.
In this paper we solve a function field analogue of classical problems in analytic number theory, concerning the auto-correlations of divisor functions, in the limit of a large finite field.
Asymptotic formulae for Titchmarsh-type divisor sums are obtained with strong error terms that are uniform in the shift parameter. This applies to more general arithmetic functions such as sums of two squares, improving the error term in the representation of the number as a sum of a prime and two squares, and to Fourier coefficients of cusp forms, generalizing a result of Pitt.
Let $\mathbb{F}_q[t]$ be the polynomial ring over the finite field $\mathbb{F}_{q}$. For arithmetic functions $ψ_{1}, ψ_{2}: \mathbb{F}_{q}[t]\rightarrow\mathbb{C}$, we establish that if a Bombieri-Vinogradov type equidistribution result holds for $ψ_{1}$ and $ψ_{2}$, then it also holds for their Dirichlet convolution $ψ_{1} \ast ψ_{2}$. As an application of this, we resolve a version of the Titchmarsh divisor problem in $\mathbb{F}_{q}[t]$. More precisely, we obtain an asymptotic for the average behaviour of the divisor function over shifted products of two primes in $\mathbb{F}_q[t]$.
Given a multiplicative function $f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum $\sum_{|h|<n\leq x} f(n) τ(n-h)$, where $τ$ denotes the divisor function and $h\in\mathbb{Z}\setminus\{0\}$. We consider in particular the special cases where $f$ is the generalized divisor function $τ_z$ with $z\in\mathbb{C}$, and the characteristic function of sums of two squares (or more generally, ideal norms of abelian extensions). As another application, we deduce a full asymptotic expansion in the generalized Titchmarsh divisor problem $\sum_{|h|<n…
A natural number $n$ is $y$-smooth if the greatest prime factor of $n$ does not exceed $y$. Let $s_{1}$ and $s_{2}$ are $y$-smooth numbers. We consider sums of smooth squares of the binary Titchmarsh divisor problem and give asymptotic formulae for $\sum_{s_{1}^{2}+s_{2}^{2}\le x}τ(s_{1}^{2}+s_{2}^{2}+1)$ for $(\log x)^{K}\le y<x^{\frac{1}{2}}$, where $K$ is large enough.
Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited.
We consider the primes which divide the denominator of the x-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive prime divisor, one that has not appeared in any earlier term. Proofs of this are known in only a few cases. Weaker results in the general direction are given, using a strong form of Siegel's Theorem and some congruence arguments. Our main result is applied to the study of prime divisors of Somos sequences.
Let $P$ be a non-torsion point on the elliptic curve $E_{a}: y^{2}=x^{3}+ax$. We show that if $a$ is fourth-power-free and either $n>2$ is even or $n>1$ is odd with $x(P)<0$ or $x(P)$ a perfect square, then the $n$-th element of the elliptic divisibility sequence generated by $P$ always has a primitive divisor.
For a prime number $p$ we study the zeros modulo $p$ of divisor polynomials of rational elliptic curves $E$ of conductor $p$. Ono made the observation that these zeros of are often $j$-invariants of supersingular elliptic curves over $\overline{\mathbb{F}_p}$. We show that these supersingular zeros are in bijection with zeros modulo $p$ of an associated quaternionic modular form $v_E$. This allows us to prove that if the root number of $E$ is $-1$ then all supersingular $j$-invariants of elliptic curves defined over $\mathbb{F}_{p}$ are zeros of the corresponding divisor polynomial. If the roo…
The modularity theorem implies that for every elliptic curve $E /\mathbb{Q}$ there exist rational maps from the modular curve $X_0(N)$ to $E$, where $N$ is the conductor of $E$. These maps may be expressed in terms of pairs of modular functions $X(z)$ and $Y(z)$ where $X(z)$ and $Y(z)$ satisfy the Weierstrass equation for $E$ as well as a certain differential equation. Using these two relations, a recursive algorithm can be used to calculate the $q$ - expansions of these parametrizations at any cusp. %These functions are algebraic over $\mathbb{Q}(j(z))$ and satisfy modular polynomials where e…
We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u^3+v^3=m, with m cube-free, all the terms beyond the first have a primitive divisor.
We introduce and study "elliptic zeta values", a two-parameter deformation of the values of Riemann's zeta function at positive integers. They are essentially Taylor coefficients of the logarithm of the elliptic gamma function, and share the SL(3,Z) modular properties of this function. Elliptic zeta values at even integers are related to Eisenstein series and thus to sums of odd powers of divisors. The elliptic zeta values at odd integers can be expressed in terms of generating series of sums of even powers of divisors.
Suppose $\mathcal{E} \to B$ is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve $B$. Let $k$ denote the function field $\overline{\mathbb{Q}}(B)$ and $E$ the associated elliptic curve over $k$. In this article, we construct adelically metrized $\mathbb{R}$-divisors $\overline{D}_X$ on the base curve $B$ over a number field, for each $X \in E(k)\otimes \mathbb{R}$. We prove non-degeneracy of the Arakelov-Zhang intersection numbers $\overline{D}_X\cdot \overline{D}_Y$, as a biquadratic form on $E(k)\otimes \mathbb{R}$. As a consequence, we have the follo…
Let $f$ be an elliptic modular form and $p$ an odd prime that is coprime to the level of $f$. We study the link between divisors of the characteristic ideal of the $p$-primary fine Selmer group of $f$ over the cyclotomic $\mathbb{Z}_p$ extension of $\mathbb{Q}$ and the greatest common divisor of signed Selmer groups attached to $f$ defined using the theory of Wach modules. One of the key ingredients of our proof is a generalization of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction at $p$ to the context of modular forms.
We examine the existence of rational divisors on modular curves of $\mathcal{D}$-elliptic sheaves and on Atkin-Lehner quotients of these curves over local fields. Using a criterion of Poonen and Stoll, we show that in infinitely many cases the Tate-Shafarevich groups of the Jacobians of these Atkin-Lehner quotients have non-square orders.
We study the existence of a Zsigmondy bound for a sequence of divisors associated to points on an elliptic curve over a function field. More precisely, let $k$ be an algebraically closed field, let $\mathcal{C}$ be a nonsingular projective curve over $k$, and let $K$ denote the function field of $\mathcal{C}$. Suppose $E$ is an ordinary elliptic curve over $K$ and suppose there does not exist an elliptic curve $E_0$ defined over $k$ that is isomorphic to $E$ over $K$. Suppose $P\in E(K)$ is a non-torsion point and $Q\in E(K)$ is a torsion point of order $r$. The sequence of points $\{nP+Q\}\su…
The congruent number elliptic curves are defined by $E_d: y^2=x^3-d^2x$, where $d\in \mathbb{N}.$ We give a simple proof of a formula for $L(\mathrm{Sym}^2(E_d),3)$ in terms of the determinant of the elliptic trilogarithm evaluated at some degree zero divisors supported on the torsion points on $E_d(\overline{\mathbb{Q}})$.
It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of Mordell curves and families of congruent number curves are given with corresponding elliptic divisibility sequences having no perfect power terms. The proofs combine primitive divisor results with modular methods for Diophantine equations.
We detail the continued fraction expansion of the square root of the general monic quartic polynomial, noting that each line of the expansion corresponds to addition of the divisor at infinity. We analyse the data yielded by the general expansion. In that way we obtain `elliptic sequences' satisfying Somos relations. I mention several new results on such sequences. The paper includes a detailed `reminder exposition' on continued fractions of quadratic irrationals in function fields.
We propose a conjectural explicit isogeny from the Jacobians of hyperelliptic Drinfeld modular curves to the Jacobians of hyperelliptic modular curves of $\mathcal{D}$-elliptic sheaves. The kernel of the isogeny is a subgroup of the cuspidal divisor group constructed by examining the canonical maps from the cuspidal divisor group into the component groups.
Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime where $E$ has good supersingular reduction. Let $F_1$ denote the characteristic power series of the Pontryagin dual of the fine Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ and let $F_2$ denote the greatest common divisor of Pollack's plus and minus $p$-adic $L$-functions or Sprung's sharp and flat $p$-adic $L$-functions attached to $E$, depending on whether $a_p(E)=0$ or $a_p(E)\ne0$. We study a link between the divisors of $F_1$ and $F_2$ in the Iwasawa algebra. This gives new insights into probl…
We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the $j$-invariant of the elliptic curve is constant. In more detail, given an elliptic curve $E$ with a point $P$ of infinite order, the sequence $D_1$, $D_2, \ldots$ of denominators of multiples $P$, $2P,\ldots$ of $P$ is a strong divisibility sequence in the sense that $\gcd(D_m, D_n) = D_{\gcd(m,n)}$. This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences. A number $N$ is called a Zsigmondy bound of the sequence if each term $D_{n}$ with $n>N$ presents a new p…
We consider the lattice point problem corresponding to a family of elliptic paraboloids in $\mathbb{R}^d$ with $d\ge3$ and we prove the expected to be optimal exponent, improving previous results. This is especially noticeable for $d=3$ because the optimal exponent is conjectural even for the sphere. We also treat some aspects of the case $d=2$, getting for a simple parabolic region an $Ω$-result that is unknown for the classical circle and divisor problems.
Let $p$ be a prime number such that the modular curve $X_0(p)$ has genus at least two. We show that the only points of the reduction mod $p$ of $X_0(p)$ with image in the reduction mod $p$ of $J_0(p)$ in the cuspidal group are the two cusps. This answers a question of Robert Coleman. For the proof we give a description of the special fibre of the Néron model of the jacobian of a semi-stable curve in terms of divisors. We also study to what extent the morphism from a semistable curve with given base point to the Néron model of its jacobian is a closed immmersion. Implicitly, logarithmic structu…
Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R in E(k(T)), write x_R=A_R/D_R^2 with relatively prime polynomials A_R(T) and D_R(T) in k[T]. The sequence {D_{nR}) for n \ge 1 is called the ``elliptic divisibility sequence of R.'' Let P,Q in E(k(T)) be independent points. We conjecture that deg (gcd(D_{nP},D_{mQ})) is bounded for m,n \ge 1, and that gcd(D_{nP},D_{nQ}) = gcd(D_{P},D_{Q}) for infinitely many n \ge 1. We prove these conjectures in the case that j(E) is in k. More generally, we prove analo…
We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in "very short intervals" of the form $I(f) := \{ f(x) + a : a \in \mathbb{F}_p \}$ for $f(x) \in \mathbb{F}_p[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the Möbius $μ$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(\sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for "generic" set of …
We compute the variances of sums in arithmetic progressions of generalised k-divisor functions related to certain L-functions in $\mathbb{F}_q(t)$, in the limit as $q\to\infty$. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when $q\to\infty$, in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual $k$-divisor function, when the L-function in question has degree one. They illustrate the role …
Let A,A' be elliptic curves or abelian varieties fully of type GSp defined over a number field K. This includes principally polarized abelian varieties with geometric endomorphism ring Z and dimension 2 or odd. We compare the number of points on the reductions of the two varieties. We prove that A and A' are K-isogenous if the following condition holds for a density-one set of primes p of K: the prime numbers dividing #A(k_p) also divide #A'(k_p). We generalize this statement to some extent for products of such varieties. This refines results of Hall and Perucca (2011) and of Ratazzi (2012).
We work out an example, for a CM elliptic curve E defined over a real quadratic field F, of Zagier's conjecture. This relates L(E,2) to values of the elliptic dilogarithm function at a divisor in the Jacobian of E which arises from K-theory.
For a discrete subgroup of an indefinite unitary group $U(1,n+1)$, $n\geq 1$, consider the attached modular variety. Using local Borcherds products, we study Heegner divisors in the local Picard group over a boundary component the compactified variety. We obtain a criterion for local Heegner divisors to be torsion elements in the local Picard group. As an application, we find that the obstructions for a local Heegner divisor to be a torsion element can be described through spaces of vector valued elliptic cusp forms spanned by certain theta-series.
Elliptic curves over finite fields with predefined conditions in the order are practically constructed using the theory of complex multiplication. The stage with longest calculations in this method reconstructs some polynomial with integer coefficients. We will prove theoretical results and give a detailed account of the method itself and how one can use a divisor of the mentioned polynomial with coefficients in some extension of the field of rational numbers.
In this paper, we investigate extreme values of $ω(E(\mathbb{F}_p))$, where $E/\mathbb{Q}$ is an elliptic curve with complex multiplication and $ω$ is the number-of-distinct-prime-divisors function. For fixed $γ> 1$, we prove that \[ \#\{p \leq x : ω(E(\mathbb{F}_p)) > γ\log\log x\} = \frac{x}{(\log x)^{2 + γ\logγ- γ+ o(1)}}. \] The same result holds for the quantity $\#\{p \leq x : ω(E(\mathbb{F}_p)) < γ\log\log x\}$ when $0 < γ< 1$. The argument is worked out in detail for the curve $E : y^2 = x^3 - x$, and we discuss how the method can be adapted for other CM elliptic curves.
By adapting the technique of David, Koukoulopoulos and Smith for computing sums of Euler products, and using their interpretation of results of Schoof à la Gekeler, we determine the average number of subgroups (or cyclic subgroups) of an elliptic curve over a fixed finite field of prime size. This is in line with previous works computing the average number of (cyclic) subgroups of finite abelian groups of rank at most 2. A required input is a good estimate for the divisor function in both short interval and arithmetic progressions, that we obtain by combining ideas of Ivić--Zhai and Blomer.
We give a geometric interpretation of the group law for Jacobian varieties by extending the geometric construction of chords and tangents on an elliptic curve. For any given algebraic curve $\mathcal X$ and reduced divisors $D_1, D_2 \in \mbox{Jac } \mathcal X$, we define curves $\mathcal X^\prime$ and $\mathcal X^{"}$ such that the intersection $\mathcal X \cap \mathcal X^\prime$ determines precisely the divisor $-(D_1+D_2)$ and the intersection $\mathcal X\cap \mathcal X^{"}$ determines $D_1+D_2$. For superelliptic curves such formulas are made explicit.
We study the existence of primes and of primitive divisors in classical divisibility sequences defined over function fields. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.
We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and Ellenberg, when the base field has characteristic zero and the supports of the conductor of the elliptic curve and of the ramification divisor of the extension are disjoint.
Let $\mathcal C_Q$ be the cyclic group of order $Q$, $n$ a divisor of $Q$ and $r$ a divisor of $Q/n$. We introduce the set of $(r,n)$-free elements of $\mathcal C_Q$ and derive a lower bound for the the number of elements $θ\in \mathbb F_q$ for which $f(θ)$ is $(r,n)$-free and $F(θ)$ is $(R,N)$-free, where $ f, F \in \mathbb F_q[x]$. As an application, we consider the existence of $\mathbb F_q$-primitive points on curves like $y^n=f(x)$ and find, in particular, all the odd prime powers $q$ for which the elliptic curves $y^2=x^3 \pm x$ contain an $\mathbb F_q$-primitive point.
We examine an elliptic curve constructed in an earlier paper from a certain representation of $\SL(2,\Z)$ on the space of convergent Dirichlet series. The curve is observed to be a modular curve for $Γ^1(15)$ and a certain orbit of modular functions is thereby associated with the Riemann zeta function. Explicit descriptions are given of these functions and of the permutation action of $\SL(2,\Z)$ on them.
In this paper we apply results from the theory of congruences of modular forms (control of reducible primes, level-lowering), the modularity of elliptic curves and Q-curves, and a couple of Frey curves of Fermat-Goldbach type, to show the existence of newforms of weight 2 and trivial nebentypus with coefficient fields of arbitrarily large degree and square-free or almost square-free level. More precisely, we prove that for any given numbers t and B, there exists a newform f of weight 2 and trivial nebentypus whose level N is square-free (almost square-free), N has exactly t prime divisors (t o…
These are the notes from a course of five lectures at the 2009 Park City Math Institute. The focus is on elliptic curves over function fields over finite fields. In the first three lectures, we explain the main classical results (mainly due to Tate) on the Birch and Swinnerton-Dyer conjecture in this context and its connection to the Tate conjecture about divisors on surfaces. This is preceded by a "Lecture 0" on background material. In the remaining two lectures, we discuss more recent developments on elliptic curves of large rank and constructions of explicit points in high rank situations.
A divisibility sequence is a sequence of integers $\{d_n\}$ such that $d_m$ divides $d_n$ if $m$ divides $n$. Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two divisibility sequences corresponding to subgroups of the multiplicative group grows in a controlled way. Silverman conjectured that a similar behaviour should appear in many algebraic groups. We extend results by Ghioca-Hsia-Tucker and Silverman for elliptic curves and prove an analogue of Silverman's conjecture over function fields for abelian and split semiabelian varieties and some generalizations of …
We show, in the large $q$ limit, that the average size of $n$-Selmer groups of elliptic curves of bounded height over $\mathbb F_q(t)$ is the sum of the divisors of $n$. As a corollary, again in the large $q$ limit, we deduce that $100\%$ of elliptic curves of bounded height over $\mathbb F_q(t)$ have rank $0$ or $1$.
Several discrete geometry problems are closely related to the arithmetic theory of elliptic curves defined on the rational fields $\mathbb{Q}$. In this paper we consider the $θ$-congruent number for $θ=\fracπ{3}$ and $\frac{2π}{3}$ and tiling number n. For the case that $n\geqslant 2$ is square-free odd integer, we determine all $n$ such that the Selmer rank of elliptic curve $E_{n,\fracπ{3}}:\ y^2=x(x-n)(x+3n)$ or/and $E_{n,\frac{2π}{3}}:\ y^2=x(x+n)(x-3n)$ is zero. From this, we provide several series of non $θ$-congruent numbers for $θ=\fracπ{3}$ and $\frac{2π}{3}$, and non tiling numbers n…
