It is shown that the number of irreducible quartic factors of the form $g(x) = x^4+ax^3+(11a+2)x^2-ax+1$ which divide the Hasse invariant of the Tate normal form $E_5$ in characteristic $l$ is a simple linear function of the class number $h(-5l)$ of the field $\mathbb{Q}(\sqrt{-5l})$, when $l \equiv 2,3$ modulo $5$. A similar result holds for irreducible quadratic factors of $g(x)$, when $l \equiv 1, 4$ modulo $5$. This implies a formula for the number of linear factors over $\mathbb{F}_p$ of the supersingular polynomial $ss_p^{(5*)}(x)$ corresponding to the Fricke group $Γ_0^*(5)$.
A formula is proved for the number of linear factors over $\mathbb{F}_l$ of the Hasse invariant of the Tate normal form $E_5(b)$ for a point of order $5$, as a polynomial in the parameter $b$, in terms of the class number of the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-l})$, proving a conjecture of the author from 2005. A similar theorem is proved for quadratic factors with constant term $-1$, and a theorem is stated for the number of quartic factors of a specific form in terms of the class number of $\mathbb{Q}(\sqrt{-5l})$. These results are shown to imply a recent conjecture of Nakaya…
In this article we develop counterexamples to the Hasse principle using only techniques from undergraduate number theory and algebra. By keeping the technical prerequisites to a minimum, we hope to provide a path for nonspecialists to this interesting area of number theory. The counterexamples considered here extend the classical counterexample of Lind and Reichardt. As discussed in an appendix, this type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in the Tate--Shafarevich group.
We show that certain Tate--Shafarevich groups are unramified which enables us to give an obstruction to the Hasse principle for torsors under tori over p-adic function fields.
We give a self-contained proof of local class field theory, via Lubin-Tate theory and the Hasse-Arf theorem, refining the arguments of Iwasawa's book. In the revised version, (i) positive characteristic case is included, (ii) the proof of base change is simplified, (iii) the construction of Artin map is simplified, and strengthened so that we start with an arbitrary Lubin-Tate group over complete maximal unramified extension. (to appear in Annales de la Faculte des Sciences de Toulouse, 17-2 (2008))
We study the distribution of Hasse's unit index $Q(L)$ for the CM-fields $L = \mathbb{Q}(\sqrt{d}, \sqrt{-1})$ as $d$ varies among positive squarefree integers. We prove that the number of $d\leq X$ such that $Q(L) = 2$ is proportional to $X/\sqrt{\log X}$.
Let A be an abelian variety over a number field k and F a finite cyclic extension of k of p-power degree for an odd prime p. Under certain technical hypotheses, we obtain a reinterpretation of the equivariant Tamagawa number conjecture (eTNC) for A, F/k and p as an explicit family of p-adic congru- ences involving values of derivatives of the Hasse-Weil L-functions of twists of A, normalised by completely explicit twisted regulators. This reinterpretation makes the eTNC amenable to numerical verification and furthermore leads to explicit predictions which refine well-known conjectures of Mazur…
We give an asymptotic expansion for the density of del Pezzo surfaces of degree four in a certain Birch Swinnerton-Dyer family violating the Hasse principle due to a Brauer-Manin obstruction. Under the assumption of Schinzel's hypothesis and the finiteness of Tate-Shafarevich groups for elliptic curves, we obtain an asymptotic formula for the number of all del Pezzo surfaces in the family, which violate the Hasse principle.
Admettant l'hypothèse de Schinzel et la finitude des groupes de Tate-Shafarevich des courbes elliptiques sur les corps de nombres, toute intersection lisse de deux quadriques dans l'espace projectif de dimension n satisfait au principe de Hasse si n>4. Le même résultat vaut pour n=4, c'est-à-dire pour les surfaces de del Pezzo de degré 4, lorsque le groupe de Brauer est réduit aux constantes et que la surface est suffisamment générale. ----- Assuming Schinzel's hypothesis and the finiteness of Tate-Shafarevich groups of elliptic curves over number fields, smooth intersections of two quadrics i…
Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $\underline{III}(E)$ be a certain group of equivalence classes of homogeneous spaces of $E$ called its Tate-Shafarevich group. We show in this paper that this group has finite cardinality and discuss its role in the Birch and Swinnerton-Dyer Conjecture. In particular, our result implies the Parity Conjecture, or the Birch and Swinnerton-Dyer Conjecture modulo 2. It also removes the finiteness condition of $\underline{III}(E)$ from previous results in the literature of this subject and makes possible for some computation problems concerning th…
Let $A$ be an abelian variety defined over a number field $k$, let $p$ be an odd prime number and let $F/k$ be a cyclic extension of $p$-power degree. Under not-too-stringent hypotheses we give an interpretation of the $p$-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points of $A$ of the ${\rm Gal}(F/k)$-valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in…
We consider the Kolyvagin cohomology classes associated to an elliptic curve $E$ defined over $\mathbb{Q}$ from a computational point of view. We explain how to go from a model of a class as an element of $(E(L)/pE(L))^{\mathrm{Gal}(L/\mathbb{Q})}$, where $p$ is prime and $L$ is a dihedral extension of $\mathbb{Q}$ of degree $2p$, to a geometric model as a genus one curve embedded in $\mathbb{P}^{p-1}$. We adapt the existing methods to compute Heegner points to our situation, and explicitly compute them as elements of $E(L)$. Finally, we compute explicit equations for several genus one curves …
We consider the Tate cohomology of the circle group acting on the topological Hochschild homology of schemes. We show that in the case of a scheme smooth and proper over a finite field, this cohomology theory naturally gives rise to the cohomological interpretation of the Hasse-Weil zeta function by regularized determinants envisioned by Deninger. In this case, the periodicity of the zeta function is reflected by the periodicity of said cohomology theory, whereas neither is periodic in general.
Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication (CM), and for each prime $p$ of good reduction, let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius. By the Hasse bound, $a_E(p) = 2\sqrt{p} \cos θ_p$ for a unique $θ_p \in [0, π]$. In this paper, we prove that the least prime $p$ such that $θ_p \in [α, β] \subset [0, π]$ satisfies \[ p \ll \left(\frac{N_E}{β- α}\right)^A, \] where $N_E$ is the conductor of $E$ and the implied constant and exponent $A > 2$ are absolute and effectively computable. Our result is an analogue for CM elliptic curves of Linnik's T…
Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair ( h^1(A/F)(1), Z[Gal(F/k)] ). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mor…
We introduce a new class of exponentials of Artin-Hasse type, called $\boldsymbolπ$-exponentials. These exponentials depends on the choice of a generator $\boldsymbolπ$ of the Tate module of a Lubin-Tate group $\mathfrak{G}$ over $\mathbb{Z}_p$. They arise naturally as solutions of solvable differential modules over the Robba ring. If $\mathfrak{G}$ is isomorphic to $\hat{\mathbb{G}}_m$ over $\mathbb{Z}_p$, we develop methods to test their over-convergence, and get in this way a stronger version of the Frobenius structure theorem for differential equations. We define a natural transformation o…
We consider refined conjectures of Birch and Swinnerton-Dyer type for the Hasse-Weil-Artin L-series of abelian varieties over general number fields. We shall, in particular, formulate several new such conjectures and establish their precise relation to previous conjectures, including to the relevant special case of the equivariant Tamagawa number conjecture. We also derive a wide range of concrete interpretations and explicit consequences of these conjectures that, in general, involve a thoroughgoing mixture of difficult archimedean considerations related to refinements of the conjecture of De…
Fix a non-CM elliptic curve $E/\mathbb{Q}$, and let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius at $p$. The Sato-Tate conjecture gives the limiting distribution $μ_{ST}$ of $a_E(p)/(2\sqrt{p})$ within $[-1, 1]$. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval $I\subseteq [-1, 1]$, let $p_{I,n}$ denote the $n$th prime such that $a_E(p)/(2\sqrt{p})\in I$. We show $\liminf_{n\to\infty}(p_{I,n+m}-p_{I,n}) < \infty$ for all $m\ge 1$ for "most" intervals, and in particular, for all $I$ with $μ_{ST}(I)\ge 0.36$. Furthe…
We establish an error term in the Sato-Tate theorem of Birch. That is, for $p$ prime, $q=p^r$ we show that $\#\{ (a,b) \in \mathbb{F}_q^2 : θ_{a,b}\in I\} =μ_{ST}(I)q^2 + O_r(q^{7/4})$ for any interval $I\subseteq[0,π]$ where for an elliptic curve $E: y^2= x^3 +ax +b$, the quantity $θ_{a,b}$ is defined by $2\sqrt{q}\cosθ_{a,b} = q+1-E(\mathbb{F}_q)$ and $μ_{ST}(I)$ denotes the Sato-Tate measure of the interval $I$.
Fix an elliptic curve E over Q. An extremal prime for E is a prime p of good reduction such that the number of rational points on E modulo p is maximal or minimal in relation to the Hasse bound. Assuming that all the symmetric power L-functions associated to E are automorphic and satisfy the Generalized Riemann Hypothesis, we give the first non-trivial upper bounds for the number of such primes when E is a curve without complex multiplication. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in the work of Rouse and Thorner (arXiv:1305.5283) and refi…
We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks $G$-Zip$^μ$. Restricting to zip data of Hodge type, we obtain a group-theoretical Hasse invariant on every Ekedahl-Oort stratum closure of a general Hodge-type Shimura variety. A key tool is the construction of a stack of zip flags $G$-ZipFlag$^μ$, fibered in flag varieties over $G$-Zip$^μ$. It provides a simultaneous generalization of the "classical case" homogeneous complex manifolds studied by Griffiths-Schmid and the "flag space" for Siegel varieties studied by Ekedahl-van der Geer. Four …
We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2\bmod 9$ or $p\equiv 5\bmod 9$. We first show that the $3$-part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their $2$-Selmer group to the $2$-rank of the ideal class group of $\mathbb{Q}(\sqrt[3]{p})$ to obtain some examples of elliptic curves with rank one and non-trivial $2$-part of the Tate-Shafarevich group.
Let $E$ be an elliptic curve over $Q$, and $τ$ an Artin representation over $Q$ that factors through the non-abelian extension $Q(\sqrt[p^n]{m},μ_{p^n})/Q$, where $p$ is an odd prime and $n,m$ are positive integers. We show that $L(E,τ,1)$, the special value at $s=1$ of the $L$-function of the twist of $E$ by $τ$, divided by the classical transcendental period $Ω_{+}^{d^+}|Ω_{-}^{d^-}|ε(τ)$ is algebraic and Galois-equivariant, as predicted by Deligne's conjecture.
Explicit formulas are proved for the $5$-torsion points on the Tate normal form $E_5$ of an elliptic curve having $(X,Y)=(0,0)$ as a point of order $5$. These formulas express the coordinates of points in $E_5[5] - \langle(0,0)\rangle$ as products of linear fractional quantities in terms of $5$-th roots of unity and a parameter $u$, where the parameter $b$ which defines the curve $E_5$ is given as $b=(\varepsilon^5 u^5- \varepsilon^{-5})/(u^5+1)$ and $\varepsilon = (-1+\sqrt{5})/2$. If $r(τ)$ is the Rogers-Ramanujan continued fraction and $b=r^5(τ)$, then the coordinates of points of order $5$…
Let $E\in\{-1, \pm 2\}$. We improve on the upper and lower densities of primes $p$ such that the equation $x^2-2py^2=E$ is solvable for $x, y\in \mathbb{Z}$. We prove that the natural density of primes $p$ such that the narrow class group of the real quadratic number field $\mathbb{Q}(\sqrt{2p})$ has an element of order $16$ is equal to $\frac{1}{64}$. We give an application of our results to the distribution of Hasse's unit index for the CM-fields $\mathbb{Q}(\sqrt{2p}, \sqrt{-1})$. Our results are consequences of a twisted joint distribution result for the $16$-ranks of class groups of $\mat…
Given a prime power $q$ and $n \gg 1$, we prove that every integer in a large subinterval of the Hasse--Weil interval $[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$ is $#A(\mathbb{F}_q)$ for some geometrically simple ordinary principally polarized abelian variety $A$ of dimension $n$ over $\mathbb{F}_q$. As a consequence, we generalize a result of Howe and Kedlaya for $\mathbb{F}_2$ to show that for each prime power $q$, every sufficiently large positive integer is realizable, i.e., $#A(\mathbb{F}_q)$ for some abelian variety $A$ over $\mathbb{F}_q$. Our result also improves upon the best known const…
We prove a new formula for the central value of the $L$-function $L(E_{D, α}, 1)$ corresponding to the family of sextic twists over $\mathbb{Q}[\sqrt{-3}]$ of elliptic curves $E_{D, α}: y^2=x^3+16D^2α^3$ for $D$ an integer and $α\in \mathbb{Q}[\sqrt{-3}]$. The formula generalizes the result of cubic twists over $\mathbb{Q}$ of Rodriguez-Villegas and Zagier for a prime $D \equiv 1 (9)$ and of Rosu for general $D$. For $α$ prime and all integers $D$, we also show that the expected value from the Birch and Swinnerton-Dyer conjecture of the order of the Tate-Shafarevich group is an integer square …
Let C be a smooth projective curve defined over a number field and let C' be a twist of C. In this article we relate the l-adic representations attached to the l-adic Tate modules of the Jacobians of C and C' through an Artin representation. This representation induces global relations between the local factors of the respective Hasse-Weil L-functions. We make these relations explicit in a particularly illustrating situation. For every Qbar-isomorphism class of genus 2 curves defined over Q with automorphism group isomorphic to D_8 or D_{12}, except for a finite number, we choose a representat…
We present novel algorithms to factor polynomials over a finite field $\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial $f(x) \in \F_q[x]$ to be factored) with respect to a Drinfeld module $ϕ$ with complex multiplication. Factors of $f(x)$ supported on prime ideals with supersingular reduction at $ϕ$ have vanishing Hasse invariant and can be separated from the rest. A Drinfeld module analogue of Deligne's congruence plays a key role in computing the Hasse invariant lift. We …
Barrucand and Cohn's theory of principal factorizations in pure cubic fields \(\mathbb{Q}(\sqrt[3]{D})\) and their Galois closures \(\mathbb{Q}(ζ_3,\sqrt[3]{D})\) with \(3\) types is generalized to pure quintic fields \(L=\mathbb{Q}(\sqrt[5]{D})\) and pure metacyclic fields \(N=\mathbb{Q}(ζ_5,\sqrt[5]{D})\) with \(13\) possible types. The classification is based on the Galois cohomology of the unit group \(U_N\), viewed as a module over the automorphism group \(\mathrm{Gal}(N/K)\) of \(N\) over the cyclotomic field \(K=\mathbb{Q}(ζ_5)\), by making use of theorems by Hasse and Iwasawa on the He…
It is shown that the quartic Fermat equation $x^4 +y^4=1$ has nontrivial integral solutions in the Hilbert class field $Σ$ of any quadratic field $K=\mathbb{Q}(\sqrt{-d})$ whose discriminant satisfies $-d \equiv 1$ (mod 8). A corollary is that the quartic Fermat equation has no nontrivial solution in $K=\mathbb{Q}(\sqrt{-p})$, for $p$ $( > 7)$ a prime congruent to $7$ (mod 8), but does have a nontrivial solution in the odd degree extension $Σ$ of $K$. These solutions arise from explicit formulas for the points of order 4 on elliptic curves in Tate normal form. The solutions are studied in deta…
Let $p\equiv 1\bmod 4$ be a prime number. We use a number field variant of Vinogradov's method to prove density results about the following four arithmetic invariants: (i) $16$-rank of the class group $\mathrm{Cl}(-4p)$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-4p})$; (ii) $8$-rank of the ordinary class group $\mathrm{Cl}(8p)$ of the real quadratic field $\mathbb{Q}(\sqrt{8p})$; (iii) the solvability of the negative Pell equation $x^2 - 2py^2 = -1$ over the integers; (iv) $2$-part of the Tate-Šafarevič group of the congruent number elliptic curve $E_p: y^2 = x^3-p^2x$. Our res…
Let $K = \mathbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A = A(q)$ denote the Gross curve over the Hilbert class field $H$ of $K$. In this note we use Magma to calculate the values $L(E/H, 1)$ for all such $q$'s up to some reasonable ranges for all primes $q$ congruent to $7$ modulo $8$. All these values are non-zero, and using the Birch and Swinnerton-Dyer conjecture, we can calculate hypothetical orders of the Tate-Shafarevich group of $A$ for all such $q$ up to $4831$.
By making use of our generalization of Barrucand and Cohn's theory of principal factorizations in pure cubic fields $\mathbb{Q}(\sqrt[3]{D})$ and their Galois closures $\mathbb{Q}(ζ_3,\sqrt[3]{D})$ with 3 possible types to pure quintic fields $L=\mathbb{Q}(\sqrt[5]{D})$ and their pure metacyclic normal fields $N=\mathbb{Q}(ζ_5,\sqrt[5]{D})$ with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields $N$ having normalized radicands in the range $2\le D<10^3$. Our classification is based on the Galois cohomology of the unit grou…
We prove a formula for the coefficients of a weight $3/2$ Cohen-Eisenstein series of square-free level $N$. This formula generalizes a result of Gross and in particular, it proves a conjecture of Quattrini. Let $l$ be an odd prime number. For any elliptic curve $E$ defined over $\mathbb{Q}$ of rank zero and square-free conductor $N$, if $l \mid |E(\mathbb{Q})|$, under certain conditions on the Shafarevich-Tate group $III_D$, we show that $l$ divides $|III_D|$ if and only if $l$ divides the class number $h(-D)$ of $\mathbb{Q}(\sqrt{-D}).$
Let $n$ be a positive square-free integer, where every odd prime factor of $n$ has form $8a\pm 1$. We determine when $n$ is non-congruent with second minimal $2$-primary Shafarevich-Tate group, in terms of the $4$-ranks of class groups and a Jacobi symbol. In particular, when every odd prime factor of $n$ has form $8a+1$, this condition is equivalent to the vanishing of the $4$-rank of the tame kernel of $\mathbb Q(\sqrt{n})$ for odd $n$, or $\mathbb Q(\sqrt{-n})$ for even $n$. This generalizes previous results.
We establish a joint distribution result concerning the fractional part of $αp^θ$ for $θ\in (0,1), \ α>0$, where $p$ is a prime satisfying a Chebotarev condition in a fixed finite Galois extension over $\mathbb{Q}$. As an application, for a fixed non-CM elliptic curve $E/\mathbb{Q}$, an asymptotic formula is given for the number of primes at the extremes of the Sato-Tate measure modulo a large prime $\ell$. These are precisely the primes $p$ for which the Frobenius trace $a_p(E)$ satisfies the congruence $a_p(E)\equiv [2\sqrt{p}] \bmod \ell$. We assume a zero-free region hypothesis for Dedekin…
We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate-Shafarevich groups of the associated elliptic curves. We also present a related criterion for primes p such that 16 divides the class number of the imaginary quadratic field Q(sqrt(-p)). Both results are based on descent methods. While we cannot show for either criterion individu…
Tate first proposed a method to determine $K_2\mathcal{O}_F,$ the tame kernel of $F,$ and gave the concrete computations for some special quadratic fields with small discriminant. After that, many examples for quadratic fields with larger discriminants are given, and similar works also have been done for cubic fields and for some special quartic fields with discriminants not large. In the present paper, we investigate the case of more general imaginary cyclic quartic field $F=\mathbb{Q}\Big(\sqrt{-(D+B\sqrt{D})}\Big)$ with class number one and large discriminants. The key problem is how to dec…
