The twisting Sato–Tate group of the curve $$y^2 = x^{8} - 14x^4 + 1$$ y 2 = x 8 - 14 x 4 + 1

2018 ◽  
Vol 290 (3-4) ◽  
pp. 991-1022 ◽  
Author(s):  
Sonny Arora ◽  
Victoria Cantoral-Farfán ◽  
Aaron Landesman ◽  
Davide Lombardo ◽  
Jackson S. Morrow
Keyword(s):  
Tate Group

scholarly journals Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent

2016 ◽  
Vol 68 (2) ◽  
pp. 361-394
Author(s):  
Francesc Fité ◽  
Josep González ◽  
Joan-Carles Lario
Keyword(s):  
Abelian Variety ◽  
Complex Multiplication ◽  
Cyclotomic Field ◽  
Explicit Method ◽  
Fermat Curve ◽  
Tate Conjecture ◽  
Tate Group ◽  
Fermat Curves ◽  
Prime Exponent ◽  
Simple Abelian Variety

AbstractLet denote the Fermat curve over ℚ of prime exponent ℓ. The Jacobian Jac() of splits over ℚ as the product of Jacobians Jac(k), 1 ≤ k ≤ ℓ −2, where k are curves obtained as quotients of by certain subgroups of automorphisms of . It is well known that Jac(k) is the power of an absolutely simple abelian variety Bk with complex multiplication. We call degenerate those pairs (ℓ, k) for which Bk has degenerate CM type. For a non-degenerate pair (ℓ, k), we compute the Sato–Tate group of Jac(Ck), prove the generalized Sato–Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of whether (ℓ, k) is degenerate, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the ℓ-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

A conjecture of Gross and Zagier: Case E(ℚ)tor≅ℤ/3ℤ

2019 ◽  
Vol 15 (09) ◽  
pp. 1793-1800 ◽  
Author(s):  
Dongho Byeon ◽  
Taekyung Kim ◽  
Donggeon Yhee
Keyword(s):  
Elliptic Curve ◽  
Infinite Order ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Roots Of Unity ◽  
Prime Divisors ◽  
Heegner Point ◽  
Tate Group ◽  
Tamagawa Numbers

Let [Formula: see text] be an elliptic curve defined over [Formula: see text] of conductor [Formula: see text], [Formula: see text] the Manin constant of [Formula: see text], and [Formula: see text] the product of Tamagawa numbers of [Formula: see text] at prime divisors of [Formula: see text]. Let [Formula: see text] be an imaginary quadratic field where all prime divisors of [Formula: see text] split in [Formula: see text], [Formula: see text] the Heegner point in [Formula: see text], and [Formula: see text] the Shafarevich–Tate group of [Formula: see text] over [Formula: see text]. Let [Formula: see text] be the number of roots of unity contained in [Formula: see text]. Gross and Zagier conjectured that if [Formula: see text] has infinite order in [Formula: see text], then the integer [Formula: see text] is divisible by [Formula: see text]. In this paper, we show that this conjecture is true if [Formula: see text].

scholarly journals On Deformations of 1-motives

2019 ◽  
Vol 62 (1) ◽  
pp. 11-22
Author(s):  
A. Bertapelle ◽  
N. Mazzari
Keyword(s):  
Abelian Variety ◽  
Deformation Theory ◽  
Positive Characteristic ◽  
Infinitesimal Deformation ◽  
Tate Group

AbstractAccording to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti–Tate group. We extend this result to 1-motives.

scholarly journals Higher-level canonical subgroups for p-divisible groups

2011 ◽  
Vol 11 (2) ◽  
pp. 363-419 ◽  
Author(s):  
Joseph Rabinoff
Keyword(s):  
Elliptic Curves ◽  
Geometric Structure ◽  
Valuation Ring ◽  
Mixed Characteristic ◽  
Tate Group ◽  
Divisible Groups

AbstractLet R be a complete rank-1 valuation ring of mixed characteristic (0, p), and let K be its field of fractions. A g-dimensional truncated Barsotti–Tate group G of level n over R is said to have a level-n canonical subgroup if there is a K-subgroup of G ⊗RK with geometric structure (Z/pnZ)g consisting of points ‘closest to zero’. We give a non-trivial condition on the Hasse invariant of G that guarantees the existence of the canonical subgroup, analogous to a result of Katz and Lubin for elliptic curves. The bound is independent of the height and dimension of G.

scholarly journals THE DISTRIBUTION OF TORSION SUBSCHEMES OF ABELIAN VARIETIES

2014 ◽  
Vol 15 (4) ◽  
pp. 693-710
Author(s):  
Jeffrey D. Achter
Keyword(s):  
Finite Fields ◽  
Abelian Varieties ◽  
Natural Numbers ◽  
Group Schemes ◽  
Tate Group ◽  
Power Group

We consider the distribution of $p$-power group schemes among the torsion of abelian varieties over finite fields of characteristic $p$, as follows. Fix natural numbers $g$ and $n$, and let ${\it\xi}$ be a non-supersingular principally quasipolarized Barsotti–Tate group of level $n$. We classify the $\mathbb{F}_{q}$-rational forms ${\it\xi}^{{\it\alpha}}$ of ${\it\xi}$. Among all principally polarized abelian varieties $X/\mathbb{F}_{q}$ of dimension $g$ with $X[p^{n}]_{\bar{\mathbb{F}}_{q}}\cong {\it\xi}_{\bar{\mathbb{F}}_{q}}$, we compute the frequency with which $X[p^{n}]\cong {\it\xi}^{{\it\alpha}}$. The error in our estimate is bounded by $D/\sqrt{q}$, where $D$ depends on $g$, $n$, and $p$, but not on ${\it\xi}$.

scholarly journals The Griffiths bundle is generated by groups

2019 ◽  
Vol 375 (3-4) ◽  
pp. 1283-1305
Author(s):  
Wushi Goldring
Keyword(s):  
Line Bundle ◽  
Reductive Group ◽  
General Setting ◽  
Arbitrary Field ◽  
Period Map ◽  
General Fiber ◽  
Tate Group

Abstract First the Griffiths line bundle of a $$\mathbf {Q}$$ Q -VHS $${\mathscr {V}}$$ V is generalized to a Griffiths character $${{\,\mathrm{grif}\,}}(\mathbf {G}, \mu ,r)$$ grif ( G , μ , r ) associated to any triple $$(\mathbf {G}, \mu , r)$$ ( G , μ , r ) , where $$\mathbf {G}$$ G is a connected reductive group over an arbitrary field F, $$\mu \in X_*(\mathbf {G})$$ μ ∈ X ∗ ( G ) is a cocharacter (over $$\overline{F}$$ F ¯ ) and $$r:\mathbf {G}\rightarrow GL(V)$$ r : G → G L ( V ) is an F-representation; the classical bundle studied by Griffiths is recovered by taking $$F=\mathbf {Q}$$ F = Q , $$\mathbf {G}$$ G the Mumford–Tate group of $${\mathscr {V}}$$ V , $$r:\mathbf {G}\rightarrow GL(V)$$ r : G → G L ( V ) the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to $${{\,\mathrm{grif}\,}}(\mathbf {G}, \mu , r)$$ grif ( G , μ , r ) . The more general setting also gives rise to the Griffiths bundle in the analogous situation in characteristic p given by a scheme mapping to a stack of $$\mathbf {G}$$ G -Zips. When $$\mathbf {G}$$ G is F-simple, we show that, up to positive multiples, the Griffiths character $${{\,\mathrm{grif}\,}}(\mathbf {G},\mu ,r)$$ grif ( G , μ , r ) (and thus also the Griffiths line bundle) is essentially independent of r with central kernel, and up to some identifications is given explicitly by $$-\mu $$ - μ . As an application, we show that the Griffiths line bundle of a projective $${{\mathbf {G}{\text{- }}{} \mathtt{Zip}}}^{\mu }$$ G - Zip μ -scheme is nef.

scholarly journals Explicit Heegner points: Kolyvagin's conjecture and non-trivial elements in the Shafarevich–Tate group

2009 ◽  
Vol 129 (2) ◽  
pp. 284-302 ◽  
Author(s):  
Dimitar Jetchev ◽  
Kristin Lauter ◽  
William Stein
Keyword(s):  
Heegner Points ◽  
Tate Group

scholarly journals SQUARENESS IN THE SPECIAL L-VALUE AND SPECIAL L-VALUES OF TWISTS

2010 ◽  
Vol 06 (05) ◽  
pp. 1091-1111
Author(s):  
AMOD AGASHE
Keyword(s):  
Quaternion Algebra ◽  
Free Abelian Group ◽  
Torsion Subgroup ◽  
Isomorphism Classes ◽  
Maximal Orders ◽  
Tate Group ◽  
Supersingular Elliptic Curves ◽  
Fundamental Discriminant ◽  
Quadratic Imaginary Fields ◽  
L Value

Let N be a prime and let A be a quotient of J0(N) over Q associated to a newform such that the special L-value of A (at s = 1) is non-zero. Suppose that the algebraic part of the special L-value of A is divisible by an odd prime q such that q does not divide the numerator of [Formula: see text]. Then the Birch and Swinnerton-Dyer conjecture predicts that the q-adic valuations of the algebraic part of the special L-value of A and of the order of the Shafarevich–Tate group are both positive even numbers. Under a certain mod q non-vanishing hypothesis on special L-values of twists of A, we show that the q-adic valuations of the algebraic part of the special L-value of A and of the Birch and Swinnerton-Dyer conjectural order of the Shafarevich–Tate group of A are both positive even numbers. We also give a formula for the algebraic part of the special L-value of A over quadratic imaginary fields K in terms of the free abelian group on isomorphism classes of supersingular elliptic curves in characteristic N (equivalently, over conjugacy classes of maximal orders in the definite quaternion algebra over Q ramified at N and ∞) which shows that this algebraic part is a perfect square up to powers of the prime two and of primes dividing the discriminant of K. Finally, for an optimal elliptic curve of arbitrary conductor E, we give a formula for the special L-value of the twist E-Dof E by a negative fundamental discriminant -D, which shows that this special L-value is an integer up to a power of 2, under some hypotheses. In view of the second part of the Birch and Swinnerton-Dyer conjecture, this leads us to the surprising conjecture that the square of the order of the torsion subgroup of E-Ddivides the product of the order of the Shafarevich–Tate group of E-Dand the orders of the arithmetic component groups of E-D, under certain mild hypotheses.

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