Torsion and Tamagawa numbers

Lorenzini, Dino

doi:10.5802/aif.2664

Cited by 24 publications

(24 citation statements)

References 53 publications

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“…In the case where A (hence A ∨ ) has everywhere semistable reduction, this conjecture was proved by Werner (cf. [Wer97, Proposition 5.1], if one takes into account the residue fields, see [Lo11]). The fourth equality follows from the fact that the pairing is non-degenerate and the fact that our base field k is finite.…”

Section: Appendix : Invariance Of Statementsmentioning

confidence: 99%

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An Analogue of the Brauer–Siegel Theorem for Abelian Varieties in Positive Characteristic

Hindry¹

2016

MMJ

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Consider a family of abelian varieties A i of fixed dimension defined over the function field of a curve over a finite field. We assume finiteness of the Shafarevic-Tate group of A i. We ask then when does the product of the order of the Shafarevic-Tate group by the regulator of A i behave asymptotically like the exponential height of the abelian variety. We give examples of families of abelian varieties for which this analogue of the Brauer-Siegel theorem can be proven unconditionally, but also hint at other situations where the behaviour is different. We also prove interesting inequalities between the degree of the conductor, the height and the number of components of the Néron model of an abelian variety. rez me: Rassmotrim seme stvo abelevyh mnogoobrazii A i nad polem funkci krivo nad koneqnym polem. Dopustim, qto gruppa Xafareviqa-Ta ta koneqna. My spraxivaem, kogda proizvedenie por dka gruppy Xafareviqa-Ta ta na regul tor dl A i asimptotiqeski bedet seb kak ksponencial na vysota abelevyh mnogoobrazi. My privodim primery mno estv abelevyh mnogoobrazii, dl kotoryh my mo em dokazat bezuslovno analog teoremy Brau ra-Zigel , no tak e rassmatrivaem i drugie situacii, kogda asimptotiqeskoe povedenie etih veliqin razliqno. Krome togo, my dokazyvaem interesnye neravenstva, sv zyva wie stepen konduktora, vysotu i qislo komponent modeli Nerona abelevogo mnogoobrazi. Contents 5. An abc theorem for semi-abelian schemes in characteristic p > 0 17 5.1. Preliminaries 17 5.2. Main statement and associated results 17 5.3. Compactification of a universal family 18 5.4. Towards the Kodaira-Spencer map 20 5.5. Case where Kod(φ U) = 0 23 6. Group of connected components 24 6.1. Rigid uniformization 24 6.2. Fourier-Jacobi expansion of theta constants 25 6.3. Jacobians and Noether's formula. 30 7. Special value at s = 1 31 7.1. The L-function of A/K 31 7.2. Lower bounds for the regulator 34 7.3. Lower bounds and small zeroes 35 7.4. An example 36 7.5. Twists 39 7.6. Twists of a constant curve 40 8. Appendix : Invariance of statements 42 References 43 Key words-Abelian varieties, heights, global fields, Brauer-Siegel theorem, Birch & Swinnerton-Dyer conjecture AMS classification-14K (abelian varieties and schemes), 11G (arithmetic algebraic geometry), 11M (Zeta and L-functions, 11R (Algebraic number theory, global fields)

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Section: Appendix : Invariance Of Statementsmentioning

confidence: 99%

“…Let B/K be an abelian variety and f : A → B an isogeny of degree a power of p. Then : Proof. For the proof of item (2) see [Lo11]) and for the others see [Mil72,§1].…”

Section: Appendix : Invariance Of Statementsmentioning

confidence: 99%

An Analogue of the Brauer–Siegel Theorem for Abelian Varieties in Positive Characteristic

Hindry¹

2016

MMJ

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“…Thus the left-hand vertical map is an isomorphism as well, which completes the proof. In the case where R ′ /R is a finite extension of discrete valuation rings, the isomorphism of Corollary 4.2 was discussed in [15,Proposition 3.19] when the associated extension of fraction fields K ′ /K is Galois, k is perfect and G ′ is the Néron model of an abelian variety over K ′ . Corollary 4.1 is proved in [2, Proposition 1.1] (see also [9, proof of Theorem 1]) when K ′ /K is separable, G ′ is as above and R is henselian.…”

Section: Some Applicationsmentioning

confidence: 99%

Galois sets of connected components and Weil restriction

Bertapelle

González-Avilés

2016

Math. Z.

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“…So assume that E d (Q) contains a Q-rational point of order 3. A theorem of Lorenzini (see[22] Proposition 1.1) implies that if E/Q is an elliptic curve with a Q-rational point of order 9, then 9 2 divides…”

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confidence: 99%

On a conjecture of Agashe

Melistas

2021

Preprint

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Let E/Q be an optimal elliptic curve, −D be a negative fundamental discriminant coprime to the conductor N of E/Q and let E −D /Q be the twist of E/Q by −D. A conjecture of Agashe predicts that if E −D /Q has analytic rank 0, then the square of the order of the torsion subgroup of E −D /Q divides the product of the order of the Shafarevich-Tate group of E −D /Q and the orders of the arithmetic component groups of E −D /Q, up to a power of 2. This conjecture can be viewed as evidence for the second part of the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero. We provide a proof of a slightly more general statement without using the optimality hypothesis. c p (E −D ), up to a power of 2. Theorem 1.2 below, which is the main theorem of this paper, implies Conjecture 1.1 (see Corollary 3.19 for the proof of this implication). We note that Conjecture 1.1 and, hence, our

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Torsion and Tamagawa numbers

Cited by 24 publications

References 53 publications

An Analogue of the Brauer–Siegel Theorem for Abelian Varieties in Positive Characteristic

An Analogue of the Brauer–Siegel Theorem for Abelian Varieties in Positive Characteristic

Galois sets of connected components and Weil restriction

On a conjecture of Agashe

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