Semistable Abelian Varieties Research Articles

Hyperelliptic $$\mathcal {S}_7$$-curves of prime conductor

An abelian threefold $$A_{/{\mathbb {Q}}}$$ of prime conductor N is favorable if its 2-division field F is an $${\mathcal {S}}_7$$ -extension over $${\mathbb {Q}}$$ with ramification index 7 over $${\mathbb {Q}}_2$$ . Let A be favorable and let B be a semistable abelian variety of conductor $$N^d$$ with B[2] filtered by d copies of A[2]. We obtain a class field theoretic criterion on F to guarantee that B is isogenous to $$A^d$$ and a fortiori, A is unique up to isogeny.

Equivariant Tamagawa number conjecture for Abelian varieties over global fields of positive characteristic

We state and prove certain cases of the equivariant Tamagawa number conjecture of a semistable Abelian variety over an everywhere unramified finite Galois extension of a global field of characteristic p > 0 p>0 under a semisimplicity hypothesis.

On the μ-invariants of abelian varieties over function fields of positive characteristic

Let $A$ be an abelian variety over a global function field $K$ of characteristic $p$. We study the $\mu$-invariant appearing in the Iwasawa theory of $A$ over the unramified $\mathbb{Z}_p$-extension of $K$. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate-Shafarevich group of $A$ and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate-Shafarevich group (which is now the $\mu$-invariant) in terms of other quantities including the Faltings height of $A$ and Frobenius slopes of the numerator of the Hasse-Weil $L$-function of $A / K$ assuming the conjectural Birch-Swinnerton-Dyer formula. Our next result is to prove this $\mu$-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the "$\mu=0$" locus of the moduli of isomorphism classes of minimal elliptic surfaces endowed with a section and with fixed large enough Euler characteristic is a dense open subset.

Grothendieck’s pairing on Néron component groups: Galois descent from the semistable case

In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This, in particular, implies Grothendieck’s conjecture on the perfectness of his pairing between the Néron component groups of an abelian variety and its dual. The point is that our formulation is well suited to Galois descent. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. We also treat coefficients in tori and, more generally, 1-motives.

Extending finite-subgroup schemes of semistable abelian varieties via log-abelian varieties

We show—for a semistable abelian variety A K over a complete discrete valuation field K —that every finite-subgroup scheme of A K extends to a log finite-flat group scheme over the valuation ring of K endowed with the canonical log structure. To achieve this, we first give a positive answer to a question of Nakayama, namely whether every weak log-abelian variety over an fs (fine and saturated) log scheme with its underlying scheme locally noetherian is a sheaf for the Kummer-flat topology. We also give several equivalent conditions defining isogenies of log-abelian varieties.

On the Non Commutative Iwasawa Main Conjecture for Abelian Varieties over Function Fields

We establish the Iwasawa main conjecture for semistable abelian varieties over a function field of characteristic p under certain restrictive assumptions. Namely we consider p -torsion free p -adic Lie extensions of the base field which contain the constant \mathbb{Z}_p -extension and are everywhere unramified. Under the usual \mu=0 hypothesis, we give a proof which mainly relies on the interpretation of the Selmer complex in terms of p -adic cohomology [F. Trihan, D. Vauclair, A comparison theorem for semi abelian schemes over a smooth curve, preprint arXiv:1505.02942, 2015] together with the trace formulas of J.-Y. Etesse and B. Le Stum [Math. Ann. 296, No. 3, 557–576 (1993; Zbl 0789.14015)].

Certain abelian varieties bad at only one prime

An abelian surface $A_{/{\mathbb Q}}$ of prime conductor $N$ is favorable if its 2-division field $F$ is an ${\mathcal S}_5$-extension with ramification index 5 over ${\mathbb Q}_2$. Let $A$ be favorable and let $B$ be any semistable abelian variety of dimension $2d$ and conductor $N^d$ such that $B[2]$ is filtered by copies of $A[2]$. We give a sufficient class field theoretic criterion on $F$ to guarantee that $B$ is isogenous to $A^d$. As expected from our paramodular conjecture, we conclude that there is one isogeny class of abelian surfaces for each conductor in $\{277, 349,461,797,971\}$. The general applicability of our criterion is discussed in the data section.

Variation of Tamagawa numbers of semistable abelian varieties in field extensions

AbstractWe investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on thep-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the formy2=f(x), under some simplifying hypotheses.

Campana points, Vojta’s conjecture, and level structures on semistable abelian varieties

We introduce a qualitative conjecture, in the spirit of Campana, to the effect that certain subsets of rational points on a variety over a number field, or a Deligne-Mumford stack over a ring of S-integers, cannot be Zariski dense. The conjecture interpolates, in a way that we make precise, between Lang's conjecture for rational points on varieties of general type over number fields, and the conjecture of Lang and Vojta that asserts that S-integral points on a variety of logarithmic general type are not Zariski-dense. We show our conjecture follows from Vojta's conjecture. Assuming our conjecture, we prove the following theorem: Fix a number field K, a finite set S of places of K containing the infinite places, and a positive integer g. Then there is an integer m_0 such that, for any m > m_0, no principally polarized abelian variety A/K of dimension g with semistable reduction outside of S has full level-m structure.

Paramodular abelian varieties of odd conductor

A precise and testable modularity conjecture for rational abelian surfaces $A$ with trivial endomorphisms, $\mathrm {End}_{\mathbb {Q}} A = \mathbb {Z}$, is presented. It is consistent with our examples, our non-existence results and recent work of C. Poor and D. S. Yuen on weight 2 Siegel paramodular forms. We obtain fairly precise information on $\ell$-division fields of semistable abelian varieties, mainly when $A[\ell ]$ is reducible, by considering extension problems for group schemes of small rank.

Arithmetic of division fields

We study the arithmetic of division fields of semistable abelian varieties A over the rationals. The Galois group of the 2-division field of A is analyzed when the conductor is odd and squarefree. The irreducible semistable mod 2 representations of small conductor are determined under GRH. These results are used in Paramodular abelian varieties of odd conductor, arXiv:1004.4699.

Cohomologie log plate, actions modérées et structures galoisiennes

Let X be a fine and saturated log scheme, and let G be a commutative finite flat group scheme over the underlying scheme of X . If G -torsors for the fppf topology can be thought of as being unramified objects by nature, then G -torsors for the log flat topology allow us to consider tame ramification. Using the results of Kato, we define a concept of Galois structure for these torsors, then we generalize the author's previous constructions (class-invariant homomorphism for semi-stable abelian varieties) in this new setting, thus dropping some restrictions.

Semistable abelian varieties with good reduction outside 15

We show that there are no non-zero semi-stable abelian varieties over $${{\bf Q}(\sqrt{5})}$$ with good reduction outside 3 and we show that the only semi-stable abelian varieties over Q with good reduction outside 15 are, up to isogeny over Q, powers of the Jacobian of the modular curve X 0(15).

Calabi–Yau varieties with semi-stable fibre structures

Motivated by the Strominger–Yau–Zaslow conjecture, we study Calabi–Yau varieties with semi-stable fibre structures. We use Hodge theory to study the higher direct images of wedge products of relative cotangent sheaves of certain semi-stable families over higher dimensional quasi-projective bases, and obtain some results on positivity. We then apply these results to study non-isotrivial Calabi–Yau varieties fibred by semi-stable Abelian varieties (or hyperkahler varieties).

Semistable abelian varieties over Q

We prove that for N=6 and N=10, there do not exist any non-zero semistable abelian varieties over ℚ with good reduction outside primes dividing N. Our results are contingent on the GRH discriminant bounds of Odlyzko. Combined with recent results of Brumer-Kramer and of Schoof, this result is best possible: if N is squarefree, there exists a non-zero semistable abelian variety over ℚ with good reduction outside primes dividing N precisely when N∉{1,2,3,5,6,7,10,13}.

Non-existence of certain semistable abelian varieties

We show that there is no semistable abelian variety defined over ℚ with bad reduction at exactly one prime p≤ 7.

Formal sections and de Rham cohomology of semistable Abelian varieties

We give a geometric description of the unit root splitting of the Hodge filtration of the first de Rham cohomology of an ordinary Abelian variety over a local field, as the splitting determined by a formal completion of the universal vectorial extension of the Abelian variety.

Endomorphisms of Semi-Stable Abelian Varieties Over Number Fields

(End Jo(N)) (0 Q of all endomorphisms of Jo(N) is equal to the algebra generated by the Hecke operators, regarded as Q-endomorphisms of J0(N). Our initial proof that this is indeed the case combined P. Deligne's theorem that Jo(N) is semi-stable over Q (in the sense of [6]) with a rather elaborate argument involving l-adic representations. This argument gave a proof that all endomorphisms of a semi-stable abelian variety over Q are rational whenever the variety has a certain endomorphism structure (similar to that of [4, Section 5]). With Shimura's help we were able to simplify the argument and at the same time make it more general. Eventually the principle emerged that all endomorphisms of a semi-stable variety are unramifted. (See (1.1) and (1.3) for precise statements.) Since Q has no unramified extensions, it follows that every endomorphism of a semi-stable abelian variety over Q is rational (i.e., defined over Q). We apply these results to Jo(N) in Section 3 and to certain other modular varieties in Section 4. The applications require an auxiliary result (based on an idea of W. Casselman) concerning the endomorphism algebra of an abelian variety with many real endomorphisms; the result is proved in Section 2.