Frobenius and the Hodge Spectral Sequence

Ogus, Arthur

doi:10.1006/aima.2001.1975

Cited by 6 publications

(10 citation statements)

References 7 publications

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“…Modulo p, the filtration F 8 can be identified with the filtration A 8 . Then the previous result shows that the hypotheses of Theorem (4.7) of [9] are satisfied, so that (E, F 8 ) and (E, F 8 ) are cohomologically concentrated in degree 1. …”

Section: If M$mentioning

confidence: 79%

“…We are now in position to apply the results of [9] to analyze the Frobenius Hodge numbers of H 1 (XÂW, E). Let N 8 denote the conjugate filtration of E, let F 8 denote the de cale of the filtered complex Ru XÂW* (E, N 8 ), and let F 8 be the conjugate of F 8 , as defined in [9].…”

Section: If M$mentioning

confidence: 99%

“…Let N 8 denote the conjugate filtration of E, let F 8 denote the de cale of the filtered complex Ru XÂW* (E, N 8 ), and let F 8 be the conjugate of F 8 , as defined in [9]. Modulo p, the filtration F 8 can be identified with the filtration A 8 .…”

Section: If M$mentioning

confidence: 99%

“…The second difficulty is that the Frobenius Hodge spectral sequence fails to degenerate, and the statements of [11] are in that case very weak. Fortunately, more refined results (which were in fact obtained to deal with the current situation) have recently become available [9] and enable us to extricate some nontrivial information. In fact, we are able to prove a remarkable p-adic continuity property of some slopes in some (but infinitely many) cases.…”

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confidence: 98%

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Elliptic Crystals and Modular Motives

Ogus

2001

Advances in Mathematics

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The goal of this paper is to investigate the crystalline properties of families of elliptic curves or, more precisely, to study the families of crystals (elliptic crystals) attached to families of elliptic curves.Elliptic crystals over a point can be defined and classified very simply in terms of semilinear algebra, and this is how we begin (1.1). Our definition is cast in the logarithmic setting, so that it applies also to semistable (degenerate) elliptic curves. Roughly speaking, an elliptic crystal over a (logarithmic) perfect field k of characteristic p is a free W-module E of rank two equipped with a Frobenius-linear endomorphism 8, a nilpotent linear endomorphism N, a two-step Hodge filtration A on k E, and an isomorphism tr: 4 2 E Ä W. These data are required to satisfy various compatibilities: for example, A 1 (k E) is the kernel of id k 8. When N is not zero, such a crystal is quite rigid, and in particular its canonical coordinates are very precisely determined (1.3). Liftings of elliptic crystals from k to its Witt ring are determined by specifying a lifting B of the Hodge filtration A of k E, and the classification is made explicit in (1.5).Our first main result (2.2) concerns the relationship between deformation theory and elliptic crystals on a curve. Let XÂk be the reduction modulo p of a smooth log curve YÂW over W, let (E, B) be an elliptic crystal on YÂW, and let (E, A) be the restriction of (E, B) to XÂk. Following Mochizuki [8] and Gunning [6], we shall say that E is indigenous if its Kodaira-Spencer mapping is an isomorphism, a condition which can be checked either on YÂW or on XÂk. If f: X$ Ä X is a morphism of smooth log curves over k then a lifting g: Theorem (2.2) shows that if p is odd and E is indigenous, the resulting map from the set of liftings of f to the set of liftings of the mod p Hodge filtration of f *E X is a bijection. As a consequence we show (reprising some of the results of Mochizuki) in Theorem (2.4) that (when the degree of A 1 E X is positive and p is odd) an indigenous bundle on XÂk determines a unique lifting YÂW, characterized by the fact

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Section: If M$mentioning

confidence: 79%

Section: If M$mentioning

confidence: 99%

Section: If M$mentioning

confidence: 99%

mentioning

confidence: 98%

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Elliptic Crystals and Modular Motives

Ogus

2001

Advances in Mathematics

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“…The reason why it is reasonable to do this is that for all primes p of good reduction for X, the eigen-values of Frobenius in the original untwisted representation are algebraic integers divisible by p (since h 0,3 = 0: this is a consequence, for example, of a conjecture of Katz that is now a theorem; see specifically [4]; for explicit results regarding divisibility, see [6]; for related issues see also [15], [16]) and therefore the downward twisted representation has the property that these eigenvalues divided by p remain algebraic integers, and therefore-in the usual sense-are "Weil numbers" of absolute value p/2.…”

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confidence: 99%

Open problems: Descending cohomology, geometrically

Mazur¹

2014

Notices of the International Congress of Chinese Mathematicians

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Symmetric powers do not stabilize

Litt

2014

Proc. Amer. Math. Soc.

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We discuss the stabilization of symmetric products Sym n (X) of a smooth projective variety X in the Grothendieck ring of varieties. For smooth projective surfaces X with non-zero h 0 (X, ω X ), these products do not stabilize; we conditionally show that they do not stabilize in another related sense, in response to a question of R. Vakil and M. Wood [20]. There are analogies between such stabilization, the Dold-Thom theorem, and the analytic class number formula. Finally, we discuss Hodge-theoretic obstructions to the stabilization of symmetric products, and provide evidence for these obstructions in terms of a relationship between the Newton polygon of a certain "motivic zeta function" associated to a curve, and its Hodge polygon.

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Frobenius and the Hodge Spectral Sequence

Cited by 6 publications

References 7 publications

Elliptic Crystals and Modular Motives

Elliptic Crystals and Modular Motives

Open problems: Descending cohomology, geometrically

Symmetric powers do not stabilize

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