Francesco Baldassarri

Geometric aspects of Dwork theory

This two-volume book collects the lectures given during the three months cycle of lectures held in Northern Italy between May and July of 2001 to commemorate Professor Bernard Dwork (1923 - 1998). It presents a wide-ranging overview of some of the most active areas of contemporary research in arithmetic algebraic geometry, with special emphasis on the geometric applications of thep-adic analytic techniques originating in Dworks work, their connection to various recent cohomology theories and to modular forms. The two volumes contain both important new research and illuminating survey articles written by leading experts in the field. The book willprovide an indispensable resource for all those wishing to approach the frontiers of research in arithmetic algebraic geometry.

On algebraic solutions of Lamé's differential equation

In two previous papers [ 1, 21 we reconsidered and improved Klein’s method for establishing whether a given second order linear differential equation with rational function coefficients (over an algebraic curve) has a full set of algebraic solutions. That procedure being rather complicated, we were interested in checking whether, when applied to classical examples, it would lead to nontrivial new results. The hypergeometric equation being in that respect well understood, Lame’s equation (see (0.1)) was the simplest type of differential equation for which we could hope to get interesting information by our method. Before explaining in detail the content of this paper, it may be useful to the reader if we recall a few facts, proved in [ 1, 21, of which constant use will be made throughout this paper. Let C be a nonsingular algebraic curve over the complex field C with function field K, and let L be a second order linear differential operator on C; that is,

Algebraic versis Rigid Cohomology with Logarithmic Coefficients

Publisher Summary This chapter discusses algebraic versus rigid cohomology with logarithmic coefficients. If the field is assumed to be algebraically closed, so that Γ is a dense subgroup of the multiplicative group and a discretely valued complete subfield K of K, with valuation ring, uniformizing parameter, and residue field is fixed, the field K will play the role of field of definition for all the algebraic objects. Under the assumptions of the theorem, also if it is assumed that the eigenvalues are p-adically non-Liouville, the formally merornorphic solutions at 0 are merornorphic. The morphisms induce isomorphisms of hypercohomology groups. The chapter presents the classification of systems with logarithmic singularities along the coordinate divisor on an open polydisk, relative to a smooth affinoid base-space. The two notions of local overconvergence and of non-Liouvilleness of the exponents of monodromy, play a crucial role. The spectral sequence of relative cohomology is used to reduce to comparison of de Rham cohomology.

On Dwork cohomology and algebraic D-modules

After works by Katz, Monsky, and Adolphson-Sperber, a comparison theorem between relative de Rham cohomology and Dwork cohomology is established in a paper by Dimca-Maaref-Sabbah-Saito in the framework of algebraic D-modules. We propose here an alternative proof of this result. The use of Fourier transform techniques makes our approach more functorial.

Soluzioni algebriche dell’equazione di Lamé e torsione delle curve ellittiche

SuntoSi discute il legame esistente tra equazioni di LaméLn conn intero e soluzioni tutte algebriche, e punti di torsione di curve ellittiche. In particolare si esibisce una equazioneL1 con gruppo di monodromia proiettivo su diedrale di ordine 6, legata ai punti di 3-divisione della curvay2=4x3−1.SummaryWe discuss the relation between Lamé equationsLn with integraln and only algebraic solutions, and torsion points of elliptic curves. In particular we exhibit an equationL1 with projective monodromy group over dihedral of order 6, related to the 3-division points of the curvey2=4x3−1.

Inequalities related to Lefschetz pencils and integrals of Chern classes

In n n ne e e eq q q qu u u ua a a al l l li i i it t t ti i i ie e e es s s s r r r re e e el l l la a a at t t te e e ed d d d t t t to o o o L L L Le e e ef f f fs s s sc c c ch h h he e e et t t tz z z z p p p pe e e en n n nc c c ci i i il l l ls s s s a a a an n n nd d d d i i i in n n nt t t te e e eg g g gr r r ra a a al l l ls s s s o o o of f f f C C C Ch h h he e e er r r rn n n n c c c cl l l la a a as s s ss s s se e e es s s s Nicholas M. Katz and Rahul Pandharipande I I I In n n nt t t tr r r ro o o od d d du u u uc c c ct t t ti i i io o o on n n n We work over an algebraically closed field k, in which a prime number … is invertible. We fix a projective, smooth, connected k-scheme X/k, of dimension n ≥ 1. We also fix a projective embedding i : X fi @. This allows us to speak of smooth hyperplane sections X€L of X, or more generally of smooth hypersurface sections X€H d of X of any degree d ≥ 1 (i.e., H d is a degree d hypersurface in the ambient @, and the scheme-theoretic intersection X€H d is smooth over k, and of codimension one in X). The paper [Ka-LAM] applied results of Larsen to the problem of determining the monodromy of the universal family of smooth hypersurface sections X€H d of X of fixed degree d. Consider the lisse ä

Special values of symmetric hypergeometric functions

…-sheaf Ï …,d on the parameter space, given by H d ÿ H n-1 (X€H d , ä

Direct images (the Gauss-Manin connection)

…)/H n-1 (X, ä

Regularity in several variables

…). Denote by N d …

Irregularity in several variables

We discuss the p-adic formula (0.3) of P. Th. Young, in the framework of Dwork’s theory of the hypergeometric equation. We show that it gives the value at 0 of the Frobenius automorphism of the unit root subcrystal of the hypergeometric crystal. The unit disk at 0 is in fact singular for the differential equation under consideration, so that it’s not a priori clear that the Frobenius structure should extend to that disk. But the singularity is logarithmic, and it extends to a divisor with normal crossings relative to Zp in P1Zp . We show that whenever the unit root subcrystal of the hypergeometric system has generically rank 1, it actually extends as a logarithmic F -subcrystal to the unit disk at 0. So, in these optics, “singular classes are not supersingular”. If, in particular, the holomorphic solution at 0 is bounded, the extended logarithmic F -crystal has no singualrity in the residue class of 0, so that it is an F -crystal in the usual sense and the Frobenius operation is holomorphic. We examine in detail its analytic form. 0. Introduction In a recent article P. Th. Young, on the line of previous work by N. Koblitz [Ko] and J. Diamond [D], used some combinatorial identities and a principle of p-adic continuity in all variables to compute special values of a certain function F(a, b, c;λ) related to the classical Gauss hypergeometric function F (a, b, c;λ). As in [p-DE IV], the function F(a, b, c;λ), for a, b, c ∈ Zp, c / ∈ Z≤0, is defined to be the maximal p-adic analytic extension of the ratio (0.1) F (a, b, c;λ) F (a′, b′, c′;λp) , where for a ∈ Zp, a′ ∈ Zp is uniquely defined by the condition that (0.2) pa′ − a = μa ∈ {0, 1, . . . , p− 1}. (We also recursively define a = a, and a = (a(i))′, for i = 0, 1, . . . .) In particular, Young obtained the formula (see [Y1, Th. 3.1] and (5.3.1) below) (0.3) F( 2α p− 1 , β p− 1 , 1 + 2α− β p− 1 ;−1) = (−1) α Γp( α p−1 )Γp( β−α p−1 ) Γp( 2α p−1 )Γp( β−2α p−1 ) Received by the editors November 15, 1994. 1991 Mathematics Subject Classification. Primary 11T23, 11S31, 12H25, 14F30. c ©1996 American Mathematical Society 2249 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2250 FRANCESCO BALDASSARRI (where Γp denotes the Morita p-adic gamma function), for any α, β ∈ Z such that (0.4) 0 max(μa(i) , μb(i)) , for any i = 0, 1, . . . . This condition guarantees that the holomorphic solution of the hypergeometric operator at 0 (that is, F (a, b, c;λ)) is a bounded holomorphic function on D(0, 1−) = {λ ∈ Cp | |λ| 1. This might be the best approach (cf. [LDE, Chap. 9], [Dw2]) when a, b, c are in 1 q−1Z∩ (0, 1). In that case one would consider (0.6) F (a, b, c;λ) = F (a, b, c;λ) F (a(f), b(f), c(f);λq) . The condition for F (a, b, c;λ) to extend to an admissible domain of analyticity would then consist of two separate assumptions. The first is (0.5) for i = f − 1, which already guarantees 1-dimensionality of the space of bounded solutions of the system at a generic point. A second condition is needed to guarantee nonsupersingularity of the class of 0, that is, the existence of a bounded solution of the system in D(0, 1−). The condition says that the order of zero at λ = 0 of the lowerright entry of the Frobenius matrix shouldn’t exceed c(q−1) [Dw2, Def. 1.11]. This is certainly the case if we insist that condition (0.5) holds for all i = 0, . . . , f − 1. This is the viewpoint of this article and of [p-DE IV]. A separate question is the one of non-supersingularity of the non-singular point λ0 at which we want to evaluate F(a, b, c;λ). In the case considered here, λ0 = −1 and c ± (a − b) ∈ Z, this is automatic (cf. (2.31) and (2.36)). In the case considered by [Ko] and [D], λ0 = 1 is singular, so condition (0.5) appears combined with a similar condition [D, (iii) of Thm. 1.1] at 1, to guarantee that the above mentioned unit root F -crystal also extends to a p-adic formal neighbourhood of 1. Again, non-supersingularity of the class of 1 follows from that condition. While problem 5 also becomes more interesting in the case of a singular fiber, where Frobenius operates on the “eigenvectors of local monodromy” (cf. [GHF, Lemmas 24.3 and 24.5.8]) in this paper, motivated by formula (0.3), we only discuss the non-singular fiber at λ0 = −1. We carefully analyze the relation between the value F( 2α p− 1 , β p− 1 , 1 + 2α− β p− 1 ;−1) = F( β p− 1 , 2α p− 1 , 1 + 2α− β p− 1 ;−1) and the unit root of the L-function (0.7) L(t) = exp ∞ ∑