29 results found
Pal A, Krishnamoorthy R, 2020, Rank 2 local systems and abelian varieties II, Publisher: arXiv
LetX/Fq be a smooth, geometrically connected, quasi projective scheme. Let Ebe a semisimple over convergent F-isocrystal on X. Suppose that irreducible summands Ei of E have rank 2, determinant ̄Qp (−1), and infinite monodromy at∞. Suppose further that for each closed point x of X, the characteristic polynomial of E at x is in Q[t]⊂Qp[t]. Then there exists a non-trivial open set U⊂X such that E|U comes from a family of abelian varieties on U. As an application, let L1 be an irreducible lisse ̄Ql sheaf on X that has rank 2, determinant ̄Ql(−1), and infinite monodromy at∞. Then all crystalline companions to L1 exist (as predicted by Deligne’s crystalline companions conjecture) if and only if there exists a non-trivial open set U⊂X and an abelian scheme πU: AU→U such that L1|U is a summand of R1(πU)∗ ̄Ql.
Lazda C, Pal A, 2019, Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz (1,1) theorem, Compositio Mathematica, Vol: 155, Pages: 1025-1045, ISSN: 0010-437X
In this article we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, stating that a rational (logarithmic) line bundle on the special fibre of a semistable scheme over kJtK lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham–Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter-example to show that the variational Tate conjecture for divisors cannot hold with Qp-coefficients.
Pal A, Endre S, 2018, The fibration method over real function fields
Let R(C) be the function field of a smooth, irreducible projective curve over R. Let X be a smooth, projective, geometrically irreducible variety equipped with a dominant morphism f onto a smooth projective rational variety with a smooth generic fibre over R(C). Assume that the cohomological obstruction introduced by Colliot-Thélène is the only one to the local-global principle for rational points for the smooth fibres of f over R(C)-valued points. Then we show that the same holds for X, too, by adopting the fibration method similarly to Harpaz--Wittenberg. We also show that the strong vanishing conjecture for n-fold Massey products holds for fields of virtual cohomological dimension at most 1 using a theorem of Haran.
Krishnamoorthy R, Pal A, 2018, Rank 2 local systems and abelian varieties
LetX/Fqbe a smooth geometrically connected variety. Inspired by work of Corlette-Simpson overC, we formulate a conjecture that absolutely irreducible rank 2 local systems withinfinite monodromy onX“come from families of abelian varieties”. WhenXis a projective variety,we prove that ap-adic variant of this conjecture reduces to the case of projective curves. If oneassumes a strong form of Deligne’s (p-adic)companions conjecturefrom Weil II, this implies that thel-adic version of our conjecture for projective varieties also reduces to the case of projective curves.Along the way we prove Lefschetz theorems for homomorphismsof abelian schemes and Barsotti-Tategroups. We also answer affirmitavely a question of Grothendieck on extending abelian schemes viatheirp-divisible groups.
Pal A, 2018, Iterated line integrals over Laurent series fields of characteristic p, Publications mathématiques de Besançon, Vol: 2017, Pages: 109-126
Pal A, Schlank TM, 2017, The Brauer-Manin obstruction to the local-global principle for the embedding problem, Publisher: arXiv
We study an analogue of the Brauer-Manin obstruction to the local-global principle for embedding problems over global fields. We will prove the analogues of several fundamental structural results. In particular we show that the (algebraic) Brauer-Manin obstruction is the only one to weak approximation when the embedding problem has abelian kernel. As a part of our investigations we also give a new, elegant description of the Tate duality pairing and prove a new theorem on the cup product.
Lazda C, Pal A, 2017, A homotopy exact sequence for overconvergent isocrystals
n this article we prove exactness of the homotopy sequenceof overconvergentp-adic fundamentalgroups for a smooth and projective morphism in characteristicp. We do so by first proving a correspondingresult for rigid analytic varieties in characteristic 0, following dos Santos [dS15] in the algebraic case. Incharacteristicp, we then proceed by a series of reductions to the case of a liftable family of curves, where wecan apply the rigid analytic result.
Lazda C, Pál A, 2016, Rigid cohomology over Laurent series fields, Publisher: Springer, ISBN: 9783319309514
Rigid Cohomology over Laurent Series Fields will provide a useful tool for anyone interested in the arithmetic of varieties over local fields of positive characteristic.
Colliot-Thelene JL, Pal A, Skorobogatov AN, 2016, Pathologies of the Brauer-Manin obstruction, Mathematische Zeitschrift, Vol: 282, Pages: 799-817, ISSN: 1432-1823
Pal A, 2015, The $p$-adic monodromy group of abelian varieties over global function fields of characteristic $p$, Publisher: arXiv
We prove an analogue of the Tate isogeny conjecture and the semi-simplicity conjecture for overconvergent crystalline Dieudonné modules of abelian varieties defined over global function fields of characteristic p. As a corollary we deduce that monodromy groups of such overconvergent crystalline Dieudonné modules are reductive, and after a finite base change of coefficients their connected components are the same as the connected components of monodromy groups of Galois representations on the corresponding l-adic Tate modules, for l different from p. We also show such a result for general compatible systems incorporating overconvergent F-isocrystals, conditional on a result of Abe.
Pal A, 2015, Etale homotopy equivalence of rational points on algebraic varieties, ALGEBRA & NUMBER THEORY, Vol: 9, Pages: 815-873, ISSN: 1937-0652
Pal A, 2014, On the Chow groups of certain geometrically rational 5-folds, JOURNAL OF NUMBER THEORY, Vol: 139, Pages: 53-79, ISSN: 0022-314X
Pal A, 2014, Hodge theory and the Mordell-Weil rank of elliptic curves over extensions of function fields, JOURNAL OF NUMBER THEORY, Vol: 137, Pages: 166-178, ISSN: 0022-314X
Pal A, 2014, ON THE NILPOTENT SECTION CONJECTURE FOR FINITE GROUP ACTIONS ON CURVES, MATHEMATIKA, Vol: 60, Pages: 183-200, ISSN: 0025-5793
Pal A, 2013, SOLVABLE POINTS ON GENUS-ONE CURVES OVER LOCAL FIELDS, JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU, Vol: 12, Pages: 31-42, ISSN: 1474-7480
Pal A, 2013, Curves which do not Become Semi-Stable After any Solvable Extension, RENDICONTI DEL SEMINARIO MATEMATICO DELLA UNIVERSITA DI PADOVA, Vol: 129, Pages: 265-276, ISSN: 0041-8994
Karolyi G, Pal A, 2012, The cyclomatic number of connected graphs without solvable orbits, JOURNAL OF THE RAMANUJAN MATHEMATICAL SOCIETY, Vol: 27, Pages: 213-240, ISSN: 0970-1249
Pal A, 2011, The real section conjecture and Smith's fixed-point theorem for pro-spaces, JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, Vol: 83, Pages: 353-367, ISSN: 0024-6107
Pal A, 2010, On the Kernel and the Image of the Rigid Analytic Regulator in Positive Characteristic, PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES, Vol: 46, Pages: 255-288, ISSN: 0034-5318
Pal A, 2010, A Rigid Analytical Regulator for the K-2 of Mumford Curves, PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES, Vol: 46, Pages: 219-253, ISSN: 0034-5318
Pal A, 2010, The Rigid Analytical Regulator and K-2 of Drinfeld Modular Curves, PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES, Vol: 46, Pages: 289-334, ISSN: 0034-5318
Pal A, 2010, On the torsion of Drinfeld modules of rank two, JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, Vol: 640, Pages: 1-45, ISSN: 0075-4102
Pal A, 2010, The Manin constant of elliptic curves over function fields, ALGEBRA & NUMBER THEORY, Vol: 4, Pages: 509-545, ISSN: 1937-0652
Ambrus Pal, 2007, On the Eisenstein ideal of Drinfeld modular curves, International Journal of Number Theory
Pal A, 2006, Proof of an exceptional zero conjecture forelliptic curves over function fields, Mathematische Zeitschrift, Vol: 254, Pages: 461-483, ISSN: 0025-5874
Pal A, 2005, On the torsion of the Mordell-Weil group of the Jacobian of Drinfeld modular curves, Documenta Math., Vol: 10, Pages: 131-198, ISSN: 1431-0643
Pal A, 2004, Solvable points on projective algebraic curves, Canad. J. Math., Vol: 56, Pages: 612-637, ISSN: 0008-414X
Pal A, 2004, Theta series, Eisenstein series and Poincareseries over function fields, Canad. J. Math., Vol: 56, Pages: 406-430, ISSN: 0008-414X
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