scholarly journals Non-tautological Hurwitz cycles

2021 ◽  
Author(s):  
Carl Lian
Keyword(s):  
Modular Form ◽  
Elliptic Curves ◽  
Fourier Coefficient ◽  
Algebraic Cycles ◽  
Stable Curves ◽  
Large Genus ◽  
Positive Genus

AbstractWe show that various loci of stable curves of sufficiently large genus admitting degree d covers of positive genus curves define non-tautological algebraic cycles on $${\overline{{\mathcal {M}}}}_{g,N}$$ M ¯ g , N , assuming the non-vanishing of the d-th Fourier coefficient of a certain modular form. Our results build on those of Graber-Pandharipande and van Zelm for degree 2 covers of elliptic curves; the main new ingredient is a method to intersect the cycles in question with boundary strata, as developed recently by Schmitt-van Zelm and the author.

scholarly journals Modular compactifications of the space of pointed elliptic curves II

2011 ◽  
Vol 147 (6) ◽  
pp. 1843-1884 ◽  
Author(s):  
David Ishii Smyth
Keyword(s):  
Elliptic Curves ◽  
Moduli Spaces ◽  
Minimal Model ◽  
Model Program ◽  
Minimal Model Program ◽  
Stable Curves ◽  
Log Minimal Model Program

AbstractWe prove that the moduli spaces of n-pointed m-stable curves introduced in our previous paper have projective coarse moduli. We use the resulting spaces to run an analogue of Hassett’s log minimal model program for $\overline {M}_{1,n}$.

Modular Parametrizations of Elliptic Curves

1985 ◽  
Vol 28 (3) ◽  
pp. 372-384 ◽  
Author(s):  
D. Zagier
Keyword(s):  
Modular Form ◽  
Elliptic Curves ◽  
Modular Curve ◽  
Holomorphic Differential ◽  
Branched Covering ◽  
Modular Parametrization ◽  
Relationship Of ◽  
The Relationship

AbstractMany — conjecturally all — elliptic curves E/ have a "modular parametrization," i.e. for some N there is a map φ from the modular curve X0(N) to E such that the pull-back of a holomorphic differential on E is a modular form (newform) f of weight 2 and level N. We describe an algorithm for computing the degree of φ as a branched covering, discuss the relationship of this degree to the "congruence primes" for f (the primes modulo which there are congruences between f and other newforms), and give estimates for the size of this degree as a function of N.

scholarly journals POWER RESIDUES OF FOURIER COEFFICIENTS OF ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION

2009 ◽  
Vol 05 (01) ◽  
pp. 109-124
Author(s):  
TOM WESTON ◽  
ELENA ZAUROVA
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Fourier Coefficient ◽  
Fourier Coefficients ◽  
Complex Multiplication ◽  
Roots Of Unity ◽  
Modulo P

Fix m greater than one and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these densities differ from the naive expectation of 1/m. We also prove our conjectures for m dividing the number of roots of unity lying in the CM field of E; the most involved case is m = 4 and complex multiplication by Q(i).

scholarly journals The $${\mathcal {H}}$$-tautological ring

2021 ◽  
Vol 27 (5) ◽  
Author(s):  
Carl Lian
Keyword(s):  
Elliptic Curves ◽  
Moduli Spaces ◽  
General Setting ◽  
Rich Source ◽  
Stable Curves ◽  
Moduli Spaces Of Curves ◽  
Tautological Ring ◽  
Galois Covers ◽  
New Feature ◽  
New Framework

AbstractWe extend the theory of tautological classes on moduli spaces of stable curves to the more general setting of moduli spaces of admissible Galois covers of curves, introducing the so-called $${\mathcal {H}}$$ H -tautological ring. The main new feature is the existence of restriction-corestriction morphisms remembering intermediate quotients of Galois covers, which are a rich source of new classes. In particular, our new framework includes classes of Harris–Mumford admissible covers on moduli spaces of curves, which are known in some (and speculatively many more) examples to lie outside the usual tautological ring. We give additive generators for the $${\mathcal {H}}$$ H -tautological ring and show that their intersections may be algorithmically computed, building on work of Schmitt-van Zelm. As an application, we give a method for computing integrals of Harris-Mumford loci against tautological classes of complementary dimension, recovering and giving a mild generalization of a recent quasi-modularity result of the author for covers of elliptic curves.

Algebraic cycles on products of elliptic curves over p-adic fields

2007 ◽  
Vol 339 (2) ◽  
pp. 241-249 ◽  
Author(s):  
Andreas Rosenschon ◽  
V. Srinivas
Keyword(s):  
Elliptic Curves ◽  
Algebraic Cycles

p-ADIC AND COMBINATORIAL PROPERTIES OF MODULAR FORM COEFFICIENTS

2006 ◽  
Vol 02 (02) ◽  
pp. 305-328 ◽  
Author(s):  
PO-RU LOH ◽  
ROBERT C. RHOADES
Keyword(s):  
Modular Form ◽  
Elliptic Curves ◽  
Modular Forms ◽  
Cusp Forms ◽  
Combinatorial Properties ◽  
Hecke Operator ◽  
Combinatorial Objects ◽  
Apéry Numbers ◽  
Space Of Cusp Forms

For two particular classes of elliptic curves, we establish congruences relating the coefficients of their corresponding modular forms to combinatorial objects. These congruences resemble a supercongruence for the Apéry numbers conjectured by Beukers and proved by Ahlgren and Ono in [1]. We also consider the trace Tr 2k(Γ0(N), n) of the Hecke operator Tn acting on the space of cusp forms S2k(Γ0(N)). We show that for (n, N) = 1, these traces interpolate p-adically in the weight aspect.

scholarly journals On the Kohnen plus space for Hilbert modular forms of half-integral weight I

2013 ◽  
Vol 149 (12) ◽  
pp. 1963-2010 ◽  
Author(s):  
Kaoru Hiraga ◽  
Tamotsu Ikeda
Keyword(s):  
Modular Form ◽  
Fourier Coefficient ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Hilbert Modular Forms ◽  
Jacobi Forms ◽  
Hecke Operator ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular

AbstractIn this paper, we construct a generalization of the Kohnen plus space for Hilbert modular forms of half-integral weight. The Kohnen plus space can be characterized by the eigenspace of a certain Hecke operator. It can be also characterized by the behavior of the Fourier coefficients. For example, in the parallel weight case, a modular form of weight $\kappa + (1/ 2)$ with $\xi \mathrm{th} $ Fourier coefficient $c(\xi )$ belongs to the Kohnen plus space if and only if $c(\xi )= 0$ unless $\mathop{(- 1)}\nolimits ^{\kappa } \xi $ is congruent to a square modulo $4$. The Kohnen subspace is isomorphic to a certain space of Jacobi forms. We also prove a generalization of the Kohnen–Zagier formula.

scholarly journals On the Asymptotic Growth of Bloch-Kato-Shafarevich-Tate Groups of Modular Forms Over Cyclotomic Extensions

2017 ◽  
Vol 69 (4) ◽  
pp. 826-850 ◽  
Author(s):  
Antonio Lei ◽  
David Loeffler ◽  
Zerbes Sarah Livia
Keyword(s):  
Asymptotic Behaviour ◽  
Modular Form ◽  
Elliptic Curves ◽  
Modular Forms ◽  
Upper Bounds ◽  
Selmer Groups ◽  
Asymptotic Growth ◽  
Supersingular Elliptic Curves ◽  
Cyclotomic Extensions ◽  
Iwasawa Invariants

AbstractWe study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic ℤp-extension of ℚ under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara, and Sprung for supersingular elliptic curves.

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