scholarly journals Explicit moduli spaces for congruences of elliptic curves

2019 ◽  
Vol 295 (3-4) ◽  
pp. 1337-1354
Author(s):  
Tom Fisher
Keyword(s):  
Elliptic Curves ◽  
Moduli Spaces ◽  
Elliptic Surface ◽  
Picard Number

Abstract We determine explicit birational models over $${{\mathbb {Q}}}$$ Q for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell–Weil lattice and the geometric Picard number.

Involutions of Higgs moduli spaces over elliptic curves and pseudo-real Higgs bundles

2019 ◽  
Vol 142 ◽  
pp. 47-65 ◽  
Author(s):  
Indranil Biswas ◽  
Luis Angel Calvo ◽  
Emilio Franco ◽  
Oscar García-Prada
Keyword(s):  
Elliptic Curves ◽  
Moduli Spaces ◽  
Higgs Bundles

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 $\overline {M}_{1,n}$.

scholarly journals COHERENT SYSTEMS ON ELLIPTIC CURVES

2005 ◽  
Vol 16 (07) ◽  
pp. 787-805 ◽  
Author(s):  
H. LANGE ◽  
P. E. NEWSTEAD
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Moduli Spaces ◽  
Coherent Systems

In this paper we consider coherent systems (E,V) on an elliptic curve which are α-stable with respect to some value of a parameter α. We show that the corresponding moduli spaces, if non-empty, are smooth and irreducible of the expected dimension. Moreover we give precise conditions for non-emptiness of the moduli spaces. Finally we study the variation of the moduli spaces with α.

scholarly journals On the Modularity of Three Calabi–Yau Threefolds With Bad Reduction at 11

2006 ◽  
Vol 49 (2) ◽  
pp. 296-312 ◽  
Author(s):  
Matthias Schütt
Keyword(s):  
Elliptic Curves ◽  
Galois Representations ◽  
Elliptic Surface ◽  
Two Dimensional ◽  
Cohomology Groups ◽  
Elliptic Surfaces ◽  
Fibre Products ◽  
Fibre Product ◽  
Rational Elliptic Surfaces

AbstractThis paper investigates the modularity of three non-rigid Calabi–Yau threefolds with bad reduction at 11. They are constructed as fibre products of rational elliptic surfaces, involving the modular elliptic surface of level 5. Their middle ℓ-adic cohomology groups are shown to split into two-dimensional pieces, all but one of which can be interpreted in terms of elliptic curves. The remaining pieces are associated to newforms of weight 4 and level 22 or 55, respectively. For this purpose, we develop a method by Serre to compare the corresponding two-dimensional 2-adic Galois representations with uneven trace. Eventually this method is also applied to a self fibre product of the Hesse-pencil, relating it to a newform of weight 4 and level 27.

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.

scholarly journals CLASS NUMBERS VIA 3-ISOGENIES AND ELLIPTIC SURFACES

2012 ◽  
Vol 09 (01) ◽  
pp. 125-137
Author(s):  
CAM McLEMAN ◽  
DUSTIN MOODY
Keyword(s):  
Elliptic Curves ◽  
Number Field ◽  
Class Number ◽  
Elliptic Surface ◽  
Class Numbers ◽  
Character Sum ◽  
Elliptic Surfaces ◽  
Higher Dimensional ◽  
Imaginary Number

We show that a character sum attached to a family of 3-isogenies defined on the fibers of a certain elliptic surface over 𝔽p relates to the class number of the quadratic imaginary number field [Formula: see text]. In this sense, this provides a higher-dimensional analog of some recent class number formulas associated to 2-isogenies of elliptic curves.

scholarly journals Bagger–Witten line bundles on moduli spaces of elliptic curves

2016 ◽  
Vol 31 (35) ◽  
pp. 1650188 ◽  
Author(s):  
Wei Gu ◽  
Eric Sharpe
Keyword(s):  
Line Bundle ◽  
Elliptic Curves ◽  
Moduli Spaces ◽  
Line Bundles ◽  
Deformation Parameters ◽  
Special Cases ◽  
Moduli Stack ◽  
Upper Half Plane ◽  
Ordinary Line ◽  
Special Case

In this paper, we discuss Bagger–Witten line bundles over moduli spaces of SCFTs. We review how in general they are “fractional” line bundles, not honest line bundles, twisted on triple overlaps. We discuss the special case of moduli spaces of elliptic curves in detail. There, the Bagger–Witten line bundle does not exist as an ordinary line bundle, but rather is necessarily fractional. As a fractional line bundle, it is nontrivial (though torsion) over the uncompactified moduli stack, and its restriction to the interior, excising corners with enhanced stabilizers, is also fractional. It becomes an honest line bundle on a moduli stack defined by a quotient of the upper half plane by a metaplectic group, rather than [Formula: see text]. We review and compare to results of recent work arguing that well-definedness of the worldsheet metric implies that the Bagger–Witten line bundle admits a flat connection (which includes torsion bundles as special cases), and gives general arguments on the existence of universal structures on moduli spaces of SCFTs, in which superconformal deformation parameters are promoted to nondynamical fields ranging over the SCFT moduli space.

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