scholarly journals Speeding up the Arithmetic on Koblitz Curves of Genus Two

2001 ◽  
pp. 106-117 ◽  
Author(s):  
Christian Günther ◽  
Tanja Lange ◽  
Andreas Stein
Keyword(s):  
Koblitz Curves ◽  
Genus Two

scholarly journals Correction to: Classification of non-local rings with genus two zero-divisor graphs

2021 ◽  
Vol 25 (4) ◽  
pp. 3355-3356
Author(s):  
T. Asir ◽  
K. Mano ◽  
T. Tamizh Chelvam
Keyword(s):  
Zero Divisor ◽  
Local Rings ◽  
Non Local ◽  
Genus Two

scholarly journals Two-loop superstring five-point amplitudes. Part II. Low energy expansion and S-duality

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Eric D’Hoker ◽  
Carlos R. Mafra ◽  
Boris Pioline ◽  
Oliver Schlotterer
Keyword(s):  
Moduli Space ◽  
Effective Interaction ◽  
Type Ii ◽  
Leading Order ◽  
Effective Interactions ◽  
Heterotic String Theory ◽  
Genus Two ◽  
Genus One ◽  
Graph Function ◽  
Point Amplitude

Abstract In an earlier paper, we constructed the genus-two amplitudes for five external massless states in Type II and Heterotic string theory, and showed that the α′ expansion of the Type II amplitude reproduces the corresponding supergravity amplitude to leading order. In this paper, we analyze the effective interactions induced by Type IIB superstrings beyond supergravity, both for U(1)R-preserving amplitudes such as for five gravitons, and for U(1)R-violating amplitudes such as for one dilaton and four gravitons. At each order in α′, the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states. To leading and sub-leading orders, the coefficients of the effective interactions D2ℛ5 and D4ℛ5 are found to match those of D4ℛ4 and D6ℛ4, respectively, as required by non-linear supersymmetry. To the next order, a D6ℛ5 effective interaction arises, which is independent of the supersymmetric completion of D8ℛ4, and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D6ℛ5, the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude. We check that the supergravity limit of U(1)R-violating amplitudes is free of UV divergences to this order, consistently with the known structure of divergences in Type IIB supergravity. Our results give strong consistency tests on the full five-point amplitude, and pave the way for understanding S-duality beyond the BPS-protected sector.

A CATALOGUE OF ORIENTABLE 3-MANIFOLDS TRIANGULATED BY 30 COLORED TETRAHEDRA

2008 ◽  
Vol 17 (05) ◽  
pp. 579-599 ◽  
Author(s):  
MARIA RITA CASALI ◽  
PAOLA CRISTOFORI
Keyword(s):  
Automatic Generation ◽  
Computational Approach ◽  
Equivalence Classes ◽  
Colored Graphs ◽  
One To One ◽  
Genus Two ◽  
Jsj Decompositions

The present paper follows the computational approach to 3-manifold classification via edge-colored graphs, already performed in [1] (with respect to orientable 3-manifolds up to 28 colored tetrahedra), in [2] (with respect to non-orientable 3-manifolds up to 26 colored tetrahedra), in [3] and [4] (with respect to genus two 3-manifolds up to 34 colored tetrahedra): in fact, by automatic generation and analysis of suitable edge-colored graphs, called crystallizations, we obtain a catalogue of all orientable 3-manifolds admitting colored triangulations with 30 tetrahedra. These manifolds are unambiguously identified via JSJ decompositions and fibering structures. It is worth noting that, in the present work, a suitable use of elementary combinatorial moves yields an automatic partition of the elements of the generated crystallization catalogue into equivalence classes, which turn out to be in one-to-one correspondence with the homeomorphism classes of the represented manifolds.

Computing Galois representations of modular abelian surfaces

2014 ◽  
Vol 17 (A) ◽  
pp. 36-48 ◽  
Author(s):  
Jinxiang Zeng
Keyword(s):  
Abelian Variety ◽  
Efficient Algorithm ◽  
Galois Representations ◽  
Abelian Surfaces ◽  
Genus Two

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f\in S_2(\Gamma _0(N))$ be a normalized newform such that the abelian variety $A_f$ attached by Shimura to $f$ is the Jacobian of a genus-two curve. We give an efficient algorithm for computing Galois representations associated to such newforms.

scholarly journals On genus-two solutions for the ILW equation

2018 ◽  
Vol 59 (2) ◽  
pp. 023511
Author(s):  
Y. Tutiya
Keyword(s):  
Genus Two

scholarly journals Tautological rings of spaces of pointed genus two curves of compact type

2016 ◽  
Vol 152 (7) ◽  
pp. 1398-1420 ◽  
Author(s):  
Dan Petersen
Keyword(s):  
Hodge Structure ◽  
Galois Representation ◽  
Mixed Hodge Structure ◽  
Poincaré Duality ◽  
General Study ◽  
Compact Type ◽  
Poincare Duality ◽  
Tautological Ring ◽  
Genus Two ◽  
Tautological Rings

We prove that the tautological ring of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$, the moduli space of $n$-pointed genus two curves of compact type, does not have Poincaré duality for any $n\geqslant 8$. This result is obtained via a more general study of the cohomology groups of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$. We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of $H^{k}({\mathcal{M}}_{2,n}^{\mathsf{ct}})$ for any $k$ and $n$ considered both as $\mathbb{S}_{n}$-representation and as mixed Hodge structure/$\ell$-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of $\overline{{\mathcal{M}}}_{2,n}$ is tautological for $n<20$, and that the tautological ring of $\overline{{\mathcal{M}}}_{2,n}$ fails to have Poincaré duality for all $n\geqslant 20$. This improves and simplifies results of the author and Orsola Tommasi.

Twofold unbranched coverings of genus two 3-manifolds are hyperelliptic

2001 ◽  
Vol 123 (1) ◽  
pp. 149-155 ◽  
Author(s):  
Alexander Mednykh ◽  
Marco Reni
Keyword(s):  
Genus Two
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