97 research outputs found

    Holomorphic maps between configuration spaces of Riemann surfaces

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    We prove a suite of results classifying holomorphic maps between configuration spaces of Riemann surfaces; we consider both the ordered and unordered setting as well as the cases of genus zero, one, and at least two. We give a complete classification of all holomorphic maps Confn(C)Confm(C)\operatorname{Conf}_n(\mathbb{C})\to \operatorname{Conf}_m(\mathbb{C}) provided that n5n\ge 5 and m2nm\le 2n extending the Tameness Theorem of Lin, which is the case m=nm = n. We also give a complete classification of holomorphic maps between ordered configuration spaces of Riemann surfaces of genus at most one (answering a question of Farb), and show that the higher genus setting is closely linked to the still-mysterious ``effective de Franchis problem''. The main technical theme of the paper is that holomorphicity allows one to promote group-theoretic rigidity results to the space level.Comment: 28 page

    Faster Complete Formulas for the GLS254 Binary Curve

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    GLS254 is an elliptic curve defined over a finite field of characteristic 2; it contains a 253-bit prime order subgroup, and supports an endomorphism that can be efficiently computed and helps speed up some typical operations such as multiplication of a curve element by a scalar. That curve offers on x86 and ARMv8 platforms the best known performance for elliptic curves at the 128-bit security level. In this paper we present a number of new results related to GLS254: - We describe new efficient and complete point doubling formulas (2M+4S) applicable to all ordinary binary curves. - We apply the previously described (x,s) coordinates to GLS254, enhanced with the new doubling formulas. We obtain formulas that are not only fast, but also complete, and thus allow generic constant-time usage in arbitrary cryptographic protocols. - Our strictly constant-time implementation multiplies a point by a scalar in 31615 cycles on an x86 Coffee Lake, and 77435 cycles on an ARM Cortex-A55, improving previous records by 13% and 11.7% on these two platforms, respectively. - We take advantage of the completeness of the formulas to define some extra operations, such as canonical encoding with (x, s) compression, constant-time hash-to-curve, and signatures. Our Schnorr signatures have size only 48 bytes, and offer good performance: signature generation in 18374 cycles, and verification in 27376 cycles, on x86; this is about four times faster than the best reported Ed25519 implementations on the same platform. - The very fast implementations leverage the carryless multiplication opcodes offered by the target platforms. We also investigate performance on CPUs that do not offer such an operation, namely a 64-bit RISC-V CPU (SiFive-U74 core) and a 32-bit ARM Cortex-M4 microcontroller. While the achieved performance is substantially poorer, it is not catastrophic; on both platforms, GLS254 signatures are only about 2x to 2.5x slower than Ed25519

    Elliptic curves over Galois number fields

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    This thesis is concerned with the statistical behaviour of elliptic curves over extension fields. That is, if K/Q is a finite extension, we study the arithmetic of E/K as E ranges in natural families of elliptic curves defined over Q. We study the statistical properties of the action of the group Aut(K) on E(K) and on the p-Selmer groups Selp(E/K) where p is a prime number. We construct special generalised Selmer groups, and show that these are related to certain representation-theoretic invariants of Selp(E/K). The sizes of these groups are related to the cokernels of the norm maps over the completions of K, which we go on to compute in several cases. In the statistical component of this thesis, we study quadratic twist families of elliptic curves and the family of ‘all elliptic curves’. For quadratic twist families we consider the behaviour over quadratic extensions. Using methods similar to those of Heath-Brown [HB93, HB94] and of Fouvry–Klüners [FK07], we determine the complete distribution of the 2-Selmer groups as Galois modules. This also allows us to determine representation-theoretic properties for the Mordell–Weil groups of 100% of twists. For the family of all elliptic curves over Q, we consider the behaviour with respect to a general finite Galois extension K/F. Writing G = Gal(K/F), our first main result is that the difference in dimension between Selp(E/K) G and Selp(E/F) has bounded average in this family. Using this we are able, with additional assumptions on K/F and p, to bound the average dimension of Selp(E/K) and so the average rank of the Mordell–Weil group E(K). Our methods also allow us to bound how often certain Z[G]-lattices occur as summands of E(K), with additional assumptions on F. We refine our results in the setting where K/Q is multiquadratic and p = 2, and prove strong upper and lower bounds for the average dimension of the 2-Selmer group

    Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.54 (2022)

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    Lagrangian fibration structure on the cotangent bundle of a del Pezzo surface of degree 4

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    In this paper, we show that there is a natural Lagrangian fibration structure on the map Φ\Phi from the cotangent bundle of a del Pezzo surface XX of degree 4 to C2\mathbb C^2. Moreover, we describe explicitly all level surfaces of the above natural map Φ\Phi.Comment: 23 page

    Geometric and analytic methods for quadratic Chabauty

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    Let X be an Atkin-Lehner quotient of the modular curve X_0(N) whose Jacobian J_f is a simple quotient of J_0(N)^{new} over Q. We give analytic methods for determining the rational points of X using quadratic Chabauty by explicitly computing two p-adic Gross--Zagier formulas for the newform f of level N and weight 2 associated with J_f when f has analytic rank 1. Combining results of Gross-Zagier and Waldspurger, one knows that for certain imaginary quadratic fields K, there exists a Heegner divisor in J_0(N)(K) whose image is finite index in J_f(Q) under the action of Hecke. We give an algorithm to compute the special value of the anticyclotomic p-adic L-function of f constructed by Bertolini, Darmon, and Prasanna, assuming some hypotheses on the prime p and on K. This value is proportional to the logarithm of the Heegner divisor on J_f with respect to the differential form f dq/q. We also compute the p-adic height of the Heegner divisor on J_f using a p-adic Gross-Zagier formula of Perrin-Riou. Additionally, we give algorithms for the geometric quadratic Chabauty method of Edixhoven and Lido. Our algorithms describe how to translate their algebro-geometric method into calculations involving Coleman-Gross heights, logarithms, and divisor arithmetic. We achieve this by leveraging a map from the Poincaré biextension to the trivial biextension

    F-Theory Realizations Of Exact Mssm Matter Spectra

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    F-theory is remarked by its powerful phenomenological model building potential due to geometric descriptions of compactifications. It translates physics quantities in the effective low energy theory to mathematical objects extracted from the geometry of the compactifications. The connection is built upon identifying the varying axio-dilaton field in type IIB supergravity theory with the complex structure modulus of an elliptic curve, that serves as the fiber of an elliptic fibration. This allows us to capture the non-perturbative back-reactions of seven branes onto the compactification space B3B_3 of an elliptically fibered Calabi--Yau fourfold Y4Y_4. The ingredients of Standard model physics, including gauge symmetries, charged matter, and Yukawa couplings, are then encoded beautifully by Y4Y_4\u27s singularity structures in codimensions one, two, and three, respectively. Moreover, many global consistency conditions, including the D3-tadpole cancellation, can be reduced to simple criteria in terms of the intersection numbers of base divisors. In this thesis, we focus on searching for explicit models in the language of F-theory geometry that admit exact Minimal Supersymmetric Standard Model (MSSM) matter spectra. We first present a concrete realization of the Standard Model (SM) gauge group with Z2\mathbb{Z}_2 matter parity, which admits three generations of chiral fermions. The existence of this discrete symmetry beyond the SM gauge group forbids proton decay. We then construct a family of O(1015)\mathcal{O}(10^{15}) F-theory vacua. These are the largest currently known class of globally consistent string constructions that admit exactly three chiral families and gauge coupling unification. We advance to study the vector-like spectra in 4d F-theory SMs. The 4-form gauge background G4G_4 controls the chiral spectra. This is the field strength of 3-form gauge potential C3C_3, which impacts the vector-like spectra. It is well known that these massless zero modes are counted by line bundle cohomologies over matter curves induced by the F-theory gauge background. In order to understand the line bundle cohomology\u27s dependence on the moduli of the compactification geometry, we pick a simple geometry and create the database consisted of matter curves, the line bundles and the vector-like spectra. We analyze this database by machine learning techniques and ugain full understanding it via the Brill-Nother theory. Subsequently, we present the appearance of root bundles and how they enter as significant ingredients of realistic F-theory geometries. The algebraic geometry approaches to root bundles allow combinatoric descriptions, which facilitate the analyze of statistics on the vector-like spectra at the end of this thesis

    Automorphisms of Algebraic Curves

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    Στην παρούσα διατριβή μελετώνται οι αυτομορφισμοί αλγεβρικών καμπυλών επί σωμάτων θετικής χαρακτηριστικής. Μετά από μια σύντομη περιγραφή μιας κλάσης καμπυλών γνωστών ως καμπύλες Harbater-Katz-Gabber αποδεικνύεται μια σειρά νέων αποτελεσμάτων για αυτές τις καμπύλες. Στη συνέχεια υπολογίζεται το κανονικό ιδεώδες μιας τέτοια καμπύλης. Στις τελευταίες παράγραφους δίνονται αποτελέσματα που αφορούν γενικές αλγεβρικές καμπύλες και σχετίζονται με το κανονικό ιδεώδες και με τη θεωρία των συζυγιών.In this dissertation automorphisms of algebraic curves in positive characteristic are studied. After a brief outline of a class of curves known as Harbater-Katz-Gabber curves, some new results regarding these curves are proven. After that the canonical ideal of such a curve is calculated. In the last paragraphs we present results concerning general algebraic curves and are related to the canonical ideal and the theory of syzygies
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