Seshadri constants on principally polarized abelian surfaces with real multiplication

Seshadri constants on abelian surfaces are fully understood in the case of Picard number one. Little is known so far for simple abelian surfaces of higher Picard number. In this paper we investigate principally polarized abelian surfaces with real multiplication. They are of Picard number two and might be considered the next natural case to be studied. The challenge is to not only determine the Seshadri constants of individual line bundles, but to understand the whole Seshadri function on these surfaces. Our results show on the one hand that this function is surprisingly complex: on surfaces with real multiplication in $$\mathbb {Z}[\sqrt{e}]$$ Z [ e ] it consists of linear segments that are never adjacent to each other—it behaves like the Cantor function. On the other hand, we prove that the Seshadri function is invariant under an infinite group of automorphisms, which shows that it does have interesting regular behavior globally.

Monotonicity, Convexity, Continuity, and Isomorphism: The Psychological Scaffolding of Arithmetic

With the natural numbers as our starting point, we obtain the arithmetic structure of real (as in R) addition and multiplication without relying on any algebraic tools; in particular, we leverage monotonicity, convexity, continuity, and isomorphism. Natural addition arises by minimizing against monotonicity. Rational addition arises from natural addition by minimizing against convexity. Real addition arises from rational addition via any one of three methods; unique convex extension, unique continuous extension, and unique monotonic extension. Real multiplication arises from real addition via isomorphism. Following these mathematical developments, we argue that each of the leveraged mathematical concepts ---monotonicity, convexity, continuity, and isomorphism --- enjoys, prior to its formal mathematical existence, an intuitive psychological existence. Taken together, these lines of argument suggest a way for psychological representation of algebraic structure to emerge from non-algebraic --- and psychologically plausible --- ingredients.

An iterative method of statistical tolerancing based on the unified Jacobian–Torsor model and Monte Carlo simulation

This paper focuses on exploring an iterative method of statistical tolerance design to guide designers to select tolerances more economically and effectively. After having identified the assembly functional requirement (FR) and the functional elements (FEs) of corresponding tolerance chain, the expression of a unified Jacobian–Torsor model can be derived. Monte Carlo simulation is employed to generate random variables simulating the variations of small displacement torsor associated with the FE pairs with all the generated random values being within the intervals constrained by the corresponding tolerance zones. Then, the real multiplication operations are repeatedly executed to this model, a large number of real torsor component values of FR will be obtained and we can perform statistical analysis for these simulated data to get the statistical limits of the assembly FR in the desired direction. The tolerances of critical FEs may need to be adjusted to satisfy the assembly FR imposed by the designer, and the percentage contribution of each FE to the assembly FR can help determine these critical tolerances that need to be tightened or loosened. Once the calculated FR is in close agreement with the imposed FR, the iterative process can be stopped, and the statistical tolerance redesign is achieved. The effectiveness of the proposed method is illustrated with a case study. Compared with the deterministic tolerancing method, the results show that the proposed method is more economical and that can relax significantly the precision required, manufacturing and inspection costs can then be reduced considerably.

Quadratic Twists of Abelian Varieties With Real Multiplication

Let $F$ be a totally real number field and $A/F$ an abelian variety with real multiplication (RM) by the ring of integers $\mathcal {O}$ of a totally real field. Assuming $A$ admits an $\mathcal {O}$-linear 3-isogeny over $F$, we prove that a positive proportion of the quadratic twists $A_d$ have rank 0. If moreover $A$ is principally polarized and $III(A_d)$ is finite, then a positive proportion of $A_d$ have $\mathcal {O}$-rank $1$. Our proofs make use of the geometry-of-numbers methods from our previous work with Bhargava, Klagsbrun, and Lemke Oliver and develop them further in the case of RM. We quantify these results for $A/\mathbb {Q}$ of prime level, using Mazur’s study of the Eisenstein ideal. For example, suppose $p \equiv 10$ or $19 \pmod {27}$, and let $A$ be the unique optimal quotient of $J_0(p)$ with a rational point $P$ of order 3. We prove that at least $25\%$ of twists $A_d$ have rank 0 and the average $\mathcal {O}$-rank of $A_d(F)$ is at most 7/6. Using the presence of two different 3-isogenies in this case, we also prove that roughly $1/8$ of twists of the quotient $A/\langle P\rangle$ have nontrivial 3-torsion in their Tate–Shafarevich groups.

WEIERSTRASS PRYM EIGENFORMS IN GENUS FOUR

We prove the connectedness of the Prym eigenforms loci in genus four (for real multiplication by some order of discriminant $D$), for any $D$. These loci were discovered by McMullen in 2006.