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We study algebraic cycles in the moduli space of PGL2-shtukas, arising from the diagonal torus. Our main result shows that their intersection pairing with the Heegner--Drinfeld cycle is the product of the r-th central derivative of an automorphic L-function (π, s) and Waldspurger's toric period integral. When L(π, 1/2) ≠ 0, this gives a new geometric interpretation for the Taylor series expansion. When L(π, 1/2) 0, the pairing vanishes, suggesting higher order analogues of the vanishing of cusps in the modular Jacobian, as well as other new phenomena. Our proof sheds new light on the algebraic correspondence introduced by Yun and Zhang, which is the geometric incarnation of "differentiating the L-function". We realize it as the Lie algebra action of e + ƒ ∈ sl2 on (ℚℓ²)⊗2d. The comparison of relative trace formulas needed to prove our formula is then a consequence of Schur--Weyl duality. [ABSTRACT FROM AUTHOR]

Fix an elliptic curve E over a number field F and an integer n which is a power of 3. We study the growth of the Mordell–Weil rank of E after base change to the fields K_d = F(\!\sqrt [2n]{d}). If E admits a 3-isogeny, then we show that the average "new rank" of E over K_d, appropriately defined, is bounded as the height of d goes to infinity. When n = 3, we moreover show that for many elliptic curves E/\mathbb {Q}, there are no new points on E over \mathbb {Q}(\sqrt [6]d), for a positive proportion of integers d. This is a horizontal analogue of a well-known result of Cornut and Vatsal [ Nontriviality of Rankin-Selberg L-functions and CM points, L -functions and Galois representations, vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 121–186]. As a corollary, we show that Hilbert's tenth problem has a negative solution over a positive proportion of pure sextic fields \mathbb {Q}(\sqrt [6]{d}). The proofs combine our recent work on ranks of abelian varieties in cyclotomic twist families with a technique we call the "correlation trick", which applies in a more general context where one is trying to show simultaneous vanishing of multiple Selmer groups. We also apply this technique to families of twists of Prym surfaces, which leads to bounds on the number of rational points in sextic twist families of bielliptic genus 3 curves. [ABSTRACT FROM AUTHOR]

We give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, the central value of the L-function of the relevant motive is non-vanishing and the Ceresa cycle is torsion in the Griffiths group, consistent with the conjectures of Beilinson and Bloch. We speculate on a possible explanation for the existence of these torsion Ceresa classes, based on some computations with cyclic Fermat quotients. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The elliptic curve Ek:y2=x3+k admits a natural 3‐isogeny ϕk:Ek→E−27k. We compute the average size of the ϕk‐Selmer group as k varies over the integers. Unlike previous results of Bhargava and Shankar on n‐Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on k; this sensitivity can be precisely controlled by the Tamagawa numbers of Ek and E−27k. As a consequence, we prove that the average rank of the curves Ek, k∈Z, is less than 1.21 and over 23% (respectively, 41%) of the curves in this family have rank 0 (respectively, 3‐Selmer rank 1). [ABSTRACT FROM AUTHOR]