Douglas L Ulmer

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• Cais, B., & Ulmer, D. (2025). P-TORSION FOR UNRAMIFIED ARTIN–SCHREIER COVERS OF CURVES. Transactions of the American Mathematical Society Series B, 12(Issue). doi:10.1090/btran/236
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  Let Y → X be an unramified Galois cover of curves over a perfect field k of characteristic p > 0 with Gal(Y/X) ≅ ℤ/pℤ, and let JXand JYbe the Jacobians of X and Y respectively. We consider the p-torsion subgroup schemes JX[p] and JY [p], analyze the Galois-module structure of JY[p], and find restrictions this structure imposes on JY[p] (for example, as manifested in its Ekedahl–Oort type) taking JX[p] as given.
• Ulmer, D. (2025). Report of the Treasurer (2024). Notices of the American Mathematical Society, 72(Issue 8). doi:10.1090/noti3234
• Pries, R., & Ulmer, D. (2024). Correction to: “On BT1group schemes and Fermat curves” (Electronic Journal of Combinatorics). New York Journal of Mathematics, 30(Issue).
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  We correct an error in Proposition 5.6(3) of [PU21] and revise other statements in the paper accordingly.
• Ulmer, D. (2024). Report of the Treasurer (2023). Notices of the American Mathematical Society, 71(9). doi:10.1090/noti3033
• Ulmer, D. (2023). Report of the Treasurer (2022). Notices of the American Mathematical Society, 70(8).
• Pries, R., & Ulmer, D. (2022). EVERY BT1 GROUP SCHEME APPEARS in A JACOBIAN. Proceedings of the American Mathematical Society, 150(2). doi:10.1090/proc/15681
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  Let p be a prime number and let k be an algebraically closed field of characteristic p. A BT1 group scheme over k is a finite commutative group scheme which arises as the kernel of p on a p-divisible (Barsotti–Tate) group. Our main result is that every BT1 group scheme over k occurs as a direct factor of the p-torsion group scheme of the Jacobian of an explicit curve defined over Fp. We also treat a variant with polarizations. Our main tools are the Kraft classification of BT1 group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves.
• Ulmer, D. (2022). Report of the Treasurer (2021). Notices of the American Mathematical Society, 69(9).
• Ulmer, D., & Urzúa, G. (2022). Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces. Selecta Mathematica, New Series, 28(2). doi:10.1007/s00029-021-00747-x
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  We consider elliptic surfaces E over a field k equipped with zero section O and another section P of infinite order. If k has characteristic zero, we show there are only finitely many points where O is tangent to a multiple of P. Equivalently, there is a finite list of integers such that if n is not divisible by any of them, then nP is not tangent to O. Such tangencies can be interpreted as unlikely intersections. If k has characteristic zero or p> 3 and E is very general, then we show there are no tangencies between O and nP. We apply these results to square-freeness of elliptic divisibility sequences and to geography of surfaces. In particular, we construct mildly singular surfaces of arbitrary fixed geometric genus with K ample and K2 unbounded.
• Pries, R., & Ulmer, D. (2021). On bt1 group schemes and fermat curves. New York Journal of Mathematics, 27(Issue).
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  Let p be a prime number and let k be an algebraically closed field of characteristic p. A BT1 group scheme over k is a finite commutative group scheme which arises as the kernel of p on a p-divisible (Barsotti–Tate) group. We compare three classifications of BT1 group schemes, due in large part to Kraft, Ekedahl, and Oort, and defined using words, canonical filtrations, and permutations. Using this comparison, we determine the Ekedahl–Oort types of Fermat quotient curves and we compute four invariants of the p-torsion group schemes of these curves.
• Ulmer, D. (2021). Report of the treasurer (2020). Notices of the American Mathematical Society, 68(9).
• Ulmer, D., & Urzua, G. (2021). Bounding tangencies of sections on elliptic surfaces. International Mathematics Research Notices, 2021(6), 4768-4802. doi:10.1093/imrn/rnaa222
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  Given an elliptic surface $\mathcal{E}\to\mathcal{C}$ over a field $k$ of characteristic zero equipped with zero section $O$ and another section $P$ of infinite order, we give a simple and explicit upper bound on the number of points where $O$ is tangent to a multiple of $P$.
• Berger, L., Hall, C., Pannekoek, R., Park, J., Pries, R., Sharif, S., Silverberg, A., & Ulmer, D. (2020). Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields. Memoirs of the American Mathematical Society, 266(1295), 0-0. doi:10.1090/memo/1295
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  Author(s): Berger, Lisa; Hall, Chris; Pannekoek, Rene; Park, Jennifer; Pries, Rachel; Sharif, Shahed; Silverberg, Alice; Ulmer, Douglas | Abstract: We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^{r-1}(x + 1)(x + t)$ over the function field $\mathbb{F}_p(t)$, when $p$ is prime and $r\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, we compute the $L$-function of $J$ over $\mathbb{F}_q(t^{1/d})$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\mathbb{F}_q(t^{1/d})$. When $d$ is divisible by $r$ and of the form $p^\nu +1$, and $K_d := \mathbb{F}_p(\mu_d,t^{1/d})$, we write down explicit points in $J(K_d)$, show that they generate a subgroup $V$ of rank $(r-1)(d-2)$ whose index in $J(K_d)$ is finite and a power of $p$, and show that the order of the Tate-Shafarevich group of $J$ over $K_d$ is $[J(K_d):V]^2$. When $rg2$, we prove that the "new" part of $J$ is isogenous over $\overline{\mathbb{F}_p(t)}$ to the square of a simple abelian variety of dimension $\phi(r)/2$ with endomorphism algebra $\mathbb{Z}[\mu_r]^+$. For a prime $\ell$ with $\ell \nmid pr$, we prove that $J[\ell](L)=\{0\}$ for any abelian extension $L$ of $\overline{\mathbb{F}}_p(t)$.
• Griffon, R., & Ulmer, D. (2020). On the arithmetic of a family of twisted constant elliptic curves. Pacific Journal of Mathematics, 305(2), 597-640. doi:10.2140/pjm.2020.305.597
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  Let $\mathbb{F}_r$ be a finite field of characteristic $p>3$. For any power $q$ of $p$, consider the elliptic curve $E=E_{q,r}$ defined by $y^2=x^3 + t^q -t$ over $K=\mathbb{F}_r(t)$. We describe several arithmetic invariants of $E$ such as the rank of its Mordell--Weil group $E(K)$, the size of its N\'eron--Tate regulator $\text{Reg}(E)$, and the order of its Tate--Shafarevich group $III(E)$ (which we prove is finite). These invariants have radically different behaviors depending on the congruence class of $p$ modulo 6. For instance $III(E)$ either has trivial $p$-part or is a $p$-group. On the other hand, we show that the product $|III(E)|\text{Reg}(E)$ has size comparable to $r^{q/6}$ as $q\to\infty$, regardless of $p\pmod{6}$. Our approach relies on the BSD conjecture, an explicit expression for the $L$-function of $E$, and a geometric analysis of the N\'eron model of $E$.
• Ulmer, D. (2019). On the brauer–siegel ratio for abelian varieties over function fields. Algebra & Number Theory, 13(5), 1069-1120. doi:10.2140/ant.2019.13.1069
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  Hindry has proposed an analog of the classical Brauer–Siegel theorem for abelian varieties over global fields. Roughly speaking, it says that the product of the regulator of the Mordell–Weil group and the order of the Tate–Shafarevich group should have size comparable to the exponential differential height. Hindry–Pacheco and Griffon have proved this for certain families of elliptic curves over function fields using analytic techniques. Our goal in this work is to prove similar results by more algebraic arguments, namely by a direct approach to the Tate–Shafarevich group and the regulator. We recover the results of Hindry–Pacheco and Griffon and extend them to new families, including families of higher-dimensional abelian varieties.
• Pries, R., & Ulmer, D. (2016). Arithmetic of abelian varieties in Artin-Schreier extensions. Transactions of the American Mathematical Society, 368(12), 8553-8595. doi:10.1090/tran6641
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  We study abelian varieties defined over function fields of curves in positive characteristic p, focusing on their arithmetic within the system of Artin-Schreier extensions. First, we prove that the L-function of such an abelian variety vanishes to high order at the center point of its functional equation under a parity condition on the conductor. Second, we develop an Artin-Schreier variant of a construction of Berger. This yields a new class of Jacobians over function fields for which the Birch and Swinnerton-Dyer con- jecture holds. Third, we give a formula for the rank of the Mordell-Weil groups of these Jacobians in terms of the geometry of their fibers of bad reduction and homomorphisms between Jacobians of auxiliary Artin-Schreier curves. We illustrate these theorems by computing the rank for explicit examples of Jaco- bians of arbitrary dimension g, exhibiting Jacobians with bounded rank and others with unbounded rank in the tower of Artin-Schreier extensions. Fi- nally, we compute the Mordell-Weil lattices of an isotrivial elliptic curve and a family of non-isotrivial elliptic curves. The latter exhibits an exotic phenom- enon whereby the angles between lattice vectors are related to point counts on elliptic curves over finite fields. Our methods also yield new results about supersingular factors of Jacobians of Artin-Schreier curves. MSC2000: Primary 11G10, 11G40, 14G05; Secondary 11G05, 11G30, 14H25, 14J20, 14K15
• Ulmer, D. (2016). Rational curves on elliptic surfaces. Journal of Algebraic Geometry, 26(2), 357-377. doi:10.1090/jag/680
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  We prove that a very general elliptic surfaceE ! P 1 over the complex numbers with a section and with geometric genus pg 2 contains no rational curves other than the section and components of singular bers. Equivalently, if E=C(t) is a very general elliptic curve of height d 3 and if L is a nite extension of C(t) with L = C(u), then the Mordell-Weil group E(L) = 0.
• Occhipinti, T., & Ulmer, D. (2015). Low-dimensional factors of superelliptic Jacobians. European Journal of Mathematics, 1(2), 279-285. doi:10.1007/s40879-014-0023-3
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  Given a polynomial f ∈ C(x), we consider the family of superelliptic curves y d = f (x) and their Jacobians Jd for varying integers d. We show that for any integer g the number of abelian varieties up to isogeny of dimension ≤g which appear in any Jd is finite and their multiplicities are bounded.
• Ulmer, D. (2015). Conductors of l-adic representations. Proceedings of the American Mathematical Society, 144(6), 2291-2299. doi:10.1090/proc/12880
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  We give a new formula for the Artin conductor of an $\ell$-adic representation of the Weil group of a local field of residue characteristic $p\neq\ell$.
• Conceicao, R. P., Hall, C., & Ulmer, D. (2014). Explicit points on the Legendre curve II. Mathematical Research Letters, 21(2), 261-280. doi:10.4310/mrl.2014.v21.n2.a5
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  Let E be the elliptic curve y 2 = x(x + 1)(x + t) over the field Fp(t) ,w here p is an odd prime. We study the arithmetic of E over extensions Fq(t 1/d ), where q is a power of p and d is an integer prime to p. The rank of E is given in terms of an elementary property of the subgroup of (Z/dZ) × generated by p. We show that for many values of d the rank is large. For example, if d divides 2(p f − 1) and 2(p f − 1)/d is odd, then the rank is at least d/2. When d =2 (p f − 1), we exhibit explicit points generating a subgroup of E(Fq(t 1/d )) of finite index in the "2-new" part, and we bound the index as well as the order of the "2-new" part of the Tate-Shafarevich group.
• Ulmer, D. (2014). Explicit points on the Legendre curve III. Algebra & Number Theory, 8(10), 2471-2522. doi:10.2140/ant.2014.8.2471
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  We study the elliptic curve E given by y^2=x(x+1)(x+t) over the rational function field k(t) and its extensions K_d=k(\mu_d,t^{1/d}). When k is finite of characteristic p and d=p^f+1, we write down explicit points on E and show by elementary arguments that they generate a subgroup V_d of rank d-2 and of finite index in E(K_d). Using more sophisticated methods, we then show that the Birch and Swinnerton-Dyer conjecture holds for E over K_d, and we relate the index of V_d in E(K_d) to the order of the Tate-Shafarevich group \sha(E/K_d). When k has characteristic 0, we show that E has rank 0 over K_d for all d.
• Ulmer, D. (2014). Explicit points on the Legendre curve. Journal of Number Theory, 136(Issue), 165-194. doi:10.1016/j.jnt.2013.09.010
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  Abstract We study the elliptic curve E given by y 2 = x ( x + 1 ) ( x + t ) over the rational function field k ( t ) and its extensions K d = k ( μ d , t 1 / d ) . When k is finite of characteristic p and d = p f + 1 , we write down explicit points on E and show by elementary arguments that they generate a subgroup V d of rank d − 2 and of finite index in E ( K d ) . Using more sophisticated methods, we then show that the Birch and Swinnerton-Dyer conjecture holds for E over K d , and we relate the index of V d in E ( K d ) to the order of the Tate–Shafarevich group Ш ( E / K d ) . When k has characteristic 0, we show that E has rank 0 over K d for all d .
• Ulmer, D. (2013). On Mordell–Weil groups of Jacobians over function fields. Journal of The Institute of Mathematics of Jussieu, 12(1), 1-29. doi:10.1017/s1474748012000618
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  We study the arithmetic of abelian varieties over $K=k(t)$ where $k$ is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield completely explicit points on elliptic curves with unbounded rank over $\Fpbar(t)$ and a new construction of elliptic curves with moderately high rank over $\C(t)$.
• Conceicao, R., Ulmer, D., & Voloch, J. F. (2012). Unboundedness of the number of rational points on curves over function fields. The New York Journal of Mathematics, 18, 291-293.
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  We construct sequences of smooth nonisotrivial curves of every genus at least two, defined over a rational function field of positive characteristic, such that the (finite) number of rational points of the curves in the sequence cannot be uniformly bounded. The question of whether there is a uniform bound for the number of rational points on curves of fixed genus greater than one over a fixed number field has been considered by several authors. In particular, Caporaso et al. [CHM97] showed that this would follow from the Bombieri–Lang conjecture that the set of rational points on a variety of general type over a number field is not Zariski dense. Abramovich and the third author [AV96] extended this result to get other uniform boundedness consequences of the Bombieri– Lang conjecture and gave some counterexamples for function fields. These counterexamples are singular curves that “change genus”. They behave like positive genus curves (and, in particular, have finitely many rational points), but are parametrizable over an inseparable extension of the ground field. In [AV96] it is shown that, for this class of equation, uniform boundedness does not hold. Specifically, one gets a one-parameter family of equations which, for suitable choice of the parameter, have a finite but arbitrarily large number of solutions. However, a negative answer to the original uniform boundedness question for smooth curves of genus at least two remained open in the function field case. In this paper we provide counterexamples to this uniform boundedness, extending constructions of the first two authors [Con09, Ulm09] for elliptic curves. Theorem. Let p > 3 be a prime number and let r be an odd number coprime to p. The number of rational points over Fp(t) of the curve Xa with equation y2 = x(xr + 1)(xr + ar) is unbounded as a varies in Fp(t) \ Fp. Proof. Let d = pn + 1 and a = td. If m divides n and n/m is odd, then d′ = pm + 1 divides d. Setting e = d/d′, we have the rational point (x, y) = (te, te(r+1)/2(tre + 1)d /2) on Xa. Thus if we take n to be odd with many factors, we have many points. Received April 9, 2012. 2010 Mathematics Subject Classification. 11G30.
• Ulmer, D., Voloch, J., & Conceição, R. (2012). Unboundedness of the number of rational points on curves over function fields. New York Journal of Mathematics, 18(Issue).
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  We construct sequences of smooth nonisotrivial curves of every genus at least two, defined over a rational function field of positive characteristic, such that the (finite) number of rational points of the curves in the sequence cannot be uniformly bounded.
• Ulmer, D., & Zarhin, Y. G. (2010). Ranks of Jacobians in towers of function fields. Mathematical Research Letters, 17(4), 637-645. doi:10.4310/mrl.2010.v17.n4.a5
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  .2. Endomorphismalgebrasof superelliptic Jacobians2.1. Notation. In this section, we collect some results on the endomorphism al-gebras of certain superelliptic Jacobians. Throughout the paper, k will denote analgebraic closure of k.If X and Y are abelian varieties over k, then we write End(X) and Hom(X,Y)for the corresponding ring and group of homomorphisms over k.Let f be a non-constant polynomial with coefficients in k and without multipleroots. We write m for the degree of f, R
• Ulmer, D. (2007). Jacobi sums, fermat jacobians, and ranks of abelian varieties over towers of function fields. Mathematical Research Letters, 14(3), 453-467. doi:10.4310/mrl.2007.v14.n3.a10
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  Our goal in this note is to give a number of examples of abelian varieties over function fields k(t) which have bounded ranks in towers of extensions such as k(t^{1/d}) for varying d. Along the way we prove some new results on Fermat curves which may be of independent interest.
• Ulmer, D. (2007). L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields. Inventiones Mathematicae, 167(2), 379-408. doi:10.1007/s00222-006-0018-x
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  The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. Conjectures of Birch and Swinnerton-Dyer, Bloch, and Beilinson relate the orders of vanishing of some of these L-functions to Mordell-Weil groups and other groups of algebraic cycles. For certain abelian varieties of high analytic rank, we are also able to prove the conjecture of Birch and Swinnerton-Dyer thus establishing the existence of large Mordell-Weil groups in those cases. In the rest of this section we state the main results of the paper.
• Ulmer, D. (2005). Geometric non-vanishing. Inventiones Mathematicae, 159(1), 133-186. doi:10.1007/s00222-004-0386-z
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  We consider L-functions attached to representations of the Galois group of the function field of a curve over a finite field. Under mild tameness hypotheses, we prove non-vanishing results for twists of these L-functions by characters of order prime to the characteristic of the ground field and more generally by certain representations with solvable image. We also allow local restrictions on the twisting representation at finitely many places. Our methods are geometric, and include the Riemann-Roch theorem, the cohomological interpretation of L-functions, and monodromy calculations of Katz. As an application, we prove a result which allows one to deduce the conjecture of Birch and Swinnerton-Dyer for non-isotrivial elliptic curves over function fields whose L-function vanishes to order at most 1 from a suitable Gross-Zagier formula.
• Ulmer, D. (2002). Elliptic curves with large rank over function fields. Annals of Mathematics, 155(1), 295-315. doi:10.2307/3062158
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  We produce explicit elliptic curves over Fp(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptotically these curves have maximal rank for their conductor. Motivated by this fact, we make a conjecture about the growth of ranks of elliptic
• Ulmer, D. L. (1996). On the Fourier coefficients of modular forms. II. Mathematische Annalen, 304(1), 363-422. doi:10.1007/bf01446299
• Ulmer, D. L. (1996). Slopes of modular forms and congruences. Annales de l'Institut Fourier, 46(1), 1-32. doi:10.5802/aif.1504
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  Our aim in this paper is to prove congruences between on the one hand certain eigenforms of level pN and weight greater than 2 and on the other hand twists of eigenforms of level pN and weight 2. One knows a priori that such congruences exist; the novelty here is the we determine the character of the form of weight 2 and the twist in terms of the slope of the higher weight form, i.e., in terms of the valuation of its eigenvalue for Up. Curiously, we also find a relation between the leading terms of the p-adic expansions of the eigenvalues for Up of the two forms. This allows us to determine the restriction to the decomposition group at p of the Galois representation modulo p attached to the higher weight form.
• Ulmer, D. (1995). A construction of local points on elliptic curves over modular curves. International Mathematics Research Notices, 1995(7), 349-363. doi:10.1155/s1073792895000262
• Ulmer, D. L. (1995). On the Fourier coefficients of modular forms. Annales Scientifiques De L Ecole Normale Superieure, 28(2), 129-160. doi:10.24033/asens.1711
• Cukierman, F., & Ulmer, D. (1993). Curves of genus ten on K3 surfaces. Compositio Mathematica, 89(1), 81-90.
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  which sends s∧t to s dt−t ds. J. Wahl made the striking observation that if C is embeddable in a K3 surface then ΦL is not onto for L = ΩC ([W], Thm. 5.9); this raises the natural problem of studying the stratification of the moduli space of curves Mg by the rank of the Wahl map Φ(C) = ΦΩ1C . Roughly speaking, our main theorem says that the closure of the locus of curves of genus 10 which lie on a K3 is equal to the locus where Φ(C) fails to be surjective. In order to state the theorem precisely and explain what is special about the case of genus 10, we need to introduce some spaces. Let Fg be the moduli space of K3 surfaces with a polarization of genus g, Pg the union, over all S ∈ Fg of the linear series |OS(1)|. Let K be the closure of the image of the natural rational map μ : Pg →Mg. As the dimension of Pg is 19+g and the dimension ofMg is 3g− 3, one might naively expect μ to be dominant for g ≤ 10 and finite onto its image for g ≥ 11. These expectations hold for g ≤ 9 ([M], Thm. 6.1) and for odd g ≥ 11 and even g ≥ 20 ([M-M], Thm. 1), but for g = 10, Mukai showed that μ is not dominant ([M], Thm. 0.7). This exceptional behavior is due to the fact that the general K3 surface of genus 10 is a codimension 3 plane section of a certain 5-fold, so that when a curve lies on a general K3, it in fact lies on a 3-dimensional family of them. One of our first tasks is to show that K is a divisor when g = 10. Over the open subset M10 of M10 of curves without automorphisms we have the relative Wahl map; let W denote its degeneracy locus and W the closure of W in M10. It is a theorem of Ciliberto-Harris-Miranda [C-H-M] that W is a divisor (i.e. the Wahl map does not degenerate everywhere), and by Wahl’s theorem K ≤ W. Our result can then be stated as follows.
• Ulmer, D. L. (1991). p-descent in characteristic p. Duke Mathematical Journal, 62(2), 237-265. doi:10.1215/s0012-7094-91-06210-1
• Ulmer, D. L. (1990). L-Functions of Universal Elliptic Curves Over Igusa Curves. American Journal of Mathematics, 112(5), 687. doi:10.2307/2374803
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  This paper is concerned with computing the L-functions of the title in terms of modular forms. Because one can produce zeros of these L-functions, they seem to provide interesting test cases for various conjectures relating L-functions to cycles. In the first three sections we state the theorem, review some consequences, and give a brief sketch of the proof. This proof is based on recent results of Katz and Mazur, so in sections 4 through 6 we establish notation by reviewing the relevant parts of their work. In sections 7 and 8 we prove an Eichler-Shimura style result on a Hecke operator. The proof proper occupies sections 9 through 12. The paper [11] provides an introduction to the universal curves over Igusa curves.
• Ulmer, D. L. (1990). On universal elliptic curves over Igusa curves. Inventiones Mathematicae, 99(1), 377-391. doi:10.1007/bf01234424
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  The purpose of this note is to introduce the arithmetic, study of the universal elliptic curve over Igusa curves. Specifically, its Hasse-WeilL-function is computed in terms of modular forms and is shown to have interesting zeros. Explicit examples are presented for which the Birch and Swinnerton-Dyer conjecture is verified.