The Chowla–Selberg Formula and The Colmez Conjecture

2010 ◽  
Vol 62 (2) ◽  
pp. 456-472 ◽  
Author(s):  
Tonghai Yang
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Abelian Varieties ◽  
Siegel Modular Forms ◽  
Abelian Surface ◽  
Intersection Numbers ◽  
Abelian Surfaces ◽  
Elliptic Modular Form ◽  
Faltings Height

AbstractIn this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.

scholarly journals Irreducibility of limits of Galois representations of Saito–Kurokawa type

2021 ◽  
Vol 7 (3) ◽  
Author(s):  
Tobias Berger ◽  
Krzysztof Klosin
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Galois Representations ◽  
Siegel Modular Forms ◽  
Selmer Groups ◽  
Two Dimensional ◽  
Elliptic Modular Form

AbstractWe prove (under certain assumptions) the irreducibility of the limit $$\sigma _2$$ σ 2 of a sequence of irreducible essentially self-dual Galois representations $$\sigma _k: G_{{\mathbf {Q}}} \rightarrow {{\,\mathrm{GL}\,}}_4(\overline{{\mathbf {Q}}}_p)$$ σ k : G Q → GL 4 ( Q ¯ p ) (as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to $$1 \oplus \rho \oplus \chi $$ 1 ⊕ ρ ⊕ χ with $$\rho $$ ρ irreducible, two-dimensional of determinant $$\chi $$ χ , where $$\chi $$ χ is the mod p cyclotomic character. More precisely, we assume that $$\sigma _k$$ σ k are crystalline (with a particular choice of weights) and Siegel-ordinary at p. Such representations arise in the study of p-adic families of Siegel modular forms and properties of their limits as $$k\rightarrow 2$$ k → 2 appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by p-adic L-values of an elliptic modular form (giving rise to $$\rho $$ ρ ) which we assume are non-zero.

scholarly journals Iwasawa Theory of Elliptic Modular Forms Over Imaginary Quadratic Fields at Non-ordinary Primes

2019 ◽  
Author(s):  
Kâzım Büyükboduk ◽  
Antonio Lei
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quadratic Field ◽  
Euler System ◽  
Base Change ◽  
Iwasawa Theory ◽  
Imaginary Quadratic Field ◽  
Selmer Groups ◽  
Imaginary Quadratic Fields ◽  
Elliptic Modular Form

AbstractThis article is a continuation of our previous work [7] on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field $K$, where the modular form in question was assumed to be ordinary at a fixed odd prime $p$. We formulate integral Iwasawa main conjectures at non-ordinary primes $p$ for suitable twists of the base change of a newform $f$ to an imaginary quadratic field $K$ where $p$ splits, over the cyclotomic ${\mathbb{Z}}_p$-extension, the anticyclotomic ${\mathbb{Z}}_p$-extensions (in both the definite and the indefinite cases) as well as the ${\mathbb{Z}}_p^2$-extension of $K$. In order to do so, we define Kobayashi–Sprung-style signed Coleman maps, which we use to introduce doubly signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson–Flach elements (out of the collection of unbounded Beilinson–Flach elements of Loeffler–Zerbes), which we use to define doubly signed $p$-adic $L$-functions. The main conjecture then relates these two sets of objects. Furthermore, we show that the integral Beilinson–Flach elements form a locally restricted Euler system, which in turn allow us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here (which may be turned into equalities in favorable circumstances).

scholarly journals Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms

2018 ◽  
Vol 30 (4) ◽  
pp. 887-913 ◽  
Author(s):  
Kâzım Büyükboduk ◽  
Antonio Lei
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quadratic Field ◽  
Iwasawa Theory ◽  
Imaginary Quadratic Field ◽  
Main Conjecture ◽  
Elliptic Modular Form

Abstract This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic {\mathbb{Z}_{p}} -tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson–Flach elements.

ON THE FOURIER COEFFICIENTS OF SIEGEL MODULAR FORMS

2016 ◽  
Vol 234 ◽  
pp. 1-16
Author(s):  
SIEGFRIED BÖCHERER ◽  
WINFRIED KOHNEN
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Main Tool ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Growth Properties

One can characterize Siegel cusp forms among Siegel modular forms by growth properties of their Fourier coefficients. We give a new proof, which works also for more general types of modular forms. Our main tool is to study the behavior of a modular form for $Z=X+iY$ when $Y\longrightarrow 0$.

scholarly journals All modular forms of weight 2 can be expressed by Eisenstein series

2020 ◽  
Vol 6 (3) ◽  
Author(s):  
Martin Raum ◽  
Jiacheng Xia
Keyword(s):  
Modular Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Linear Combinations ◽  
Integral Weight ◽  
Elliptic Modular Form

Abstract We show that every elliptic modular form of integral weight greater than 1 can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central $$\mathrm{L}$$ L -values present in all previous work. For weights greater than 2, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.

scholarly journals Siegel modular forms and theta series attached to quaternion algebras II

1997 ◽  
Vol 147 ◽  
pp. 71-106 ◽  
Author(s):  
S. Böcherer ◽  
R. Schulze-Pillot
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quaternion Algebra ◽  
Automorphic Forms ◽  
Multiplicative Group ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Quaternion Algebras

AbstractWe continue our study of Yoshida’s lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms on the quaternion algebra correspond to modular forms of arbitrary even weights and square free levels; in particular we obtain a construction of Siegel modular forms of weight 3 attached to a pair of elliptic modular forms of weights 2 and 4.

scholarly journals Antisymmetric Paramodular Forms of Weights 2 and 3

2019 ◽  
Vol 2020 (20) ◽  
pp. 6926-6946 ◽  
Author(s):  
Valery Gritsenko ◽  
Cris Poor ◽  
David S Yuen
Keyword(s):  
Projective Space ◽  
Modular Forms ◽  
Differential Forms ◽  
Rational Points ◽  
Siegel Modular Forms ◽  
Abelian Surfaces ◽  
Algebraic Set ◽  
Minimal Weight ◽  
Dimensional Projective Space

Abstract We define an algebraic set in $23$-dimensional projective space whose ${{\mathbb{Q}}}$-rational points correspond to meromorphic, antisymmetric, paramodular Borcherds products. We know two lines inside this algebraic set. Some rational points on these lines give holomorphic Borcherds products and thus construct examples of Siegel modular forms on degree 2 paramodular groups. Weight $3$ examples provide antisymmetric canonical differential forms on Siegel modular three-folds. Weight $2$ is the minimal weight and these examples, via the paramodular conjecture, give evidence for the modularity of some rank 1 abelian surfaces defined over $\mathbb{Q}$.

scholarly journals Construction of siegel modular forms of degree three and commutation relations of Hecke operators

1985 ◽  
Vol 100 ◽  
pp. 83-96 ◽  
Author(s):  
Yoshio Tanigawa
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Siegel Modular Forms ◽  
Hecke Operators ◽  
Weil Representation ◽  
Commutation Relations ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Shimura Correspondence

In connection with the Shimura correspondence, Shintani [6] and Niwa [4] constructed a modular form by the integral with the theta kernel arising from the Weil representation. They treated the group Sp(1) × O(2, 1). Using the special isomorphism of O(2, 1) onto SL(2), Shintani constructed a modular form of half-integral weight from that of integral weight. We can write symbolically his case as “O(2, 1)→ Sp(1)” Then Niwa’s case is “Sp(l)→ O(2, 1)”, that is from the halfintegral to the integral. Their methods are generalized by many authors. In particular, Niwa’s are fully extended by Rallis-Schiffmann to “Sp(l)→O(p, q)”.

scholarly journals Restriction of Siegel modular forms to modular curves

2002 ◽  
Vol 65 (2) ◽  
pp. 239-252 ◽  
Author(s):  
Cris Poor ◽  
David S. Yuen
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Congruence Subgroup ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Modular Curves ◽  
Linear Relations

We study homomorphisms form the ring of Siegel modular forms of a given degree to the ring of elliptic modular forms for a congruence subgroup. These homomorphisms essentially arise from the restriction of Siegel modular forms to modular curves. These homomorphisms give rise to linear relations among the Fourier coefficients of a Siegel modular form. We use this technique to prove that dim .

scholarly journals Hilbert modular forms and p-adic Hodge theory

2009 ◽  
Vol 145 (5) ◽  
pp. 1081-1113 ◽  
Author(s):  
Takeshi Saito
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Galois Representation ◽  
Hodge Theory ◽  
Hilbert Modular Forms ◽  
Hilbert Modular Form ◽  
Langlands Correspondence ◽  
Hilbert Modular ◽  
Elliptic Modular Form ◽  
Local Langlands Correspondence

AbstractFor the p-adic Galois representation associated to a Hilbert modular form, Carayol has shown that, under a certain assumption, its restriction to the local Galois group at a finite place not dividing p is compatible with the local Langlands correspondence. Under the same assumption, we show that the same is true for the places dividing p, in the sense of p-adic Hodge theory, as is shown for an elliptic modular form. We also prove that the monodromy-weight conjecture holds for such representations.

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